Abstract
We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau–Turaev associated to the fundamental groups of surfaces.
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Alekseev, A., Kawazumi, N., Kuno, Y., Naef, F.: The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem. Adv. Math. 326, 1–53 (2018). arXiv:1703.05813
Alekseev, A., Kawazumi, N., Kuno, Y., Naef, F.: The Goldman-Turaev Lie bialgebras and the Kashiwara-Vergne problem in higher genera. Preprint; arXiv:1804.09566
Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. 54(1), 3–29 (2002). arXiv:math/0006168
Bielawski, R.: Quivers and Poisson structures. Manuscripta Math. 141(1-2), 29–49 (2013). arXiv:1108.3222
Crawley-Boevey, W.: Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities. Comment. Math. Helv. 74(4), 548–574 (1999)
Crawley-Boevey, W.: Poisson structures on moduli spaces of representations. J. Algebra 325, 205–215 (2011)
Crawley-Boevey, W., Shaw, P.: Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem. Adv. Math. 201(1), 180–208 (2006). arXiv:math/0404186
Chalykh, O., Fairon, M.: Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models. J. Geom. Phys. 121, 413–437 (2017). arXiv:1704.05814
Cuntz, J., Quillen, D.: Algebra extensions and nonsingularity. J. Amer. Math. Soc. 8(2), 251–289 (1995)
Iyudu, N., Kontsevich, M.: Pre-Calabi-Yau algebras as noncommutative Poisson structures; IHES/m/18/04 (2018)
Kontsevich, M.: Formal (Non)-Commutative symplectic geometry. In: Gelfand, I. M., Corwin, L., Lepowsky, J. (eds.) The Gelfand Mathematical Seminars, 1990–1992, pp 173–187. Birkhauser, Boston (1993)
Kontsevich, M., Rosenberg, A.L.: Noncommutative smooth spaces. The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., pp 85–108. Birkhäuser, Boston (2000)
Le Bruyn, L., Procesi, C.: Semisimple representations of quivers. Trans. Amer. Math. Soc. 317(2), 585–598 (1990)
Massuyeau, G., Turaev, V.: Quasi-Poisson structures on representation spaces of surfaces. Int. Math. Res. Not. IMRN, no. 1, 1–64 (2014). arXiv:1205.4898
Odesskii, A.V., Rubtsov, V.N., Sokolov, V.V.: Double Poisson brackets on free associative algebras. Noncommutative birational geometry, representations and combinatorics, Contemp. Math, vol. 592, pp 225–239. Amer. Math. Soc., Providence (2013). arXiv:1208.2935
Odesskii, A.V., Rubtsov, V.N., Sokolov, V.V.: Parameter-dependent associative Yang-Baxter equations and Poisson brackets. Int. J. Geom. Methods Mod. Phys. 11 (9), 1460036 (2014). 18 pages, arXiv:1311.4321
Pichereau, A., Van de Weyer, G.: Double Poisson cohomology of path algebras of quivers. J. Algebra 319(5), 2166–2208 (2008). arXiv:math/0701837
Powell, G.: On double Poisson structures on commutative algebras. J. Geom. Phys. 110, 1–8 (2016). arXiv:1603.07553
Sokolov, V.V.: Classification of constant solutions of the associative Yang-Baxter equation on Mat3. Theoret. and Math. Phys. 176(3), 1156–1162 (2013). Russian version appears in Teoret. Mat. Fiz. 176, no. 3, 385–392 (2013). arXiv:1212.6421
Van den Bergh, M.: Double Poisson algebras. Trans. Amer. Math. Soc. 360(11), 5711–5769 (2008). arXiv:math/0410528
Van den Bergh, M.: Non-commutative quasi-Hamiltonian spaces. In: Poisson geometry in mathematics and physics, vol. 450 of Contemp. Math. arXiv:math/0703293, pp 273–299. Amer. Math. Soc., Providence (2008)
Van de Weyer, G.: Double Poisson structures on finite dimensional semi-simple algebras. Algebr. Represent. Theory 11(5), 437–460 (2008)
Acknowledgements
The author is grateful to O. Chalykh for introducing him to the theory of double brackets, and for valuable comments on an earlier draft of this work which greatly improved the presentation of the present paper. The author also thanks A. Alekseev for useful discussions, and the referees for their comments. Part of this work was supported by a University of Leeds 110 Anniversary Research Scholarship.
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Presented by: Iain Gordon
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Appendices
Appendix A: Vanishing of the map κ
In this appendix, we prove Lemma 2.20. Note that κ is a linear combination of triple brackets, so it is itself a triple bracket. By definition, it is a derivation in its last argument and is cyclically anti-symmetric. Thus, to show that κ vanishes, it suffices to show that it is equal to zero when applied to generators of Af. Before tackling this task, we use Eq. ?? and remark that we can write
where 1A is the identity map. Therefore, evaluated on some elements a,b,c ∈ Af, we can write
so that we will write down the terms \(A,B,C,A^{\prime },B^{\prime },C^{\prime }\) for the different types of generators. Using the cyclicity, we only have twenty cases to check. We will only detail the computations in the first few cases, and we will give the final form of the terms \(A,B,C,A^{\prime },B^{\prime },C^{\prime }\) in the remaining cases so that the reader can check that they sum up to zero.
Before beginning with the calculations, we remark that identities involving the double bracket \( \left \{\!\!\left \{-,-\right \}\!\!\right \} \) follow from extension from A to Af which respects the derivation property in each variable. That is, given e+,f+ ∈{𝜖,e12} and e−,f−∈{𝜖,e21}, we have for any a = e+αe−, b = f+βf− with α,β∈ A that
Here, in the left-hand side we have the induced double bracket on Af, while the double bracket in the right-hand side is the original one on A. Recall that we can choose generators a,b ∈ Af that admit such a decomposition by Lemma 2.11.
1.1 A.1 All generators of the same type
We drop the idempotent 𝜖 in our computations since this is the unit in Af.
- Generators of the second type. :
-
Write a = e12α, b = e12β and c = e12γ for α,β,γ ∈ e2A𝜖. Using Eq. ??, then the derivation property for the outer bimodule structure in the second entry of the double bracket on Af together with Eq. A.2, we get that
$$ \begin{aligned} A(a,b,c)=&-\frac12 \left\{\!\!\left\{e_{12}\alpha,e_{12}\gamma e_{12}\upbeta\right\}\!\!\right\} \otimes e_{1} =-\frac12 \left( e_{12}\gamma \left\{\!\!\left\{e_{12}\alpha,e_{12}\upbeta\right\}\!\!\right\} + \left\{\!\!\left\{e_{12}\alpha,e_{12}\gamma \right\}\!\!\right\} e_{12}\upbeta\right)\otimes e_{1} \\ =&-\frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime}e_{12}\upbeta \otimes e_{1} -\frac12 e_{12}\gamma e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}\otimes e_{1}{\kern1.7pt}. \end{aligned} $$Similarly we obtain
$$ \begin{aligned} B(a,b,c) =&-\frac12 \tau_{(123)}(e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime}e_{12}\gamma \otimes e_{1} + e_{12}\alpha e_{12}\left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\} ^{\prime} \otimes e_{12}\left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\}^{\prime\prime}\otimes e_{1}){\kern1.7pt} \\ =&-\frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime}e_{12}\gamma -\frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ C(a,b,c) =&-\frac12 \tau_{(132)}(e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{12}\alpha \otimes e_{1} + e_{12}\upbeta e_{12}\left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12}\left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime}\otimes e_{1}){\kern1.7pt} \\ =&-\frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} - \frac12 e_{12}\left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime}\otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}{\kern1.7pt}. \end{aligned} $$Now, remark that Eq. A.2 gives \( \left \{\!\!\left \{e_{12} \upbeta ,e_{12} \gamma \right \}\!\!\right \} =e_{12} \left \{\!\!\left \{\upbeta ,\gamma \right \}\!\!\right \}^{\prime }\otimes e_{12} \left \{\!\!\left \{\upbeta ,\gamma \right \}\!\!\right \}^{\prime \prime }\), so that the element in the first copy of A⊗2 is also a generator of the second type. Using Eq. ?? for the expression of \( \left \{\!\!\left \{-,-\right \}\!\!\right \}_{fus}\), we get
$$ \begin{aligned} A^{\prime}(a,b,c)=& \left\{\!\!\left\{e_{12}\alpha,e_{12}\left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\}^{\prime}\right\}\!\!\right\}_{fus}\otimes e_{12}\left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\}^{\prime\prime} \\ =&\frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}. \end{aligned} $$In the same way, we find
$$ \begin{aligned} B^{\prime}(a,b,c)=& \frac12 \tau_{(123)}(e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} - e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{12}\upbeta \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} ) \\ =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{12}\upbeta \otimes e_{1} {\kern1.7pt}, \\ C^{\prime}(a,b,c)=& \frac12 \tau_{(132)}(e_{1} \otimes e_{12}\gamma e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} - e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{12}\gamma \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} ) \\ =& \frac12 e_{12}\gamma e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} - \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{12}\gamma{\kern1.7pt}. \end{aligned} $$Summing all terms, we obtain after obvious cancellations
$$ \begin{aligned} \kappa(a,b,c)=& -\frac12 (e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime}e_{12}\upbeta \otimes e_{1} + e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{12}\upbeta \otimes e_{1}) \\ &-\frac12 (e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime}e_{12}\gamma + e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{12}\gamma) \\ &-\frac12 (e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} + e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} ){\kern1.7pt}. \end{aligned} $$It remains to notice in the last expression that all lines vanish using the cyclic antisymmetry of the double bracket.
- Generators of the third type. :
-
Write a = αe21, b =β e21 and c = γe21 for α,β,γ ∈ 𝜖Ae2. From Eqs. ?? and A.2 we get that
$$ \begin{aligned} A(a,b,c)=&\frac12 \left\{\!\!\left\{\alpha e_{21},\gamma e_{21}\upbeta e_{21}\right\}\!\!\right\} \otimes e_{1} \\ =&\frac12 \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} \upbeta e_{21} \otimes e_{1} +\frac12 \gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1}{\kern1.7pt}. \end{aligned} $$Similarly we obtain
$$ \begin{aligned} B(a,b,c) =&\frac12 e_{1} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime}e_{21} \gamma e_{21} +\frac12 e_{1} \otimes \alpha e_{21} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} {\kern1.7pt}, \\ C(a,b,c) =&\frac12 \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21} +\frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21}{\kern1.7pt}. \end{aligned} $$Noticing from Eq. A.2 that \( \left \{\!\!\left \{b,c\right \}\!\!\right \}^{\prime }= \left \{\!\!\left \{\upbeta ,\gamma \right \}\!\!\right \}^{\prime } e_{21}\) is a generator of the third type, we get again from Eq. ??
$$ \begin{aligned} A^{\prime}(a,b,c)=& \left\{\!\!\left\{\alpha e_{21},\left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\}^{\prime} e_{21}\right\}\!\!\right\}_{fus}\otimes \left\{\!\!\left\{{\upbeta,\gamma}\right\}\!\!\right\}^{\prime\prime} e_{21} \\ =&\frac12 \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} - \frac12 e_{1} \otimes \alpha e_{21} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt}. \end{aligned} $$Analogously
$$ \begin{aligned} B^{\prime}(a,b,c) =& \frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} \upbeta e_{21} \otimes e_{1} - \frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} {\kern1.7pt}, \\ C^{\prime}(a,b,c) =&\frac12 e_{1} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \gamma e_{21} - \frac12 \gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} {\kern1.7pt}. \end{aligned} $$Summing all terms yield
$$ \begin{aligned} \kappa(a,b,c)=& +\frac12 (\left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} \upbeta e_{21} \otimes e_{1} + \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} \upbeta e_{21} \otimes e_{1} ) \\ &+\frac12 (e_{1} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime}e_{21} \gamma e_{21} + e_{1} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \gamma e_{21}) \\ &+\frac12 (\left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21} + \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} ){\kern1.7pt}, \end{aligned} $$which is zero using the cyclic antisymmetry.
- Generators of the first type. :
-
For a,b,c ∈ 𝜖A𝜖, we have by Eq. A.2 that the double bracket \( \left \{\!\!\left \{-,-\right \}\!\!\right \} \) evaluated on any two of these elements belongs to (𝜖A𝜖)⊗2. At the same time, Eq. ?? gives that \(\left \{\!\!\left \{{\epsilon A \epsilon ,\epsilon A \epsilon }\right \}\!\!\right \}_{fus}=0\). Hence all terms in Eq. A.1 trivially vanish and κ(a,b,c) = 0.
- Generators of the fourth type. :
-
As in the first type case, we use Eq. A.2 to get that \( \left \{\!\!\left \{e_{12} A e_{21},e_{12} A e_{21}\right \}\!\!\right \} \subset (e_{12} A e_{21})^{\otimes 2}\) and Eq. ?? to obtain \(\left \{\!\!\left \{{e_{12} A e_{21},e_{12} A e_{21}}\right \}\!\!\right \} _{fus}=0\), so that all terms vanish.
1.2 A.2 Two generators of the first type
Let a,b ∈ 𝜖A𝜖.
- With one generator of the second type. :
-
Consider c = e12γ for some γ ∈ e2A𝜖. Using Eqs. ?? and ??,
$$ \begin{aligned} A=&-\frac12 e_{1} \left\{\!\!\left\{a, b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a, b\right\}\!\!\right\}^{\prime\prime}\otimes e_{12}\gamma{\kern1.7pt}, \\ B=& \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime} -\frac12 e_{1} \otimes \left\{\!\!\left\{b,a \right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,a \right\}\!\!\right\}^{\prime\prime} e_{12}\gamma -\frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime} {\kern1.7pt}. \end{aligned} $$By Eq. ??, C trivially vanishes. It is also the case for \(B^{\prime }\) because \( \left \{\!\!\left \{e_{12}\gamma ,a\right \}\!\!\right \}^{\prime }\in \epsilon A \epsilon \). Next we get by Eqs. ?? and ?? that
$$ \begin{aligned} A^{\prime}=& \frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ C^{\prime}=&\frac12 e_{1} \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes e_{12}\gamma - \frac12 e_{1} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime}e_{12}\gamma{\kern1.7pt}, \end{aligned} $$so that all terms cancel out together (after using the cyclic antisymmetry, which we will need in each of the remaining cases).
- With one generator of the third type. :
-
Consider c = γe21 for some γ ∈ 𝜖Ae2. We get from Eqs. ?? and ?? that
$$ \begin{array}{@{}rcl@{}} A&=&\frac12 \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}b \otimes e_{1} + \frac12 \gamma e_{21} \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes e_{1}\\ &&- \frac12 \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes b e_{1}{\kern1.7pt}, \\ B&=& \frac12 \gamma e_{21} \otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime\prime} e_{1}{\kern1.7pt}. \end{array} $$Again using Eq. ?? we have C = 0, and \(A^{\prime }=0\) since \( \left \{\!\!\left \{b,\gamma e_{21}\right \}\!\!\right \}^{\prime } \in \epsilon A \epsilon \). Finally, from Eqs. ?? and ?? we get
$$ \begin{aligned} B^{\prime} =& \frac12 \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21}b \otimes e_{1} - \frac12 \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21}\otimes b e_{1}{\kern1.7pt},\\ C^{\prime} =&\frac12 \gamma e_{21} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} e_{1} - \frac12 \gamma e_{21} \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes e_{1}{\kern1.7pt}, \end{aligned} $$and all terms sum up to zero.
- With one generator of the fourth type. :
-
Consider c = e12γe21 for some γ ∈ e2Ae2. First, using Eqs. ?? and ?? we get
$$ \begin{aligned} A=& \frac12 e_{12} \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}b \otimes e_{1} + \frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \\ &-\frac12 e_{12} \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes b e_{1} - \frac12 e_{1} \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,b\right\}\!\!\right\}^{\prime\prime} \otimes e_{12}\gamma e_{21}{\kern1.7pt}, \\ B=& \frac12 e_{12}\gamma e_{21} \otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime\prime} e_{1} + \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} \\ &-\frac12 e_{1} \otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime}\otimes \left\{\!\!\left\{b,a\right\}\!\!\right\}^{\prime\prime} e_{12}\gamma e_{21} - \frac12 e_{1} \otimes ae_{12} \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{b,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt}. \end{aligned} $$
Again, C = 0 by Eq. ??. Meanwhile, we find from Eqs. ??, ?? and ??
Summing terms together, we get κ = 0.
1.3 A.3 Two generators of the second type
Let a = e12α,b = e12β for α,β∈ e2A𝜖. We only collect the final form of the terms \(A,B,C,A^{\prime },B^{\prime },C^{\prime }\) from now on, and the reader can check that they sum up to zero.
- With one generator of the first type. :
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Consider c ∈ 𝜖A𝜖.
$${} \begin{aligned} A \!=&\frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}\!\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \!\otimes\! e_{1} c - \frac12 \left\{\!\!\left\{\alpha,c\right\}\!\!\right\}^{\prime} \!\otimes\! e_{12} \!\left\{\!\!\left\{\!\alpha,c\!\right\}\!\!\right\}^{\prime\prime}\!e_{12}\upbeta \otimes e_{1} {\kern1.7pt}-{\kern1.7pt} \frac12 ce_{12} \left\{\!\!\left\{\!\alpha,\upbeta\!\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\!\alpha,\upbeta\!\right\}\!\!\right\}^{\prime\prime} \!\otimes e_{1}{\kern1.7pt}, \\ B =& -\frac12 e_{12}\alpha \otimes e_{1} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ C =&-\frac12 \left\{\!\!\left\{c,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{12} \alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{c, \upbeta\right\}\!\!\right\}^{\prime} - \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime}{\kern1.7pt}, \\ A^{\prime} =& \frac12 e_{12}\alpha \otimes e_{1} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime} -\frac12 \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime}e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ B^{\prime} =& \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} -\frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} e_{12}\upbeta \otimes e_{1} {\kern1.7pt}, \\ C^{\prime} =& \frac12 c e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} -\frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1}c{\kern1.7pt}. \end{aligned} $$ - With one generator of the third type. :
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Consider c = γe21 for some γ ∈ 𝜖Ae2.
$$ \begin{aligned} A =&\frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \gamma e_{21} - \frac12 \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\otimes e_{12}\upbeta e_{1}{\kern1.7pt}, \\ B =&\frac12 \gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta, \alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{1} - \frac12 e_{12} \alpha \otimes e_{1} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} {\kern1.7pt}, \\ C =&- \frac12 \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21} -\frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}e_{21}{\kern1.7pt}, \\ A^{\prime} =& \frac12 e_{12}\alpha \otimes e_{1} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} -\frac12 \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21}{\kern1.7pt}, \\ B^{\prime} =& \frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12} \upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} - \frac12 \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \upbeta e_{1}{\kern1.7pt}, \\ C^{\prime} =& \frac12 \gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{1} - \frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \gamma e_{21}{\kern1.7pt}. \end{aligned} $$ - With one generator of the fourth type. :
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Consider c = e12γe21 for some γ ∈ e2Ae2.
$$ \begin{aligned} A =&-\frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \upbeta e_{1}{\kern1.7pt}, \\ B =&\frac12 e_{12} \gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{1} - \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{12}\gamma e_{21} \\ &- \frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ C =&- \frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21} -\frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12}\upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}e_{21}{\kern1.7pt}, \\ A^{\prime} =& \frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} -\frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{12}\alpha \otimes e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21}{\kern1.7pt}, \\ B^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes e_{12} \upbeta e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \upbeta e_{1}{\kern1.7pt}, \\ C^{\prime} =& \frac12 e_{12}\gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{1} - \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{12}\gamma e_{21}{\kern1.7pt}. \end{aligned} $$
1.4 A.4 Two generators of the third type
Let a = αe21,b =β e21 for α,β∈ 𝜖Ae2.
- With one generator of the first type. :
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Consider c ∈ 𝜖A𝜖.
$${} \begin{aligned} A=&\frac12 \left\{\!\!\left\{\alpha ,c\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha ,c\right\}\!\!\right\}^{\prime\prime} e_{1}\otimes \upbeta e_{21}{\kern1.7pt}, \\ B =&\frac12 e_{1} \!\otimes\! \left\{\!\!\left\{\!\upbeta,\alpha\!\right\}\!\!\right\}^{\prime} \!e_{21} \!\otimes\! \left\{\!\!\left\{\!\upbeta,\alpha\!\right\}\!\!\right\}^{\prime\prime} e_{21}c + \frac12 e_{1} \otimes \alpha e_{21} \left\{\!\!\left\{\upbeta,c\!\right\}\!\!\right\}^{\prime} e_{21} \otimes\! \left\{\!\!\left\{\!\upbeta,c\!\right\}\!\!\right\}^{\prime\prime} {\kern1.7pt}-{\kern1.7pt} \frac12 c e_{1} \!\otimes\! \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21}\! \otimes\! \left\{\!\!\left\{\!\upbeta,\alpha\!\right\}\!\!\right\}^{\prime\prime}\! e_{21} {\kern1.7pt}, \\ C =& \frac12 \left\{\!\!\left\{c,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{c,\upbeta\right\}\!\!\right\}^{\prime} + \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} {\kern1.7pt}, \\ A^{\prime} =&\frac12 \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime}e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{1} \otimes \alpha e_{21} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ B^{\prime} =&\frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime}e_{1} \otimes \upbeta e_{21} - \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} {\kern1.7pt}, \\ C^{\prime} =&\frac12 e_{1} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} c - \frac12 c e_{1} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} {\kern1.7pt}. \end{aligned} $$ - With one generator of the second type. :
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Consider c = e12γ for some γ ∈ e2A𝜖.
$$ \begin{aligned} A =& \frac12 e_{12} \left\{\!\!\left\{\alpha ,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha ,\gamma\right\}\!\!\right\}^{\prime\prime} e_{1} \otimes \upbeta e_{21} - \frac12 e_{1} \left\{\!\!\left\{\alpha ,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha ,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \gamma{\kern1.7pt}, \\ B =&\frac12 e_{1} \alpha e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta ,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta ,\gamma\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{12} \gamma e_{1} \otimes \left\{\!\!\left\{\upbeta ,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta ,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ C =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} + \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} {\kern1.7pt}, \\ A^{\prime} =&\frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{1} \alpha e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} {\kern1.7pt},\\ B^{\prime} =&\frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}e_{1} \otimes \upbeta e_{21} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} {\kern1.7pt},\\ C^{\prime} =&\frac12 e_{1} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \gamma - \frac12 e_{12} \gamma e_{1} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} {\kern1.7pt}. \end{aligned} $$ - With one generator of the fourth type. :
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Consider c = e12γe21 for some γ ∈ e2Ae2.
$$ \begin{aligned} A =& \frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\upbeta e_{21} \otimes e_{1} + \frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \\ &- \frac12 e_{1} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12}\gamma e_{21}{\kern1.7pt}, \\ B =&\frac12 e_{1} \alpha e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt}, \\ C =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21} + \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} {\kern1.7pt}, \\ A^{\prime} =&\frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \alpha e_{21} \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} - \frac12 e_{1} \alpha e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt},\\ B^{\prime} =&\frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}e_{21} \upbeta e_{21} \otimes e_{1} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} {\kern1.7pt},\\ C^{\prime} =&\frac12 e_{1} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12}\gamma e_{21} - \frac12 e_{12} \gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} {\kern1.7pt}. \end{aligned} $$
1.5 A.5 Two generators of the fourth type
Let a = e12αe21,b = e12βe21 for α,β∈ e2Ae2.
- With one generator of the first type. :
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Consider c ∈ 𝜖A𝜖. We get C = 0, while
$$ \begin{aligned} A =& \frac12 \left\{\!\!\left\{\alpha,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,c\right\}\!\!\right\}^{\prime\prime} e_{1} \otimes e_{12} \upbeta e_{21} + \frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} c \\ &- \frac12 \left\{\!\!\left\{\alpha,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,c\right\}\!\!\right\}^{\prime\prime} e_{12} \upbeta e_{21} \otimes e_{1} - \frac12 c e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} {\kern1.7pt}, \\ B =& \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} c + \frac12 e_{1} \otimes e_{12}\alpha e_{21} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime} \\ &- \frac12 c e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} - \frac12 e_{12} \alpha e_{21} \otimes e_{1} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,c\right\}\!\!\right\}^{\prime\prime} {\kern1.7pt}, \end{aligned} $$$$ \begin{aligned} A^{\prime} =& \frac12 e_{12}\alpha e_{21} \otimes e_{1} \left\{\!\!\left\{\upbeta ,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta ,c\right\}\!\!\right\}^{\prime\prime} - \frac12 e_{1} \otimes e_{12} \alpha e_{21} \left\{\!\!\left\{\upbeta ,c\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta ,c\right\}\!\!\right\}^{\prime\prime}{\kern1.7pt}, \\ B^{\prime} =& \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} e_{1} \otimes e_{12} \upbeta e_{21} - \frac12 \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{c,\alpha\right\}\!\!\right\}^{\prime} e_{12} \upbeta e_{21} \otimes e_{1}{\kern1.7pt}, \\ C^{\prime} =&\frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} c +\frac12 c e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} \\ &- \frac12 c e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} - \frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} c {\kern1.7pt}. \end{aligned} $$ - With one generator of the second type. :
-
Consider c = e12γ for some γ ∈ e2A𝜖. We get C = 0, \(A^{\prime }=0\), while
$$ \begin{aligned} A =&\frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{1} \otimes e_{12}\upbeta e_{21} - \frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{12} \upbeta e_{21} \otimes e_{1} \\ &-\frac12 e_{12}\gamma e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{21}\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1}{\kern1.7pt},\\ B =&-\frac12 e_{12} \gamma e_{1}\otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ B^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{1} \otimes e_{12} \upbeta e_{21} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{12} \upbeta e_{21} \otimes e_{1}{\kern1.7pt}, \\ C^{\prime} =& \frac12 e_{12}\gamma e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} -\frac12 e_{12}\gamma e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} {\kern1.7pt}. \end{aligned} $$ - With one generator of the third type. :
-
Consider c = γe21 for some γ ∈ 𝜖Ae2. We get C = 0, \(B^{\prime }=0\), while
$$ \begin{aligned} A=& \frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21}\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21}\otimes e_{1} \gamma e_{21}{\kern1.7pt}, \\ B =&\frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{21}\gamma e_{21} + \frac12 e_{1} \otimes e_{12} \alpha e_{21} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\\ &-\frac12 e_{12}\alpha e_{21} \otimes e_{1} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21}\otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21}{\kern1.7pt},\\ A^{\prime} =& \frac12 e_{12}\alpha e_{21} \otimes e_{1} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} -\frac12 e_{1} \otimes e_{12}\alpha e_{21} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21}\otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt},\\ C^{\prime} =& \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \gamma e_{21} -\frac12 e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} \gamma e_{21} {\kern1.7pt}. \end{aligned} $$
1.6 A.6 Remaining cases
We now take three different types of generators.
- No generator of the fourth type. :
-
Let a ∈ 𝜖A𝜖, b = e12β for β ∈ e2A𝜖 and c = γe21 for γ ∈ 𝜖Ae2. We have \(A^{\prime }=0\), while
$$ \begin{aligned} A =&\frac12 e_{12} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \gamma e_{21} - \frac12 \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12} \upbeta e_{1}{\kern1.7pt},\\ B =& \frac12 \gamma e_{21}\otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime\prime} e_{1}{\kern1.7pt}, \\ C =& -\frac12 \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \upbeta \otimes e_{1} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21}{\kern1.7pt},\\ B^{\prime} =& \frac12 \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{12}\upbeta \otimes e_{1} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21} - \frac12 \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12}\upbeta e_{1}{\kern1.7pt}, \\ C^{\prime} =& \frac12 \gamma e_{21} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} e_{1} - \frac12 e_{12} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \gamma e_{21} {\kern1.7pt}. \end{aligned} $$ - No generator of the third type. :
-
Let a ∈ 𝜖A𝜖, b = e12β for β ∈ e2A𝜖 and c = e12γe21 for γ ∈ e2Ae2.
$$ \begin{aligned} A =&-\frac12e_{12} \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\otimes e_{12}\upbeta e_{1}{\kern1.7pt},\\ B =&\frac12 e_{12}\gamma e_{21} \otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime\prime} e_{1} + \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} \\ &-\frac12 e_{1} \otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime\prime} e_{12} \gamma e_{21} -\frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt}, \\ C =& - \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{12}\upbeta \otimes e_{1} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21} {\kern1.7pt}, \end{aligned} $$$$ \begin{aligned} A^{\prime} =& \frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} - \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ B^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{12}\upbeta \otimes e_{1} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21} - \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime}e_{21} \otimes e_{12}\upbeta e_{1}{\kern1.7pt}, \\ C^{\prime} =& \frac12 e_{12}\gamma e_{21} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime}e_{1} - \frac12 e_{1} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} e_{12}\gamma e_{21} {\kern1.7pt}. \end{aligned} $$ - No generator of the second type. :
-
This case and the next one are a bit tedious. We set a ∈ 𝜖A𝜖, b =β e21 for β ∈ 𝜖Ae2 and c = e12γe21 for γ ∈ e2Ae2.
$$ \begin{aligned} A =&\frac12 e_{12} \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\upbeta e_{21}\otimes e_{1} + \frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime}\otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \\ &- \frac12 e_{1} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12}\gamma e_{21}{\kern1.7pt}, \\ B =&\frac12 e_{12}\gamma e_{21} \otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime\prime} e_{1} + \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} \\ &-\frac12 e_{1} \otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime} e_{21}\otimes \left\{\!\!\left\{\upbeta,a\right\}\!\!\right\}^{\prime\prime} e_{12} \gamma e_{21} -\frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}{\kern1.7pt}, \\ C =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21}a \otimes e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime}e_{21} + \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime} e_{21} \\ &-\frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes a e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21}{\kern1.7pt}, \end{aligned} $$$$ \begin{aligned} A^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} a \otimes e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} + \frac12 e_{1} \otimes a e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} \\ &-\frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes a e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} - \frac12 e_{1} a \otimes e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime}e_{21} {\kern1.7pt}, \\ B^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime}e_{21} \upbeta e_{21}\otimes e_{1} - \frac12 e_{12} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \otimes \upbeta e_{21} \left\{\!\!\left\{\gamma,a\right\}\!\!\right\}^{\prime}e_{21} {\kern1.7pt}, \\ C^{\prime} =& \frac12 e_{12}\gamma e_{21} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} e_{1} + \frac12 e_{1} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12}\gamma e_{21} \\ &-\frac12 e_{1} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime}e_{12}\gamma e_{21} -\frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime} \otimes \left\{\!\!\left\{a,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} {\kern1.7pt}. \end{aligned} $$ - No generator of the first type. :
-
Let a = e12α for α ∈ e2A𝜖, b =β e21 for β ∈ 𝜖Ae2 and c = e12γe21 for γ ∈ e2Ae2.
$$ \begin{aligned} A =&\frac12 e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21}\upbeta e_{21}\otimes e_{1} + \frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}\otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{1} \\ &- \frac12 e_{1} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21} \otimes e_{12}\gamma e_{21}{\kern1.7pt}, \\ B =& \frac12 e_{12}\gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{1} - \frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\alpha\right\}\!\!\right\}^{\prime\prime} e_{12}\gamma e_{21}\\ &- \frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime}e_{21}\otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ C =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \upbeta e_{21} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime\prime} e_{21}\otimes e_{12} \alpha e_{1} \otimes \left\{\!\!\left\{\gamma,\upbeta\right\}\!\!\right\}^{\prime} e_{21}{\kern1.7pt}, \end{aligned} $$$$ \begin{aligned} A^{\prime} =& \frac12 e_{1} \otimes e_{12}\alpha e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} - \frac12 e_{12} \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime} e_{21} \otimes e_{12}\alpha e_{1} \otimes \left\{\!\!\left\{\upbeta,\gamma\right\}\!\!\right\}^{\prime\prime} e_{21} {\kern1.7pt}, \\ B^{\prime} =& \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime} e_{21} \upbeta e_{21}\otimes e_{1} - \frac12 e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime\prime} \otimes e_{1} \upbeta e_{21}\otimes e_{12} \left\{\!\!\left\{\gamma,\alpha\right\}\!\!\right\}^{\prime}e_{21} {\kern1.7pt},\\ C^{\prime} =& \frac12 e_{12}\gamma e_{21} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} e_{1} + \frac12 e_{1} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{12}\gamma e_{21} \\ &-\frac12 e_{1} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime}e_{12}\gamma e_{21} -\frac12 e_{12}\gamma e_{21} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime} \otimes e_{12} \left\{\!\!\left\{\alpha,\upbeta\right\}\!\!\right\}^{\prime\prime}e_{21} \otimes e_{1} {\kern1.7pt}. \end{aligned} $$
Appendix B: Proof of Lemma 2.21
Note that Tr(Φs) = 𝜖Φs𝜖 for s≠ 2, while Tr(Φ2) = e12Φ2e21. In particular, using that for s≠ 2 we have Φs = esΦses, we get Tr(Φs) = Φs by understanding that equality in Af.
1.1 B.1 Moment map condition for the non-fused idempotents
First, assume that s≠ 1,2. Then, using Lemma 2.18, we get
which gives \( \left \{\!\!\left \{\operatorname {Tr}({\Phi }_{s}),-\right \}\!\!\right \}_{fus}=0\). Therefore, if a = e+αe− is a generator of Af,
where the double bracket in the last equality is taken in A. By assumption Φs satisfies (??) for \( \left \{\!\!\left \{-,-\right \}\!\!\right \} \) on A so that
where we omitted to write the idempotents 𝜖, because with s≠ 1,2 we get es𝜖 = es = 𝜖es. It remains to see that it coincides with Eq. ?? in all four cases of generators. For example, if a = e12α𝜖 with α ∈ e2A𝜖, we obtain for e+ = e12,e− = 𝜖
because the second and last terms in Eq. B.1 disappear since e12es = 0 = e12Φs. Meanwhile, the right-hand side of Eq. ?? reads in that case
and the last two terms disappear as s≠ 1,2. Indeed esa = ese12α = 0 and Tr(Φs)a = 𝜖(esΦses)𝜖(e12α) = esΦsese12α = 0. The two expressions coincide, and the result is similar with the other types of generators.
1.2 B.2 Moment map condition at the fused idempotent
Using the derivation properties and decomposing the double bracket \( \left \{\!\!\left \{-,-\right \}\!\!\right \}^{f}\) as \( \left \{\!\!\left \{-,-\right \}\!\!\right \} + \left \{\!\!\left \{-,-\right \}\!\!\right \}_{fus}\), we obtain for a = e+αe−∈ Af, α ∈ A, that
The first two terms can easily be obtained from Eq. ??. Since Tr(Φ2) is a generator of fourth type (??), we need (??)–(??) to evaluate the third term. In the exact same way, as Tr(Φ1) is a generator of first type (??), we need (??)–(??) to evaluate the last term. Thus, we check separately the four types of generators.
- On a generator of the first type. :
-
We let a ∈ 𝜖A𝜖, hence e+ = e− = 𝜖 and a = α. We directly get by Eq. ?? that \(\left \{\!\!\left \{{{\Phi }_{2},a}\right \}\!\!\right \}=0\) since e2a = 0 = ae2, while \( \left \{\!\!\left \{\operatorname {Tr}({\Phi }_{1}),a\right \}\!\!\right \}_{fus}=0\) by Eq. ??. For the remaining two terms, we have on one hand by Eq. ??
$$ \left\{\!\!\left\{{\Phi}_{1},a\right\}\!\!\right\} =\frac12 (ae_{1}\otimes \operatorname{Tr}({\Phi}_{1})-e_{1} \otimes \operatorname{Tr}({\Phi}_{1}) a + a \operatorname{Tr}({\Phi}_{1}) \otimes e_{1}-\operatorname{Tr}({\Phi}_{1}) \otimes e_{1} a){\kern1.7pt}, $$after projecting the equality in Af where Tr(Φ1) = Φ1. On the other hand by Eq. ??
$$ \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{2}),a\right\}\!\!\right\}_{fus}=\frac12(a e_{1} \otimes \operatorname{Tr}({\Phi}_{2}) + \operatorname{Tr}({\Phi}_{2}) \otimes e_{1} a - a \operatorname{Tr}({\Phi}_{2}) \otimes e_{1} - e_{1} \otimes \operatorname{Tr}({\Phi}_{2}) a){\kern1.7pt}. $$Putting this back in Eq. B.2 yields
$${} \begin{aligned} \left\{\!\!\left\{{{\Phi}_{1}^{f}},a\right\}\!\!\right\}^{f}\!=& \frac12 (ae_{1}\operatorname{Tr}(\!{\Phi}_{2}\!)\otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!)-e_{1}\operatorname{Tr}(\!{\Phi}_{2}\!) \otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!) a + a\! \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) \otimes e_{1}-\!\operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) \otimes e_{1} a) \\ &+\frac12(a e_{1} \!\otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!)\operatorname{Tr}(\!{\Phi}_{2}\!) + \operatorname{Tr}(\!{\Phi}_{2}\!) \otimes \operatorname{Tr}(\!{\Phi}_{1}\!)e_{1} a - a \operatorname{Tr}(\!{\Phi}_{2}\!) \otimes \operatorname{Tr}(\!{\Phi}_{1}\!)e_{1} - e_{1} \!\otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!)\operatorname{Tr}(\!{\Phi}_{2}\!) a\!){\kern1.7pt}. \end{aligned} $$Using that Tr(Φ1) = e1 Tr(Φ1)e1 and Tr(Φ2) = e1 Tr(Φ2)e1 allows us to conclude after cancellation of the first and seventh terms, and the second and sixth terms.
- On a generator of the second type. :
-
Let a = e12α𝜖 with e+ = e12, e− = 𝜖, α ∈ e2A𝜖. We get from Eq. ??
$$ \left\{\!\!\left\{{\Phi}_{1},\alpha\right\}\!\!\right\} =\frac12 (\alpha e_{1}\otimes {\Phi}_{1} + \alpha {\Phi}_{1} \otimes e_{1} ){\kern1.7pt}, \quad \left\{\!\!\left\{{\Phi}_{2},\alpha\right\}\!\!\right\} =-\frac12 (e_{2} \otimes {\Phi}_{2} \alpha +{\Phi}_{2} \otimes e_{2} \alpha){\kern1.7pt}, $$because e1α = 0 and αe2 = 0. Meanwhile, Eqs. ?? and ?? give
$$ \begin{array}{@{}rcl@{}} \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{1}),a\right\}\!\!\right\}_{fus}&=&\frac12 (e_{1} \otimes \operatorname{Tr}({\Phi}_{1}) a - e_{1} \operatorname{Tr}({\Phi}_{1}) \otimes a) {\kern1.7pt}, \\ \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{2}),a\right\}\!\!\right\}_{fus}&=&\frac12 (a e_{1} \otimes \operatorname{Tr}({\Phi}_{2}) - a \operatorname{Tr}({\Phi}_{2}) \otimes e_{1}){\kern1.7pt}. \end{array} $$Hence, Eq. B.2 gives
$$ \begin{aligned} \left\{\!\!\left\{{{\Phi}_{1}^{f}},a\right\}\!\!\right\}^{f}=& -\frac12 ({\kern1.7pt} e_{12}e_{21} \otimes \operatorname{Tr}({\Phi}_{1})e_{12}{\Phi}_{2} \alpha +e_{12}{\Phi}_{2}e_{21} \otimes \operatorname{Tr}({\Phi}_{1})e_{12}e_{2} \alpha {\kern1.7pt} ) \\ &+ \frac12 ({\kern1.7pt}e_{12}\alpha e_{1} \operatorname{Tr}({\Phi}_{2})\otimes {\Phi}_{1} + e_{12}\alpha {\Phi}_{1} \operatorname{Tr}({\Phi}_{2}) \otimes e_{1} {\kern1.7pt}) \\ &+\frac12 (a e_{1} \otimes \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) - a \operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1})e_{1}) \\ &+ \frac12 (e_{1}\operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1}) a - e_{1} \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) \otimes a){\kern1.7pt}. \end{aligned} $$This equality holds in Af where Tr(Φ1) = Φ1, Tr(Φ2) = e12Φ2e21 and a = e12α. Thus it is not hard to rewrite all factors in the four first terms as Tr(Φs),a or the idempotents (we have to note for the first term that e12Φ2α = e12Φ2e2α = e12Φ2e21e12α = Tr(Φ2)a). After cancelling out the second (resp. third) with the seventh (resp. sixth) term, we get the desired result.
- On a generator of the third type. :
-
Let a = 𝜖αe21 with e+ = 𝜖, e− = e21, α ∈ 𝜖Ae2. Using Eq. ?? yields
$$ \left\{\!\!\left\{{\Phi}_{1},\alpha\right\}\!\!\right\} =-\frac12 (e_{1} \otimes {\Phi}_{1} \alpha +{\Phi}_{1} \otimes e_{1} \alpha){\kern1.7pt}, \quad \left\{\!\!\left\{{\Phi}_{2},\alpha\right\}\!\!\right\} =\frac12 (\alpha e_{2}\otimes {\Phi}_{2} + \alpha {{\Phi}}_{2} \otimes e_{2}){\kern1.7pt}, $$because αe1 = 0 and e2α = 0. From Eqs. ?? and ?? we obtain
$$ \begin{array}{@{}rcl@{}} \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{1}),a\right\}\!\!\right\}_{fus}&=&\frac12 (a \operatorname{Tr}({\Phi}_{1}) \otimes e_{1} - a \otimes \operatorname{Tr}({\Phi}_{1}) e_{1}) {\kern1.7pt}, \\ \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{2}),a\right\}\!\!\right\}_{fus}&=&\frac12 (\operatorname{Tr}({\Phi}_{2}) \otimes e_{1} a - e_{1} \otimes \operatorname{Tr}({\Phi}_{2}) a ){\kern1.7pt}. \end{array} $$Summing everything inside Eq. B.2, we get
$$ \begin{aligned} \left\{\!\!\left\{{{\Phi}_{1}^{f}},a\right\}\!\!\right\}^{f}=& \frac12 (\alpha e_{21} \otimes \operatorname{Tr}({\Phi}_{1})e_{12} {\Phi}_{2}e_{21} + \alpha {\Phi}_{2} e_{21} \otimes \operatorname{Tr}({\Phi}_{1})e_{12} e_{2}e_{21} )\\ &-\frac12 (e_{1} \operatorname{Tr}({\Phi}_{2}) \otimes {\Phi}_{1} \alpha e_{21} +{\Phi}_{1} \operatorname{Tr}({\Phi}_{2}) \otimes e_{1} \alpha e_{21}) \\ &+ \frac12 (\operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1})e_{1} a - e_{1} \otimes \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) a ) \\ &+ \frac12 (a \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2}) \otimes e_{1} - a\operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1}) e_{1}){\kern1.7pt}. \end{aligned} $$By arguments similar to the previous case, we can rewrite the four first terms using a,Tr(Φ1),Tr(Φ2) and the idempotents e1,e2 so that the second and eighth terms cancel out, while the third and fifth terms cancel out. The remaining terms give the desired result.
- On a generator of the fourth type. :
-
We let a = e12αe21 with e+ = e12, e− = e21, α ∈ e2Ae2. We directly get by Eq. ?? that \(\left \{\!\!\left \{{{\Phi }_{1},a}\right \}\!\!\right \}=0\), and by Eq. ?? that \(\left \{\!\!\left \{{\operatorname {Tr}({\Phi }_{2}),a}\right \}\!\!\right \}_{fus}=0\). For the remaining two terms, we have by Eqs. ?? and ??
$$ \begin{aligned} \left\{\!\!\left\{{\Phi}_{2},\alpha\right\}\!\!\right\} =&\frac12 (\alpha e_{2}\otimes {\Phi}_{2}-e_{2} \otimes {\Phi}_{2} \alpha + \alpha {\Phi}_{2} \otimes e_{2}-{\Phi}_{2} \otimes e_{2} \alpha){\kern1.7pt}, \\ \left\{\!\!\left\{\operatorname{Tr}({\Phi}_{1}),a\right\}\!\!\right\}_{fus}=&\frac12(a \operatorname{Tr}({\Phi}_{1}) \otimes e_{1} + e_{1} \otimes \operatorname{Tr}({\Phi}_{1}) a - a \otimes \operatorname{Tr}({\Phi}_{1}) e_{1} - e_{1} \operatorname{Tr}({\Phi}_{1}) \otimes a ){\kern1.7pt}. \end{aligned} $$Thus, we get after some easy manipulations
$$ \begin{aligned} \left\{\!\!\left\{\!{{\Phi}_{1}^{f}},a\!\right\}\!\!\right\}^{f}\!=& \frac12 (a\otimes \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2})-e_{1} \otimes \operatorname{Tr}({\Phi}_{1})\operatorname{Tr}({\Phi}_{2})a + a\operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1})-\operatorname{Tr}({\Phi}_{2}) \otimes \operatorname{Tr}({\Phi}_{1})a) \\ &+\!\frac12(a\! \operatorname{Tr}(\!{\Phi}_{1}\!)\operatorname{Tr}(\!{\Phi}_{2}\!) \!\otimes\! e_{1} {\kern1.7pt}+{\kern1.7pt} e_{1} \operatorname{Tr}(\!{\Phi}_{2}\!)\otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!) a - a\! \operatorname{Tr}(\!{\Phi}_{2}\!) \!\otimes\! \operatorname{Tr}(\!{\Phi}_{1}\!) e_{1} - e_{1}\! \operatorname{Tr}(\!{\Phi}_{2}\!) \operatorname{Tr}(\!{\Phi}_{1}\!) \!\otimes\! a\! ){\kern1.7pt}, \end{aligned} $$from which we can conclude.
Appendix C: Proof of Proposition 4.4
Note that any B-linear double bracket on A of degree at most + 4 on generators needs to satisfy
after using that t = e1te2,s = e2se1 with the cyclic antisymmetry and the derivation rules. Moreover, if \( \left \{\!\!\left \{-,-\right \}\!\!\right \} \) is a double quasi-Poisson bracket it must satisfy (??) on generators, and this is easily seen to be equivalent to require that
Lemma C.1
If Eq. C.2a holds, then either λ = l = 0 or
Proof
By Eq. ??, we have that for any a ∈ A,
We first look at the case a = t. Using Eq. C.1a, we can find that
The first four terms cancel if we take their sum under cyclic permutations, so that we can write
Therefore either λ = 0, or the different coefficients vanish i.e. γ = 0, ϕ1 = 0 while \(\alpha _{1}^{\prime }=-\alpha _{3}\) and ϕ0 = −ϕ2. Doing the computation with s instead of t, we need either l = 0 or the same four conditions. □
Lemma C.2
If λ = 0 and Eq. C.2b holds, then
The same identities are satisfied if l = 0 and Eq. C.2c holds.
Proof
When we compute \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) using Eq. ??, we get that the term (st)3 ⊗ t ⊗ e1 only appears with a factor \({\phi _{0}^{2}}\), and e2 ⊗ t ⊗ (ts)3 only appears with a factor \(-{\phi _{2}^{2}}\). Therefore, if Eq. C.2b is satisfied we need ϕ0 = ϕ2 = 0 which gives Eq. C.5a.
Under the conditions from Eq. C.5a, the only terms remaining in \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) are given by st ⊗ t ⊗ e1, e2 ⊗ t ⊗ ts, e2 ⊗ t ⊗ e1 and st ⊗ t ⊗ ts with respective coefficients \((\alpha _{1}^{\prime })^{2}-\phi _{1} \gamma \), − ((α3)2 − ϕ1γ), \((\alpha _{1}^{\prime }-\alpha _{3})\gamma \) and \((\alpha _{1}^{\prime }-\alpha _{3})\phi _{1}\). Comparing with Eq. C.2b, we get Eq. C.5b.
The method is exactly the same in the case l = 0 assuming that Eq. C.2c holds. □
We get by combining Lemmas C.1 and C.2 that if λ = l = 0 as well as \(\alpha _{1}^{\prime }\neq \alpha _{3}\), we are in the case 1.a) of Proposition 4.4. If \(\alpha _{1}^{\prime }=\alpha _{3}\) instead, we are in the case 1.b).
We now assume that at least one of the two constants λ,l is nonzero. Hence, if the double bracket (C.1a)–(C.1b) satisfies (C.2a), it must be such that
using Lemma C.1.
Lemma C.3
If Eq. C.2b holds, then ϕ0 = 0, lλ = 0 and \({\alpha _{3}^{2}}=\frac 14\). Moreover, the same statement holds if Eq. C.2c holds.
Proof
Developing \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) with Eq. ??, we get that the term e2 ⊗ tstst ⊗ ts only appears with a factor \({\phi _{0}^{2}}\). (This is also true for e2 ⊗ t ⊗ tststs, st ⊗ tstst ⊗ e1 and ststst ⊗ t ⊗ e1 with factor \(-{\phi _{0}^{2}}\).) Therefore ϕ0 = 0. Under this condition, we obtain that
and we get the remaining two equalities by comparing this expression with Eq. C.2b. The computation for \( \left \{\!\!\left \{s,s,t\right \}\!\!\right \} \) with Eq. C.2c is similar and gives the second result. □
As a consequence of this lemma, ϕ0 vanishes and \(\alpha _{3}=\pm \frac 12\) in Eq. C.6. Furthermore, either we have λ≠ 0 with l = 0, or we have l≠ 0 with λ = 0. These are respectively Case 2 and Case 3 from Proposition 4.4.
Appendix D:: Proof of Proposition 4.8
1.1 D.1 Coefficients verifying the triple brackets identities
The strategy of the proof is given after Proposition 4.8. In this subsection, we gather a list of equalities that the coefficients appearing in the double bracket must satisfy in order for the corresponding triple bracket to satisfy (??) or (??).
1.1.1 D.1.1 First conditions
Lemma D.1
If a double bracket given by Eqs. ??–?? and ?? satisfies (??), then we have \({\upbeta }_{0}={\upbeta }_{0}^{\prime }=\alpha _{1}=\alpha _{3}^{\prime }=0\).
Proof
Without computing all terms, we can remark that in \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) (obtained from Eq. ?? using Eqs. ??, ??) the element s3 ⊗ 1 ⊗ 1 appears with coefficient \({{\upbeta }_{0}^{2}}\), and so do respectively 1 ⊗ 1 ⊗ s3, t2s ⊗ 1 ⊗ 1 and 1 ⊗ 1 ⊗ st2 with coefficients \(-({\upbeta }_{0}^{\prime })^{2}\), \({\alpha _{1}^{2}}\) and \(-(\alpha _{3}^{\prime })^{2}\). None of these expressions appear Eq.??. □
We can go through a similar argument using \( \left \{\!\!\left \{s,s,t\right \}\!\!\right \} \) instead.
Lemma D.2
If a double bracket given by Eqs. ??–?? and ?? satisfies (??), then we have \(\alpha _{0}=\alpha _{0}^{\prime }=\alpha _{1}=\alpha _{3}^{\prime }=0\).
Hence, we are left to discuss the coefficients of the double bracket given by Eqs. ??–?? and
1.1.2 D.1.2 Identities verified by the coefficients when Eq. ?? is satisfied
Lemma D.3
Consider a double bracket defined on A by Eqs. ??, ?? and D.1, with ν = 0, \(\lambda \in {\Bbbk }\) and \(\mu \in \{\pm \frac 12\}\). Then Eq. ?? is satisfied if and only if the following list of identities hold :
Proof
We collect now all nonzero terms that appear in the expansion of \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) obtained from Eq. ??, leaving the cumbersome (but elementary) computations to the reader.
The coefficients for t ⊗ t ⊗ s, s ⊗ t ⊗ t, 1 ⊗ t ⊗ ts and st ⊗ t ⊗ 1 are respectively \(\gamma _{0}\gamma _{1}-{\alpha ^{2}_{2}}\), \((\alpha _{2}^{\prime })^{2}-\gamma _{0}\gamma _{1}\), \(-{\alpha _{3}^{2}}\) and \((\alpha _{1}^{\prime })^{2}\). The coefficient for t ⊗ 1 ⊗ ts and 1 ⊗ t2 ⊗ s is − α2α3 − μ(α2 + α3), while we have for st ⊗ 1 ⊗ t and s ⊗ t2 ⊗ 1 the coefficient \(\alpha _{1}^{\prime } \alpha _{2}^{\prime } - \mu (\alpha _{1}^{\prime } + \alpha _{2}^{\prime })\). Since these terms appear in Eq. ??, this gives Eqs. D.2a and D.2b. In particular, all the other coefficients in the expansion of \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) must vanish.
The coefficients for st ⊗ 1 ⊗ s, s ⊗ t ⊗ s and s ⊗ 1 ⊗ ts are respectively \(\gamma _{1} (\alpha _{1}^{\prime }- \mu )\), \(\gamma _{1} (\alpha _{2}^{\prime }-\alpha _{2})\) and γ1(α3 + μ), which yields Eq. D.2c.
The vanishing of the coefficients for st ⊗ 1 ⊗ 1, s ⊗ 1 ⊗ t and s ⊗ t ⊗ 1 gives successively the three identities in Eq. D.2d. Similarly 1 ⊗ 1 ⊗ ts, t ⊗ 1 ⊗ s and 1 ⊗ t ⊗ s imply Eq. D.2e, while 1 ⊗ t2 ⊗ t, t ⊗ t2 ⊗ 1 and t ⊗ t ⊗ t give Eq. D.2f.
The coefficients for t ⊗ t ⊗ 1,1 ⊗ t ⊗ t and 1 ⊗ t2 ⊗ 1,t ⊗ 1 ⊗ t give Eqs. D.2g and D.2h respectively. With t ⊗ 1 ⊗ 1, 1 ⊗ t ⊗ 1 and 1 ⊗ 1 ⊗ t we obtain (D.2i).
The terms 1 ⊗ 1 ⊗ s and s ⊗ 1 ⊗ 1 give Eq. D.2j. We finally get Eq. D.2k from s ⊗ 1 ⊗ s and 1 ⊗ 1 ⊗ 1. □
In the exact same way, we get the next lemma.
Lemma D.4
Consider a double bracket defined on A by Eqs. ??, ?? and D.1, with ν≠ 0 and \(\lambda , \mu \in {\Bbbk }\) satisfying 4(μ2 − λν) = 1. Then Eq. ?? is satisfied if and only if the following list of identities holds :
Remark D.5
These results are easily adapted to the case where the double bracket is Poisson, i.e. when the associated triple bracket (??) identically vanishes. In such a case, we require 4(μ2 − λν) = 0 to get \(\left \{\!\!\left \{{t,t,t}\right \}\!\!\right \}=0\) by [18, Proposition A.1].
If ν = μ = 0, then \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} =0\) when the conditions (D.2a)–(D.2k) of Lemma D.3 are satisfied with the extra requirements that all the terms containing a factor μ are removed, and that all the terms \(\pm \frac 12\) and \(\frac 14\) in Eqs. D.2a–D.2b are removed (in particular \(\alpha _{1}^{\prime }=\alpha _{3}=0\)).
If ν≠ 0 and μ2 − λν = 0 then \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} =0\) when the conditions (D.3a)–(D.3e) of Lemma D.3 are satisfied with the extra requirements that the terms \(\pm \frac 12\) and \(\frac 14\) appearing in the identities (D.3b) are removed.
1.1.3 D.1.3 Identities verified by the coefficients when Eq. ?? is satisfied
We can obtain the analogues of Lemmae D.3 and D.4 when Eq. ?? is satisfied as follows. Using the cyclic antisymmetry of the double bracket, remark that we can get from Eq. D.1
Comparing (??) and (??), then doing the same with Eqs. D.1 and D.4, one can see that to compute \( \left \{\!\!\left \{s,s,t\right \}\!\!\right \} \) one just needs to consider \( \left \{\!\!\left \{t,t,s\right \}\!\!\right \} \) in which we replace all variables s by t and vice-versa, then do the following changes in the coefficients
For n = 0, \(l \in {\Bbbk }\) and \(m \in \{\pm \frac 12\}\), we have that Eq. ?? is satisfied if and only if the list of identities obtained by applying (D.5) to (D.2a)–(D.2k) is verified.
For n≠ 0 and \(l,m \in {\Bbbk }\) satisfying 4(m2 − ln) = 1, we have that Eq. ?? is satisfied if and only if the list of identities obtained by applying (D.5) to (D.3a)–(D.3e) is verified.
1.2 D.2 Splitting the identities into cases
Lemma D.6
Consider a reduced double bracket defined on A by Eqs. ??, ?? and D.1, with ν = λ = 0 and \(\mu \in \{\pm \frac 12\}\). Then Eq. ?? is satisfied if and only if the double bracket verifies one of the following cases
-
Case A:
For \(\gamma _{0},\gamma _{1}\in {\Bbbk }^{\times }\), then \(\gamma \in {\Bbbk }\) is free while
$$ \begin{aligned} & \alpha_{1}^{\prime}=\mu, \quad \alpha_{3}=-\mu,\quad \alpha_{2}^{\prime}=\alpha_{2} {\kern1.7pt}{\kern1.7pt} \text{ with }{\kern1.7pt} {\alpha_{2}^{2}}=\frac14 + \gamma_{0} \gamma_{1}{\kern1.7pt}, \\ & {\upbeta}_{1}=\frac{\gamma_{0} {\upbeta}_{2}}{\alpha_{2}-\mu}, \quad {\upbeta}_{1}^{\prime}=\frac{\gamma_{0} {\upbeta}_{2}}{\alpha_{2}+\mu}, \quad {\upbeta}_{2}^{\prime}={\upbeta}_{2}{\kern1.7pt}{\kern1.7pt} \text{ with }{{\upbeta}_{2}^{2}}=\gamma \gamma_{1}{\kern1.7pt}. \end{aligned} $$(D.6) -
Case B:
For \(\gamma _{1}\in {\Bbbk }^{\times }\), γ0 = 0, then \({\upbeta }_{2} \in {\Bbbk }\) is free while
$$ \alpha_{1}^{\prime}=\mu, \quad \alpha_{3}=-\mu, \quad {\upbeta}_{2}^{\prime}={\upbeta}_{2}, \quad \gamma=\frac{{{\upbeta}_{2}^{2}}}{\gamma_{1}} {\kern1.7pt}, $$(D.7)and one of the following two sets of conditions holds :
$$ \begin{array}{@{}rcl@{}} \text{\textit{B1)}}&\quad \alpha_{2}^{\prime}=\alpha_{2}=\mu, \quad {\upbeta}_{1}^{\prime}=0, \quad {\upbeta}_{1}=\frac{2\mu {\upbeta}_{2}}{\gamma_{1}}{\kern1.7pt}, \end{array} $$(D.8a)$$ \begin{array}{@{}rcl@{}} \text{\textit{B2)}}&\quad \alpha_{2}^{\prime}=\alpha_{2}=-\mu, \quad {\upbeta}_{1}=0, \quad {\upbeta}_{1}^{\prime}=-\frac{2\mu {\upbeta}_{2}}{\gamma_{1}}{\kern1.7pt}. \end{array} $$(D.8b) -
Case C:
For \(\gamma _{0}\in {\Bbbk }^{\times }\), γ1 = 0, then
$$ \alpha_{1}^{\prime}=\mu, \quad \alpha_{3}=-\mu, \quad {\upbeta}_{2}^{\prime}={\upbeta}_{2}=0, \quad \gamma=0 {\kern1.7pt}, $$(D.9)and one of the following two sets of conditions holds :
$$ \begin{array}{@{}rcl@{}} \text{\textit{C1)}}&\quad \alpha_{2}^{\prime}=\alpha_{2}=\mu, \quad {\upbeta}_{1}^{\prime}=0, \quad {\upbeta}_{1}\in {\Bbbk}{\kern1.7pt}, \end{array} $$(D.10a)$$ \begin{array}{@{}rcl@{}} \text{\textit{C2)}}&\quad \alpha_{2}^{\prime}=\alpha_{2}=-\mu, \quad {\upbeta}_{1}=0, \quad {\upbeta}_{1}\in {\Bbbk}{\kern1.7pt}. \end{array} $$(D.10b) -
Case D:
For γ0 = γ1 = 0, then \({\upbeta }_{2}^{\prime }={\upbeta }_{2}=0\) and one of the following sets of conditions holds :
if \((\alpha _{1}^{\prime },\alpha _{3})=(-\mu ,\mu )\),
$$ \text{\textit{D1)}}\quad \alpha_{1}^{\prime}=\alpha_{2}=-\mu, \quad \alpha_{3}=\alpha_{2}^{\prime}=\mu, \quad {\upbeta}_{1}={\upbeta}_{1}^{\prime}=\gamma=0{\kern1.7pt}; $$(D.11)if \((\alpha _{1}^{\prime },\alpha _{3})=(\mu ,\mu )\),
$$ \begin{array}{@{}rcl@{}} \text{\textit{D2.1)}}&\quad \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{2}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{3}{\kern1.7pt}={\kern1.7pt}\mu, \quad \alpha_{2}{\kern1.7pt}={\kern1.7pt}-\mu, \quad {\upbeta}_{1}=0, \quad {\upbeta}_{1}^{\prime},\gamma\in {\Bbbk}{\kern1.7pt}, \end{array} $$(D.12a)$$ \begin{array}{@{}rcl@{}} \text{\textit{D2.2)}}&\quad \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{3}{\kern1.7pt}={\kern1.7pt}\mu, \quad \alpha_{2}=\alpha_{2}^{\prime}=-\mu, \quad {\upbeta}_{1}={\upbeta}_{1}^{\prime}=\gamma=0{\kern1.7pt}; \end{array} $$(D.12b)if \((\alpha _{1}^{\prime },\alpha _{3})=(-\mu ,-\mu )\),
$$ \begin{array}{@{}rcl@{}} \text{\textit{D3.1)}}&\quad \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{2}{\kern1.7pt}={\kern1.7pt}\alpha_{3}{\kern1.7pt}={\kern1.7pt}-\mu, \quad \alpha_{2}^{\prime}{\kern1.7pt}={\kern1.7pt}\mu, \quad {\upbeta}_{1}^{\prime}=0, \quad {\upbeta}_{1},\gamma\in {\Bbbk}{\kern1.7pt}, \end{array} $$(D.13a)$$ \begin{array}{@{}rcl@{}} \text{\textit{D3.2)}}&\quad \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{3}{\kern1.7pt}={\kern1.7pt}-\mu, \quad \alpha_{2}=\alpha_{2}^{\prime}=\mu, \quad {\upbeta}_{1}={\upbeta}_{1}^{\prime}=\gamma=0{\kern1.7pt}; \end{array} $$(D.13b)if \((\alpha _{1}^{\prime },\alpha _{3})=(\mu ,-\mu )\),
$$ \begin{array}{@{}rcl@{}} \text{\textit{D4.1)}}& \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{2}{\kern1.7pt}={\kern1.7pt}\alpha_{2}^{\prime}{\kern1.7pt}={\kern1.7pt}\mu, \quad \alpha_{3}{\kern1.7pt}={\kern1.7pt}-\mu, \quad {\upbeta}_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\gamma{\kern1.7pt}={\kern1.7pt}0, \quad {\upbeta}_{1}\in {\Bbbk}{\kern1.7pt}, \end{array} $$(D.14a)$$ \begin{array}{@{}rcl@{}} \text{\textit{D4.2)}}&\quad \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\mu, \quad \alpha_{3}{\kern1.7pt}={\kern1.7pt}\alpha_{2}{\kern1.7pt}={\kern1.7pt}\alpha_{2}^{\prime}{\kern1.7pt}={\kern1.7pt}-\mu, \quad {\upbeta}_{1}{\kern1.7pt}={\kern1.7pt}\gamma{\kern1.7pt}={\kern1.7pt}0, \quad {\upbeta}_{1}^{\prime}\in {\Bbbk}{\kern1.7pt}, \end{array} $$(D.14b)$$ \begin{array}{@{}rcl@{}} \text{\textit{D4.3)}}& \alpha_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}\alpha_{2}{\kern1.7pt}={\kern1.7pt}\mu, \quad \alpha_{3}{\kern1.7pt}={\kern1.7pt}\alpha_{2}^{\prime}{\kern1.7pt}={\kern1.7pt}-\mu, \quad \gamma{\kern1.7pt}={\kern1.7pt}0, \quad {\upbeta}_{1}^{\prime}{\kern1.7pt}={\kern1.7pt}-{\upbeta}_{1},{\kern1.7pt} {\upbeta}_{1}\!\in\! {\Bbbk},\quad \end{array} $$(D.14c)$$ \begin{array}{@{}rcl@{}} \text{\textit{D4.4)}}&\quad \alpha_{1}^{\prime}=\alpha_{2}^{\prime}=\mu, \quad \alpha_{3}=\alpha_{2}=-\mu, \quad {\upbeta}_{1}={\upbeta}_{1}^{\prime}=\gamma=0{\kern1.7pt}. \end{array} $$(D.14d)
The proof of Lemma D.6 consists in listing the possible coefficients of a reduced double bracket that satisfy Lemma D.3. The next lemma is obtained similarly from Lemma D.4.
Lemma D.7
Consider a reduced double bracket defined on A by Eqs. ??, ?? and D.1, with \(\nu \in {\Bbbk }^{\times }\) and μ = 0, \(\lambda =\frac {-1}{4\nu }\). Then Eq. ?? is satisfied if and only if the double bracket verifies
and one of the following two conditions holds :
Remark D.8
From the discussion in §D.1.3, we get that a reduced double bracket defined on A by Eqs. ??, ?? and D.1 satisfies (??) if and only if the double bracket verifies one of the cases from Lemma D.6 or Lemma D.7 after application of the mapping (D.5) on the different coefficients in each case.
1.3 D.3 Finishing the proof
We need to see which conditions from Lemma D.6 or Lemma D.7 are compatible with at least one of the conditions obtained by applying the mapping (D.5), as explained in Remark D.8.
For example, applying transformation (D.5) to the case D4.4 in Lemma D.6 yields
A quick inspection shows that this is compatible with the conditions of the cases D1, D4.3 given by Eqs. D.11, D.14c in Lemma D.6, and Aν given by Eq. D.16a in Lemma D.7. In the first two cases, and under the isomorphism t↦s, s↦t (with \(\mu \leftrightarrow m\)), the obtained double quasi-Poisson brackets satisfy Case 3 of Proposition 4.8 given by Eq. ??. In the last case, the double bracket is isomorphic to Case 6 of Proposition 4.8 given by Eq. ?? under the same isomorphism (with m↦μ, ν↦n).
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Fairon, M. Double Quasi-Poisson Brackets: Fusion and New Examples. Algebr Represent Theor 24, 911–958 (2021). https://doi.org/10.1007/s10468-020-09974-w
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DOI: https://doi.org/10.1007/s10468-020-09974-w