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Infinity-enhancing of Leibniz algebras

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Abstract

We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies (Bonezzi and Hohm in Commun Math Phys 377:2027–2077, 2020), and differential graded Lie algebras, which have been already used in this context. We explain how any Leibniz algebra gives rise to a differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra. Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically induces an \(L_\infty \)-algebra structure on the suspension of the underlying chain complex. We explicitly give the brackets to all orders and show that they agree with the partial results obtained from the infinity-enhanced Leibniz algebras in Bonezzi and Hohm (Commun Math Phys 377:2027–2077, 2020).

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Notes

  1. Unfortunately, the differential graded Lie algebra in [41] was also called ‘tensor hierarchy algebra’, but there is an important difference: the tensor hierarchy algebra in [35] is a priori not a differential graded Lie algebra, but a \(\mathbb {Z}\)-graded Lie superalgebra with a subspace at degree \(-1\) accommodating all possible embedding tensors satisfying the representation constraint. Restricting it to a one-dimensional subspace spanned by one particular embedding tensor leads to the differential graded Lie algebra called ‘tensor hierarchy algebra’ in [41].

  2. The solid lines would correspond to functors in the language of category theory. Then, given the diagram, there would exist a canonical functor from the category of Leibniz algebras to the category of \(L_\infty \)-algebras (that restricts to the identity functor on the full subcategory of Lie algebras). However, we do not address this question explicitly in this paper, because that would significantly increase its length and obscure our original motivation. See also [53] for another point of view on this question.

  3. Obviously, the suspension operator has an inverse. It is called the desuspension operator and is denoted \(s^{-1}\).

  4. By (3.5), we see that the generalized Lie derivatives are endomorphisms of \(\mathfrak {gl}(V)\), so \(T_0\) can be seen as a sub-Lie-algebra of \(\mathfrak {gl}(V)\). Moreover, by (2.22), we have that \(\mathcal {L}_x(y)=x\circ y=x_L(y)\) for every \(x,y\in V\). Hence, \(\mathcal {L}_x\) and \(x_L\) define the same endomorphism on V; however, it could well happen that for some \(x\in \mathcal {Z}\), the right-hand side of (3.6) is not vanishing, implying that \(T_0\) may be bigger than \({V}\Big /{\mathcal {Z}}\). This is consistent with the fact that the map \(x\longmapsto \mathcal {L}_x\) defines an embedding tensor \(\Theta :V\rightarrow \mathfrak {gl}(V)\). Then, by (2.27), we obtain again that \(T_0=\mathrm {Im}(\Theta )\simeq {V}\Big /{\mathrm {Ker}(\Theta )}\) may have a bigger dimension than \({V}\Big /{\mathcal {Z}}\).

  5. There are actually many more non-negatively graded dgLa than there are infinity-enhanced Leibniz algebras, because the degree 0 part of such a dgLa T is forgotten by the functor G. This opens the question of the physical information that may be contained in \(T_0\), and that cannot be captured by the associated infinity-enhanced Leibniz algebra G(T).

  6. The rigorous formula is

    $$\begin{aligned} \sum _{i+j=n+1}\ (-1)^{i(j-1)} \sum _{\sigma \in \text {Un}(i,n-i)}\epsilon ^\sigma _{x_1,\ldots ,x_n}\,l_{j}\big (l_i(x_{\sigma (1)},\ldots ,x_{\sigma (i)}),x_{\sigma (i+1)},\ldots ,x_{\sigma (n)}\big )=0\nonumber \\ \end{aligned}$$
    (4.2)

    where \(\text {Un}(i,n-i)\) is the set of \((i,n-i)\)-unshuffles and where \(\epsilon ^\sigma _{x_1,\ldots ,x_n}\) is the sign induced by the permutation of the elements \(x_1,\ldots , x_n\) in the exterior algebra of X, i.e., \(x_1\wedge \ldots x_n=\epsilon ^\sigma _{x_1,\ldots ,x_n} x_{\sigma (1)}\wedge \ldots \wedge x_{\sigma (n)}\).

  7. See for example Remark 5.4 in [52] and there replace the differential \(\partial \) with the map D defined in [51]. Unfortunately, they also made a mistake in this application: for \(n\ge 2\) the right-hand side of the formula computing the \(n+1\) bracket should inherit a minus sign.

  8. Since Getzler’s 1-bracket is \(\{a\}=\partial (a)\), we pick up a minus sign from (4.3).

  9. Getzler’s definition of higher brackets does not satisfy the definition of a \(L_\infty \)-algebra, but it does if we reverse the sign of the odd brackets of order 3 and higher (even brackets of order 4 and higher are vanishing anyway since \(B_3=B_5=\cdots =0\)). Hence, when passing to the skew-symmetric convention, the odd brackets inherit a plus sign, because the sign brought by the translation from the symmetric convention to the skew-symmetric convention (see (4.3)) cancels the sign that we added to correct Getzler’s formula for odd brackets of order 3 and higher.

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Acknowledgements

We would like to thank Roberto Bonezzi, Martin Cederwall, Olaf Hohm and Jim Stasheff for discussions and comments on the first version of this paper. In particular we are grateful to Olaf Hohm for explaining the ideas behind the work [1]. We would also like to thank the anonymous referee for suggesting some well justified clarifications on the role of the degree-zero subspace of the differential graded Lie algebras. This work was initiated during a visit at Institut des Hautes Études Scientifiques (IHÉS), and we would like to thank the institute for its hospitality. The work of JP is supported by the Swedish Research Council, project no. 2015-02468. The work of SL is supported by the Agence Nationale de la Recherche, project SINGSTAR.

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Authors and Affiliations

  1. IMJ-PRG, Université Paris Diderot, Paris, France

    Sylvain Lavau

  2. Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden

    Jakob Palmkvist

  3. School of Science and Technology, Örebro University, Örebro, Sweden

    Jakob Palmkvist

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  1. Sylvain Lavau
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Lavau, S., Palmkvist, J. Infinity-enhancing of Leibniz algebras. Lett Math Phys 110, 3121–3152 (2020). https://doi.org/10.1007/s11005-020-01324-7

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