Abstract
We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.
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Bahturin, Yu. A.: Identical Relations in Lie Algebras. VNU Science Press, Utrecht (1987)
Bahturin, Y.A., Yasumura, F.: Distinguishing simple algebras by means of polynomial identities. São Paulo J. Math. 13, 39–72 (2019)
Beidar, K.I., Martindale, III, W.S., Mikhalev, A.V.: Rings with generalized identities. Marcel Dekker, Inc, (1996)
Berele, A.: Cocharacter sequence for algebras with Hopf algebra actions. J. Algebra 185, 869–885 (1996)
Coelho, S.P., Polcino Milies, C.: Derivations of upper triangular matrix rings. Linear Algebra Appl. 187, 263–267 (1993)
Di Vincenzo, O.M., Nardozza, V.: On the ∗-Polynomial Identities of a Class of ∗-minimal algebras. Comm. Algebra 38, 3078–3093 (2010)
Drensky, V.: Free Algebras and PI-Algebras. Graduate course in Algebra. Springer, Singapore (2000)
Drensky, V., Giambruno, A.: Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 × 2 matrices with involution. Canad. J. Math. 46, 718–733 (1994)
Drensky, V., Kasparian, A.: Polynomial identities of eighth degree for 3 × 3 matrices, Annuaire de l’Univ. de Sofia. Fac. de Math. et Meca., Livre 1, Math. 77, 175–195 (1983)
Galitski, L. Yu., Timashev, D.A.: On classification of metabelian Lie algebras. J. Lie Theory 9, 125–156 (1999)
Giambruno, A., Rizzo, C.: Differential identities, 2×2 upper triangular matrices and varieties of almost polynomial growth, J. Pure Appl. Algebra. https://doi.org/10.1016/j.jpaa.2018.07.004 (2018)
Jacobson, N.: Lie algebras. Interscience Publishers (1962)
Karchenko, V.K.: Differential identities of prime rings. Algebra and Logic 17(2), 58–80 (1979)
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Presented by: Michel Van den Bergh
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Di Vincenzo, O.M., Nardozza, V. Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra. Algebr Represent Theor 25, 187–209 (2022). https://doi.org/10.1007/s10468-020-10014-w
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DOI: https://doi.org/10.1007/s10468-020-10014-w
Keywords
- Differential polynomial identity
- Upper triangular matrices
- Differential codimensions
- Lie algebras
- Derivation action