Polynomial identities with involution, superinvolutions and the Grassmann envelope

Aljadeff, Eli; Giambruno, Antonio; Karasik, Yakov

doi:10.1090/proc/13546

Polynomial identities with involution, superinvolutions and the Grassmann envelope
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by Eli Aljadeff, Antonio Giambruno and Yakov Karasik PDF

Proc. Amer. Math. Soc. 145 (2017), 1843-1857 Request permission

Abstract:

Let $A$ be an algebra with involution $*$ over a field of characteristic zero. We prove that in case $A$ satisfies a non-trivial $*$-identity, then $A$ has the same $*$-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution. As a consequence we give a positive answer to the Specht problem for algebras with involution, i.e., any T-ideal of identities of an algebra with involution is finitely generated as a T-ideal.

References

Eli Aljadeff and Alexei Kanel-Belov, Representability and Specht problem for $G$-graded algebras, Adv. Math. 225 (2010), no. 5, 2391–2428. MR 2680170, DOI 10.1016/j.aim.2010.04.025
Eli Aljadeff, Alexei Kanel-Belov, and Yaakov Karasik, Kemer’s theorem for affine PI algebras over a field of characteristic zero, J. Pure Appl. Algebra 220 (2016), no. 8, 2771–2808. MR 3471186, DOI 10.1016/j.jpaa.2015.12.008
S. A. Amitsur, Identities in rings with involutions, Israel J. Math. 7 (1969), 63–68. MR 242889, DOI 10.1007/BF02771748
Vesselin Drensky and Antonio Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for $2\times 2$ matrices with involution, Canad. J. Math. 46 (1994), no. 4, 718–733. MR 1289056, DOI 10.4153/CJM-1994-040-6
Antonio Giambruno, Antonio Ioppolo, and Daniela La Mattina, Varieties of algebras with superinvolution of almost polynomial growth, Algebr. Represent. Theory 19 (2016), no. 3, 599–611. MR 3503233, DOI 10.1007/s10468-015-9590-3
Antonino Giambruno and Amitai Regev, Wreath products and P.I. algebras, J. Pure Appl. Algebra 35 (1985), no. 2, 133–149. MR 775466, DOI 10.1016/0022-4049(85)90036-2
Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105, DOI 10.1090/surv/122
A. S. Gordienko, Amitsur’s conjecture for associative algebras with a generalized Hopf action, J. Pure Appl. Algebra 217 (2013), no. 8, 1395–1411. MR 3030542, DOI 10.1016/j.jpaa.2012.11.001
Yaakov Karasik, Kemer’s theory for $H$-module algebras with application to the PI exponent, J. Algebra 457 (2016), 194–227. MR 3490081, DOI 10.1016/j.jalgebra.2016.02.021
Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR 1108620, DOI 10.1090/mmono/087
Amitai Regev, Existence of identities in $A\otimes B$, Israel J. Math. 11 (1972), 131–152. MR 314893, DOI 10.1007/BF02762615
I. Sviridova, Finite basis problem for identities with involution, preprint, arXiv:1410.2233.