Polynomial identities with involution, superinvolutions and the Grassmann envelope
Authors:
Eli Aljadeff, Antonio Giambruno and Yakov Karasik
Journal:
Proc. Amer. Math. Soc. 145 (2017), 1843-1857
MSC (2010):
Primary 16R10, 16R50; Secondary 16W10
DOI:
https://doi.org/10.1090/proc/13546
Published electronically:
January 11, 2017
MathSciNet review:
3611301
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be an algebra with involution
over a field of characteristic zero. We prove that in case
satisfies a non-trivial
-identity, then
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.
- [1] Eli Aljadeff and Alexei Kanel-Belov, Representability and Specht problem for 𝐺-graded algebras, Adv. Math. 225 (2010), no. 5, 2391–2428. MR 2680170, https://doi.org/10.1016/j.aim.2010.04.025
- [2] 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, https://doi.org/10.1016/j.jpaa.2015.12.008
- [3] S. A. Amitsur, Identities in rings with involutions, Israel J. Math. 7 (1969), 63–68. MR 242889, https://doi.org/10.1007/BF02771748
- [4] Vesselin Drensky and Antonio Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for 2×2 matrices with involution, Canad. J. Math. 46 (1994), no. 4, 718–733. MR 1289056, https://doi.org/10.4153/CJM-1994-040-6
- [5] 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, https://doi.org/10.1007/s10468-015-9590-3
- [6] Antonino Giambruno and Amitai Regev, Wreath products and P.I. algebras, J. Pure Appl. Algebra 35 (1985), no. 2, 133–149. MR 775466, https://doi.org/10.1016/0022-4049(85)90036-2
- [7] Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105
- [8] 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, https://doi.org/10.1016/j.jpaa.2012.11.001
- [9] Yaakov Karasik, Kemer’s theory for 𝐻-module algebras with application to the PI exponent, J. Algebra 457 (2016), 194–227. MR 3490081, https://doi.org/10.1016/j.jalgebra.2016.02.021
- [10] 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
- [11] Amitai Regev, Existence of identities in 𝐴⊗𝐵, Israel J. Math. 11 (1972), 131–152. MR 314893, https://doi.org/10.1007/BF02762615
- [12] I. Sviridova, Finite basis problem for identities with involution, preprint, arXiv:1410.2233.
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16R10, 16R50, 16W10
Retrieve articles in all journals with MSC (2010): 16R10, 16R50, 16W10
Additional Information
Eli Aljadeff
Affiliation:
Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Email:
aljadeff@tx.technion.ac.il
Antonio Giambruno
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Email:
antonio.giambruno@unipa.it
Yakov Karasik
Affiliation:
Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Email:
yaakov@tx.technion.ac.il
DOI:
https://doi.org/10.1090/proc/13546
Keywords:
Polynomial identity,
involution,
superinvolution,
Grassmann algebra
Received by editor(s):
June 12, 2016
Published electronically:
January 11, 2017
Additional Notes:
The first author was partially supported by the ISRAEL SCIENCE FOUNDATION(grant No. 1017/12). The second author was partially supported by GNSAGA of INDAM
Communicated by:
Jerzy Weyman
Article copyright:
© Copyright 2017
American Mathematical Society