Star-fundamental algebras: polynomial identities and asymptotics
HTML articles powered by AMS MathViewer
- by Antonio Giambruno, Daniela La Mattina and Cesar Polcino Milies PDF
- Trans. Amer. Math. Soc. 373 (2020), 7869-7899 Request permission
Abstract:
We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution $*$.
To any star-algebra $A$ is attached a numerical sequence $c_n^*(A)$, $n\ge 1$, called the sequence of $*$-codimensions of $A$. Its asymptotic is an invariant giving a measure of the $*$-polynomial identities satisfied by $A$. It is well known that for a PI-algebra such a sequence is exponentially bounded and $\exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)}$ can be explicitly computed. Here we prove that if $A$ is a star-fundamental algebra, \begin{equation*} C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, \end{equation*} where $C_1>0,C_2, t$ are constants and $t$ is explicitly computed as a linear function of the dimension of the skew semisimple part of $A$ and the nilpotency index of the Jacobson radical of $A$. We also prove that any finite dimensional star-algebra has the same $*$-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144–167] we get that if $A$ is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, $\lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n}$ exists and is an integer or half an integer.
References
- Eli Aljadeff, Antonio Giambruno, and Yakov Karasik, Polynomial identities with involution, superinvolutions and the Grassmann envelope, Proc. Amer. Math. Soc. 145 (2017), no. 5, 1843–1857. MR 3611301, DOI 10.1090/proc/13546
- Eli Aljadeff, Geoffrey Janssens, and Yakov Karasik, The polynomial part of the codimension growth of affine PI algebras, Adv. Math. 309 (2017), 487–511. MR 3607284, DOI 10.1016/j.aim.2017.01.022
- 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
- William Beckner and Amitai Regev, Asymptotic estimates using probability, Adv. Math. 138 (1998), no. 1, 1–14. MR 1645060, DOI 10.1006/aima.1994.1503
- Allan Berele, Properties of hook Schur functions with applications to p.i. algebras, Adv. in Appl. Math. 41 (2008), no. 1, 52–75. MR 2419763, DOI 10.1016/j.aam.2007.03.002
- Allan Berele, Antonio Giambruno, and Amitai Regev, Involution codimensions and trace codimensions of matrices are asymptotically equal. part A, Israel J. Math. 96 (1996), no. part A, 49–62. MR 1432726, DOI 10.1007/BF02785533
- A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime $T$-ideals, Israel J. Math. 103 (1998), 17–28. MR 1613536, DOI 10.1007/BF02762265
- Allan Berele and Amitai Regev, Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5155–5172. MR 2415069, DOI 10.1090/S0002-9947-08-04500-5
- 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
- A. Giambruno and D. La Mattina, PI-algebras with slow codimension growth, J. Algebra 284 (2005), no. 1, 371–391. MR 2115020, DOI 10.1016/j.jalgebra.2004.09.003
- A. Giambruno, D. La Mattina, and V. M. Petrogradsky, Matrix algebras of polynomial codimension growth, Israel J. Math. 158 (2007), 367–378. MR 2342471, DOI 10.1007/s11856-007-0017-7
- Antonio Giambruno, Daniela La Mattina, and Mikhail Zaicev, Classifying the minimal varieties of polynomial growth, Canad. J. Math. 66 (2014), no. 3, 625–640. MR 3194163, DOI 10.4153/CJM-2013-016-5
- A. Giambruno, C. Polcino Milies, and A. Valenti, Star-polynomial identities: computing the exponential growth of the codimensions, J. Algebra 469 (2017), 302–322. MR 3563016, DOI 10.1016/j.jalgebra.2016.07.037
- 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
- A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), no. 2, 145–155. MR 1658530, DOI 10.1006/aima.1998.1766
- A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221–243. MR 1680198, DOI 10.1006/aima.1998.1790
- A. Giambruno and M. Zaicev, Involution codimensions of finite-dimensional algebras and exponential growth, J. Algebra 222 (1999), no. 2, 471–484. MR 1734235, DOI 10.1006/jabr.1999.8016
- A. Giambruno and M. Zaicev, Asymptotics for the standard and the Capelli identities, Israel J. Math. 135 (2003), 125–145. MR 1996399, DOI 10.1007/BF02776053
- A. Giambruno and M. Zaicev, Codimension growth and minimal superalgebras, Trans. Amer. Math. Soc. 355 (2003), no. 12, 5091–5117. MR 1997596, DOI 10.1090/S0002-9947-03-03360-9
- 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
- Antonio Giambruno and Mikhail Zaicev, Proper identities, Lie identities and exponential codimension growth, J. Algebra 320 (2008), no. 5, 1933–1962. MR 2437638, DOI 10.1016/j.jalgebra.2008.06.009
- Antonio Giambruno and Mikhail Zaicev, Growth of polynomial identities: is the sequence of codimensions eventually non-decreasing?, Bull. Lond. Math. Soc. 46 (2014), no. 4, 771–778. MR 3239615, DOI 10.1112/blms/bdu031
- A. S. Gordienko, Asymptotics of $H$-identities for associative algebras with an $H$-invariant radical, J. Algebra 393 (2013), 92–101. MR 3090060, DOI 10.1016/j.jalgebra.2013.05.032
- I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. MR 0442017
- 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
- A. R. Kemer, Finite basability of identities of associative algebras, Algebra i Logika 26 (1987), no. 5, 597–641, 650 (Russian). MR 985840
- A. R. Kemer, Representability of reduced-free algebras, Algebra i Logika 27 (1988), no. 3, 274–294, 375 (Russian); English transl., Algebra and Logic 27 (1988), no. 3, 167–184 (1989). MR 997959, DOI 10.1007/BF01978562
- 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
- Daniela La Mattina, Varieties of almost polynomial growth: classifying their subvarieties, Manuscripta Math. 123 (2007), no. 2, 185–203. MR 2306632, DOI 10.1007/s00229-007-0091-5
- Daniela La Mattina, Varieties of algebras of polynomial growth, Boll. Unione Mat. Ital. (9) 1 (2008), no. 3, 525–538. MR 2455329
- Daniela La Mattina and Fabrizio Martino, Polynomial growth and star-varieties, J. Pure Appl. Algebra 220 (2016), no. 1, 246–262. MR 3393459, DOI 10.1016/j.jpaa.2015.06.008
- D. La Mattina, S. Mauceri, and P. Misso, Polynomial growth and identities of superalgebras and star-algebras, J. Pure Appl. Algebra 213 (2009), no. 11, 2087–2094. MR 2533307, DOI 10.1016/j.jpaa.2009.03.003
- D. La Mattina and P. Misso, Algebras with involution with linear codimension growth, J. Algebra 305 (2006), no. 1, 270–291. MR 2262527, DOI 10.1016/j.jalgebra.2006.06.044
- Amitai Regev, Existence of identities in $A\otimes B$, Israel J. Math. 11 (1972), 131–152. MR 314893, DOI 10.1007/BF02762615
- Amitai Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), no. 2-3, 246–250. MR 738172, DOI 10.1007/BF02760520
- Louis H. Rowen, Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Academic Press, Inc., Boston, MA, 1988. MR 940245
- Irina Sviridova, Finitely generated algebras with involution and their identities, J. Algebra 383 (2013), 144–167. MR 3037972, DOI 10.1016/j.jalgebra.2013.03.002
Additional Information
- Antonio Giambruno
- Affiliation: Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy
- MR Author ID: 73185
- ORCID: 0000-0002-3422-2539
- Email: antonio.giambruno@unipa.it, antoniogiambr@gmail.com
- Daniela La Mattina
- Affiliation: Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy
- MR Author ID: 734661
- ORCID: 0000-0002-0714-1442
- Email: daniela.lamattina@unipa.it
- Cesar Polcino Milies
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, CEP-05315-970, São Paulo, Brazil; and Universidade Federal do ABC, Av. dos Estados 5001, Santo Andre, São Paulo, Brazil
- MR Author ID: 140680
- ORCID: 0000-0002-8389-0533
- Email: polcino@ime.usp.br, polcino@ufabc.edu.br
- Received by editor(s): April 2, 2018
- Received by editor(s) in revised form: October 29, 2019, and March 2, 2020
- Published electronically: August 28, 2020
- Additional Notes: The first author was partially supported by CNPq proc. 400439/2014-0, the second author was partially supported by GNSAGA of INdAM, and the third author was partially supported by FAPESP, proc. 2015/09162-9 and CNPq proc. 300243/79-0.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 7869-7899
- MSC (2010): Primary 16R10, 16R50; Secondary 16P90, 16W10
- DOI: https://doi.org/10.1090/tran/8182
- MathSciNet review: 4169676