Artículos de revistas
On The Polynomial Identities Of The Algebra M 11 (e)
Registro en:
Linear Algebra And Its Applications. , v. 438, n. 11, p. 4469 - 4482, 2013.
243795
10.1016/j.laa.2013.01.032
2-s2.0-84875455586
Autor
Koshlukov P.
De Mello T.C.
Institución
Resumen
Verbally prime algebras are important in PI theory. They were described by Kemer over a field K of characteristic zero: 0 and K〈T〉 (the trivial ones), Mn(K),Mn(E), Mab(E). Here K〈T〉 is the free associative algebra of infinite rank, with free generators T,E denotes the infinite dimensional Grassmann algebra over K, Mn(K) and Mn(E) are the n×n matrices over K and over E, respectively. The algebras Mab(E) are subalgebras of Ma+b(E), see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of Mn(K) and lots of its properties. Models for the generic algebras of Mn(E) and Mab(E) are also known but their structure remains quite unclear. In this paper we study the generic algebra of M 11(E) in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of M 11(E). Also we describe the polynomial identities in two variables of the algebra M11(E). © 2013 Elsevier Inc. All rights reserved. 438 11 4469 4482 Azevedo, S., Fidelis, M., Koshlukov, P., Tensor product theorems in positive characteristic (2004) J. Algebra, 276 (2), pp. 836-845 Berele, A., Generic verbally prime algebras and their GK-dimensions (1993) Commun. Algebra, 21 (5), pp. 1487-1504 Colombo, J., Koshlukov, P., Central polynomials in the matrix algebra of order two (2004) Linear Algebra Appl., 377, pp. 53-67 Drensky, V., A minimal basis for the identities of a second-order matrix algebra over a field of characteristic 0 (1981) Algebra i Logika, 20 (3), pp. 282-290. , (in Russian), Translation: Algebra Log. 20(3) (1981) 188-194 Drensky, V., (1999) Free Algebras and PI-algebras: Graduate Course in Algebra, , Springer Singapore Genov, G., Basis for identities of a third order matrix algebra over a finite field (1981) Algebra Log., 20, pp. 241-257 Genov, G., Siderov, P., A basis for identities of the algebra of fourth-order matrices over a finite field. I, II (1982) Serdica, 8, pp. 313-323. , 351-366 (in Russian) Giambruno, A., Koshlukov, P., On the identities of the Grassmann algebras in characteristic p > 0 (2001) Israel J. Math., 122, pp. 305-316 Gordienko, A., Regev's conjecture and codimensions of P.I. algebras (2009) Acta Appl. Math., 108, pp. 33-55 Kemer, A.R., Asymptotic basis of identities of algebras with unit from the variety Var (M 2 (K)) (1990) Soviet Math., 33 (6), pp. 71-76 Kemer, A.R., Ideals of Identities of Associative Algebras (1991) Translations Math. Monographs, 84. , AMS Providence, RI Kemer, A.R., The standard identity in characteristic p: A conjecture of I.B. Volichenko (1993) Israel J. Math., 81, pp. 343-355 Koshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic p ≠ 2 (2001) J. Algebra, 241, pp. 410-434 Koshlukov, P., De Mello, T.C., The centre of generic algebras of small PI algebras (2013) J. Algebra, 375, pp. 109-120 Koshlukov, P., De Mello, T.C., On the Polynomial Identities of the Algebra M 11 (E), , arxiv:1208.2185 Krakowski, D., Regev, A., The polynomial identities of the Grassmann algebra (1973) Trans. Amer. Math. Soc., 191, pp. 429-438 Maltsev, Yu.N., Kuzmin, E.N., A basis for identities of the algebra of second order matrices over a finite field (1978) Algebra Log., 17, pp. 17-21 Mishchenko, S., Valenti, A., A star-variety with almost polynomial growth (2000) J. Algebra, 223 (1), pp. 66-84 Nikolaev, R., Identities of two variables in the second-order matrix algebra over a field of characteristic zero (1984) Serdica, 10 (1), pp. 11-18. , (in Russian) Popov, A., Identities of the tensor square of a Grassmann algebra (1982) Algebra i Logika, 21 (4), pp. 442-471. , Translation: Algebra Log. 21(4) (1982) 296-316 Procesi, C., Rings with Polynomial Identities (1973) Pure and Appl. Math., 17. , Marcel Dekker New York Razmyslov, Yu.P., The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero (1973) Algebra i Logika, 12, pp. 83-113. , 121 (in Russian), Translation: Algebra Log. 12 (1973) 47-63 (1974) Razmyslov, Yu.P., Identities of Algebras and Their Representations (1994) Translations of Math. Monographs, 138. , Amer. Math. Soc. Providence, RI