A relevant problem for the study of polynomial identities satisfied by an associative algebra with involution (A, *) consists in describing the codimension sequence c_m(A,*) associated to the ideal T*(A) of all *-polynomial identities of A. If A is finite dimensional then the behavior of the exponential growth of the codimension sequence is described by the so-called *-exponent. In this paper we contribute to this research by describing a class of algebras with involution that contains all *-minimal algebras up to *-PI equivalence. More precisely, we assume the field F is of characteristic zero and all algebras are finite dimensional. Given an m-tuple (A_1,...,A_m) of finite dimensional *-simple algebras we introduce a block-triangular matrix algebra with involution, denoted as UT* (A_1,...,A_m), where each A_i can be embedded as *algebra. We describe the T*-ideal of R = UT*(A_1,...,A_m) in terms of the ideals T*(A_i) and prove that any algebra with involution which is minimal with respect to its *-exponent is *-PIequivalent to R for a suitable choice of (A_1,...,A_m). For the case t = 1 that is when R is a *-simple algebra itself we show that R is *-minimal by using arguments based on the representation theory of symmetric groups. We prove the *-minimality of R also for the case A_ 1 = ... = A_m = F.

Minimal algebras with respect to their *-exponent

DI VINCENZO, Onofrio Mario;
2007

Abstract

A relevant problem for the study of polynomial identities satisfied by an associative algebra with involution (A, *) consists in describing the codimension sequence c_m(A,*) associated to the ideal T*(A) of all *-polynomial identities of A. If A is finite dimensional then the behavior of the exponential growth of the codimension sequence is described by the so-called *-exponent. In this paper we contribute to this research by describing a class of algebras with involution that contains all *-minimal algebras up to *-PI equivalence. More precisely, we assume the field F is of characteristic zero and all algebras are finite dimensional. Given an m-tuple (A_1,...,A_m) of finite dimensional *-simple algebras we introduce a block-triangular matrix algebra with involution, denoted as UT* (A_1,...,A_m), where each A_i can be embedded as *algebra. We describe the T*-ideal of R = UT*(A_1,...,A_m) in terms of the ideals T*(A_i) and prove that any algebra with involution which is minimal with respect to its *-exponent is *-PIequivalent to R for a suitable choice of (A_1,...,A_m). For the case t = 1 that is when R is a *-simple algebra itself we show that R is *-minimal by using arguments based on the representation theory of symmetric groups. We prove the *-minimality of R also for the case A_ 1 = ... = A_m = F.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11563/2607
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