A Characterization of Superalgebras with Pseudoinvolution of Exponent 2
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A Characterization of Superalgebras with Pseudoinvolution of Exponent 2 Antonio Ioppolo1 Received: 17 May 2020 / Accepted: 16 September 2020 / © Springer Nature B.V. 2020
Abstract Let A be a superalgebra endowed with a pseudoinvolution ∗ over an algebraically closed field of characteristic zero. If A satisfies an ordinary non-trivial identity, then its graded ∗codimension sequence cn∗ (A), n = 1, 2, . . ., is exponentially bounded (Ioppolo and Martino (Linear Multilinear Algebra 66(11), 2286–2304 2018). In this paper we capture this exponential growth giving a positive answer to the Amitsur’s conjecture for this kind of algebras. More precisely, we shall see that the limn→∞ n cn∗ (A) exists and it is an integer, denoted exp∗ (A) and called graded ∗-exponent of A. Moreover, we shall characterize superalgebras with pseudoinvolution according to their graded ∗-exponent. Keywords Exponent · Pseudoinvolutions · Exponential growth · Polynomial identities Mathematics Subject Classification (2010) Primary 16R50 · 16W50; Secondary 16R10 · 16W10
1 Introduction A conjecture of Amitsur, well known to mathematicians working in the theory of polynomial identities, stated that the exponential growth of the codimension sequence of a PI-algebra (i.e., an algebra satisfying a non-trivial identity) should be an integer. Such a sequence was introduced by Regev in 1972 and, in characteristic zero, gives an actual quantitative measure of the identities satisfied by a given algebra. In [19], he proved that for a PI-algebra the sequence of codimensions is exponentially bounded. It is exactly this exponential growth that Giambruno and Zaicev captured in their celebrated results [8, 9], answering positively to the Amitsur’s conjecture. More precisely, they proved the existence of such an integer, called exponent of the PI-algebra, and they gave an explicit way to
Communicated by:Presented by: Kenneth Goodearl A. Ioppolo was supported by the Fapesp post-doctoral grant number 2018/17464-3 Antonio Ioppolo
[email protected] 1
IMECC, UNICAMP, S´ergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil
A. Ioppolo
compute it. Moreover, in [11], they characterized associative algebras of exponent equal to 2. From that moment, several authors studied the problem posed by Amitsur and, more generally, the exponential growth of the codimension sequence in the setting of algebras with some additional structure. Between them, we recall the cases of algebras with involution [7, 10, 18], superalgebras [3] and more generally algebras graded by a group [1, 2, 6, 15], algebras with a generalised Hopf action [12] and superalgebras with graded involution [20] or superinvolution [4, 5, 13]. In this paper we deal with superalgebras endowed with a pseudoinvolution which are a natural generalization of the algebras with involution. The existence of pseudoinvolutions of the first kind was proved in 2008 by Jaber in [16]. Two years later, Martinez and Zelmanov used pseudoinvolutions in order to completely classify the irreducible bimodules over simple
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