Codimensions of star-algebras and low exponential growth
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CODIMENSIONS OF STAR-ALGEBRAS AND LOW EXPONENTIAL GROWTH BY
Antonio Giambruno and Daniela La Mattina∗ Dipartimento di Matematica e Informatica, Universit` a di Palermo Via Archirafi 34, 90123, Palermo, Italy e-mail: [email protected], [email protected], [email protected]
ABSTRACT
In this paper we prove that if A is any algebra with involution ∗ satisfying a non-trivial polynomial identity, then its sequence of ∗-codimensions is eventually non-decreasing. Furthermore, by making use of the ∗-exponent we reconstruct the only two ∗-algebras, up to T ∗ -equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.
1. Introduction Let A be an algebra with involution ∗ over a field F of characteristic zero. One associates to A, in a natural way, a numerical sequence c∗n (A), n = 1, 2, . . ., called the sequence of ∗-codimensions of A which is the main tool for the quantitative investigation of the polynomial identities of the algebra A. Recall that c∗n (A), n = 1, 2, . . . , is the dimension of the space of multilinear ∗-polynomials in n fixed variables in the corresponding relatively free algebra with involution of countable rank. Such a sequence has been extensively studied (see [8, 15, 16, 17, 18, 19]) but it turns out that it can be explicitly computed only in very ∗ Partially supported by GNSAGA of INdAM.
Received September 5, 2018
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A. GIAMBRUNO AND D. LA MATTINA
Isr. J. Math.
few cases. In case A is a PI-algebra, i.e., it satisfies a non-trivial polynomial identity, it was proved in [9] that, as in the ordinary case, c∗n (A), n = 1, 2, . . ., is exponentially bounded. For this reason the interest focused on the computation of such asymptotics since they represent an invariant of the T∗ -ideal of the ∗-identities satisfied by A. Recently in [1] the authors characterized the varieties of PI-algebras with involution by proving that any such variety is generated by the Grassmann envelope of a finite-dimensional superalgebra with superinvolution. The major application of this result was obtained in [8] where it was shown that the exponent (exp∗ (A)) of a PI-algebra with involution exists and is an integer. More precisely, for general PI-algebras, it was proved that there exist constants C1 > 0, C2 , t, s such that (1)
C1 nt exp∗ (A)n ≤ c∗n (A) ≤ C2 ns exp∗ (A)n
for all n ≥ 1. The next step is to ask if the polynomial factor in (1) is uniquely determined, i.e., t = s, giving in this way a second invariant of a T∗ -ideal, after the ∗-exponent. Recently in [6] the authors gave a positive answer to this question for the class of ∗-fundamental algebras. More precisely, they proved the following: let A = A¯ + J be a ∗-fundamental algebra over an algebraically closed field where A¯ is a ∗-semisimple subalgebra and J is the Jacobson radical of A. Then c∗n (A) 1 ¯ − − r) + s, lim logn = − (dim(A) n→∞ exp∗ (A)n 2 ¯ − is the Lie algebra of skew elements of A¯ and r where J s = 0, J
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