Regev’s Conjecture and Codimensions of P.I. Algebras

Abstract

We study the periodicity of the proper cocharacters and show that Regev's conjecture holds in unitary algebras of P.I. exponent 2. Also we discuss the asymptotic behaviour of the codimensions and cocharacters of Clifford algebras and deal with other important examples of P.I. algebras.Keywords Polynomial identities · Codimensions · Cocharacters · Regev's conjecture · P.I. algebras · P.I. exponent · Commutator · Proper polynomial · Associative algebras · Clifford algebras · Grassmann envelope Mathematics Subject Classification (2000) Primary 16R10 · Secondary 05C90 · 15A66 · 16R40 · 16W55 · 20C30In the 80's conjectures about the asymptotic behaviour of the codimensions of associative algebras were made by S.A. Amitsur and A. Regev.Let A be an associative algebra and c n (A) be its codimension sequence.Amitsur's conjecture was proved in 1999 by M.V. Zaicev and A. Giambruno [1, Chap. 6, Theorem 5.2, p. 143]. Regev's conjecture was proved by V.S. Drensky for algebras of polynomial growth [2], by A. Regev for finite dimensional semisimple algebras [3], by M.V. Zaicev and A. Giambruno for algebras of block-triangular matrices [4]. Regev's conjecture in its generality still remains an open problem. 34 A.S. GordienkoV.S. Drensky has shown [2] that the multiplicity sequence of the irreducible cocharacters with fixed lower rows of every polynomial growth algebra is periodic. In Sect. 1 we show that this sequence is, from some place, a constant. Therefore, we obtain a new proof of Regev's conjecture for algebras of P.I. exponent 1. Furthermore, we study the periodicity of the proper cocharacters and show that Regev's conjecture holds in unitary algebras of P.I. exponent 2.Despite the intense activity in this field, only few examples are known in which the codimensions, cocharacters, and bases for the polynomial identities are precisely computed. The cocharacters of the algebra of 2 × 2 matrices were found by V.S. Drensky and E. Formanek [5,6]. The codimensions of this algebra were precisely computed by C. Procesi [7]. The codimensions and cocharacters of the Grassmann algebra were calculated by D. Krakowski and A. Regev [8]. The basis for the polynomial identities of the algebras UT n (F ) of upper triangular matrices was found by Yu.N. Malcev [9]. In Sect. 2 we compute the codimension, cocharacter, colength sequences, and a basis for the polynomial identities of the algebra A 1 = x a c 0 y b 0 0 x over a field of characteristic 0. In Sect. 3 we discuss the asymptotic behaviour of the codimensions and cocharacters of Clifford algebras. The multilinear identities of minimal degree for Clifford algebras of rank 1 are found. In Sect. 4 we present the Grassmann envelope of a finite dimensional superalgebra with the T -ideal of identities generated by [x 1 , x 2 , x 3 , x 4 ]. It provides us a version of its codimensions calculation.Here we give an improved account of the results obtained in [10][11][12][13].

Graded algebras with polynomial growth of their codimensions

“…We follow the scheme of proof of Gordienko, see [18,19] where the same fact was proved for the ordinary cocharacter instead of the G-graded one. We keep the notation introduced above.…”

“…Later on we prove that in fact the non-zero multiplicities may occur only for multipartitions having bounded quantity of boxes outside the first row of the first partition. In the sequel we transfer a theorem of Gordienko, see [18,19] to the graded case. Namely we prove that these multiplicities are eventually constants depending only on the shape of the multipartition below the first row.…”

“…Namely we prove that these multiplicities are eventually constants depending only on the shape of the multipartition below the first row. To this end we employ some ideas developed in [18,19] adapted to the graded case.…”