On the Growth of Identities
by
Alexander Guterman
The Moscow State University

Let R be an associative algebra with a polynomial identity (PI-algebra) over a field F of zero characteristic. Quantative description of its identities is investigated usually with the aid of codimension sequence c_n(R), see [1] for more detailes. By now, the list of algebras with known codimensions is very short. However, in all cases computed so far, the codimension sequence asymptotic behaves as c_n(R) ~ c n^g aⁿ,  n --> \infty; where 1 <= a in Z,  g in 1/2 Z, and c is a certain constant. It is still a conjecture that this holds for an arbitrary associative PI-algebra. The corresponding inverse problem is also open. Namely, it is interesting to obtain an algebra R with a given growth of identities. The available technique for constructing such algebras was to multiply T-ideals of matrix, Grassmann, commutative, and nilpotent algebras. Here we propose completly another type of constructions. In order to obtain these constructions we characterize ideals of identities for various classes of algebras. The quantative description of identities for such algebras is also given. For the more detailes see [2, 3].

References

[1] A. Regev, Existence of identities in A\otimesB, Israel J. Math. 11 (1972), 131-152.

[2] A.E. Guterman, Identities of near-triangular matrices, Matem. Sbornik 192(6) 2001, 3-15.

[3] A.Regev, A.Guterman, On the Growth of Identities, Algebra. Berlin, New York: Walter de Gruyter, 2000, 319-331

Date received: February 8, 2002