0000000000851781
AUTHOR
Francesca Saviella Benanti
Capelli identities on algebras with involution or graded involution
We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution.
ON THE ASYMPTOTICS OF CAPELLI POLYNOMIALS
Abstract. We present old and new results about Capelli polynomials, Z2-graded Capelli polynomials and Capelli polynomials with involution and their asymptotics. Let Capm = Pσ2Sm (sgnσ)tσ(1)x1tσ(2) · · · tσ(m−1)xm−1tσ(m) be the m-th Capelli polynomial of rank m. In the ordinary case (see [33]) it was proved the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial Capk2+1 and the codimensions of the matrix algebra Mk(F ). In [9] this result was extended to superalgebras proving that the Z2-graded codimensions of the T2-ideal generated by the Z2-graded Capelli polynomials Cap0 M+1 and Cap1 L+1 for some fixed M, L, are asymptotically equal to the Z2-g…
Asymptotics for multiplicities in the cocharacters of some PI-algebras
We consider associative PI-algebras over a eld of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some signi cant classes of algebras. We also give a characterization of nitely generated algebras for which this behavior is linear or quadratic.
Defining relations of minimal degree of the trace algebra of 3 X 3 matrices
The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d 2. Minimal sets of generators of Cnd are known for n = 2 and n = 3 for any d as well as for n = 4 and n = 5 and d = 2. The defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Starting with the generating set of C3d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C3d is equal to 7 for any d 3. We have determined all relations of minimal degree. For d = 3 we have also found the defining relations of degree 8. The proofs are based on methods of representati…