In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: D 2 , the algebra of 2 × 2 diagonal matrices and C 2 , the algebra of 2 × 2 matrices generated by e 11 + e 22 and e 12. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.
Let A be an associative algebra with superinvolution ∗ over a field of characteristic zero and let c_n∗(A) be its sequence of corresponding ∗-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ∗-codimensions between polynomial and exponential is allowed
Let A be a superalgebra with graded involution or superinvolution * and let c * n (A), n = 1, 2, . . . , be its sequence of * -codimensions. In case A is finite dimensional, in [6,15] it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and a 4-dimensional subalgebra of the 4×4 upper-triangular matrices with suitable graded involutions or superinvolutions.In this paper we study the general case of * -superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of * -superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in [18].
Let A be an associative algebra over a field F of characteristic zero endowed with a graded involution or a superinvolution ∗ and let c^∗_n(A) be its sequence of ∗-codimensions. In [4,12]\ud
it was proved that if A is finite dimensional such sequence is polynomially bounded if and only if A generates a variety not containing a finite number of ∗-algebras: the group\ud
algebra of Z_2 and a 4-dimensional subalgebra of the 4 × 4 upper triangular matrices with suitable graded involutions or superinvolutions.\ud
In this paper we focus our attention on such algebras since they are the only finite dimensional ∗-algebras, up to T^∗_2 -equivalence, generating varieties of almost polynomial\ud
growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. We classify the subvarieties of such varieties by giving a complete list of\ud
generating finite dimensional ∗-algebras. Along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any\ud
given growth. Finally we describe the ∗-algebras whose ∗-codimensions are bounded by a linear function
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