Superalgebras with graded involution: classifying minimal varieties of quadratic growth
All algebras considered in the paper are over a field of characteristic 0. In the study of the identities satisfied by a given algebra \(A\) a prominent role plays its codimension sequence \(c_n(A)\). Let \(F\langle X\rangle\) be the free associative algebra and denote by \(P_n\) the vector space of all multilinear polynomials of degree \(n\) in the variables \(x_1\), \dots, \(x_n\) in it. Clearly \(\dim P_n=n!\). The symmetric group \(S_n\) acts on the left on \(P_n\) by permuting the variables, thus endowing it with the structure of an \(S_n\)-module. This module is clearly isomorphic to the regular \(S_n\)-module \(FS_n\). If \(I\) is the T-ideal of the algebra \(A\) then \(P_n\cap I\) is a submodule, and one employs the representation theory of the symmetric group in the study of PI algebras. In characteristic 0 the T-ideal \(I\) is generated by all of its multilinear elements, namely by the intersections \(I\cap P_n\). It should be noted though that these intersections grow very fast, and according to a well-known theorem of \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)], \(I\cap P_n\) tends to become almost the whole of \(P_n\) for large \(n\). That is why one studies the quotient \(P_n/(I\cap P_n\). Its dimension \(c_n(I)\) is the \(n\)-th codimension of \(I\), and of any algebra \(A\) that has \(I\) as its T-ideal. According to Regev's theorem \(c_n(I)\le (d-1)^{2n}\) for every \(n\) where \(d\) is the degree of a nontrivial identity in \(I\). \par One defines in a similar way the codimension sequence of an algebra with involution, and for a superalgebra (that is a 2-graded algebra) with an involution, and with a graded involution. Of particular interest are the varieties of algebras, superalgebras, and so on, where the codimension sequence grows like a polynomial function. A variety \(V\) is said to be of growth \(n^k\) if its codimensions grow like the polynomial function \(n^k\). It is a minimal variety of polynomial growth \(n^k\) whenever it is of growth \(n^k\) and every proper subvariety is of growth \(n^t\) where \(t<k\). \par The main result of the paper under review is the classification of the superalgebras with a graded involution that generate minimal varieties of quadratic growth. They give a list of 36 superalgebras with graded involutions that generate all minimal varieties of such algebras of quadratic growth of their codimensions.
scientific article; zbMATH DE number 7337672
scientific article