Minimal varieties of PI-superalgebras with graded involution
Let \(A\) be an associative algebra over a field \(F\) of characteristic 0, it is well known that the polynomial identities of \(A\) are consequences of the multilinear ones. Hence in order to study the identities satisfied by \(A\) it suffices to consider only the multilinear identities of \(A\). Denote by \(P_n\) the vector space of the multilinear polynomials in the first \(n\) free variables in the free associative algebra, and let \(I=Id(A)\) be the ideal of identities of \(A\). Thus one might want to study the intersections \(P_n\cap I\) for every \(n\). It is clear that \(P_n\) is a left module over the symmetric group \(S_n\) which acts by permuting the variables. Since \(I\) is invariant under permutations of the variables it follows that \(P_n\cap I\) is a submodule of \(P_n\). Therefore one may apply the well developed theory of the representations of \(S_n\) to the study of polynomial identities in algebras. In fact this approach is not the best one, as it has a significant drawback. Let us denote by \(P_n(A)=P_n/(P_n/\cap I)\) and by \(c_n(A)=\dim P_n(A)\) the \(n\)th codimension of \(A\). \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)] proved that if \(A\) satisfies an identity of degree \(d\) then \(c_n(A)\le (d-1)^{2n}\) for every \(n\). This implies that for \(n\) large enough, the intersection \(P_n\cap I\) becomes very large. Hence studying the quotient module \(P_n(A)\) is better from computational point of view. The codimension sequence \(c_n(A)\), introduced by Regev in that paper, is one of the main numerical invariants of a T-ideal (that is an ideal of identities of an algebra). It is also one of the most studied ones. Let us mention that in very few occasions it is possible (and plausible) to compute it explicitly. Amitsur conjectured that the limit \(\lim_{n\to\infty} (c_n(A)^{1/n})\), called the PI exponent of \(A\), exists for every PI algebra \(A\), and is, moreover, a nonnegative integer. This conjecture was confirmed by \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 140, No. 2, 145--155 (1998; Zbl 0920.16012); Adv. Math. 142, No. 2, 221--243 (1999; Zbl 0920.16013)]. This motivated an extensive research towards the classification of the PI algebras according to the growth of their codimension sequences, and their PI exponent. A variety of algebras \(V\) is minimal of PI exponent \(d\) whenever it is of PI exponent \(d\) but every proper subvariety is of exponent \(<d\). The minimal varieties were classified by \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 174, No. 2, 310--323 (2003; Zbl 1035.16013)]. Similar notions can be defined for algebras with additional structure. These include group-graded algebras, algebras with involution, and so on. Of particular interest are the algebras graded by the group of order 2; these are often called 2-graded algebras, or superalgebras. The paper under review studies superalgebras equipped with a graded involution (that is an involution which respects the grading). These are called \(*\)-superalgebras. The main contributions of the present paper consist in the complete classification of the minimal varieties of \(*\)-superalgebras of finite basic rank. Recall here that these varieties must be generated by finitely generated \(*\)-superalgebras that satisfy ordinary identities (that is why \textsl{finite basic rank}). Let \((A_1,\ldots, A_m)\) be an \(m\)-tuple of simple \(*\)-superalgebras. The authors construct a suitable subalgebra of the upper block triangular matrix algebra, denoted by \(UT^*_2(A_1,\ldots, a_m)\). It is equipped with an appropriate involution \(*\), and with an appropriate 2-grading (which is elementary, that is the matrix units \(E_{ij}\) are all homogeneous). The involution preserves the grading. Every one of the \(A_i\) embeds into the latter subalgebra and the embedding respects the involution. The main theorem of the paper, Theorem 2.2, states that a variety of \(*\)-superalgebras of finite basic rank is minimal of PI exponent \(d\) if and only if it is generated by a \(*\)-superalgebra of the latter form satisfying \(\dim(A_1\oplus\cdots\oplus A_m)=d\). The paper is well and skillfully written, and provides a good account on the progress in the area. It provides as well insights on the main methods employed in the solution of the above important and difficult problem.
scientific article; zbMATH DE number 7341471
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