Unitary superalgebras with graded involution or superinvolution of polynomial growth
In this manuscript, the authors investigate associative unitary superalgebras with graded involution or superinvolution having a polynomial growth of the codimension sequence. Let \(F\) be a field of characteristic zero and \(A\) an associative algebra over \(F\). Recall that \(A\) is a superalgebra (or a \(\mathbb{Z}_2\)-graded algebra) if it has a decomposition \(A = A_0\oplus A_1\), where \(A_0\) and \(A_1\) are subspaces of \(A\), such that \(A_0A_0 + A_1A_1 \subseteq A_0\) and \(A_0A_1 + A_1A_0 \subseteq A_1\). An involution \(\ast\) defined on \(A\) is said to be a graded involution if \((A_0)^\ast = A_0\) and \((A_1)^\ast = A_1\). On the other hand, a super involution on \(A\) is a graded linear map \(\ast : A\to A\) such that \((a^\ast)^\ast=a\) for all \(a\in A\) and \((ab)^\ast = (-1)^{|a||b|}b^\ast a^\ast\), for any homogeneous elements \(a,b\in A\), where \(|c|\) denotes the homogeneous degree of \(c\in A_0 \cup A_1\). Let \(B(Y_0)\) be the subalgebra of \(F\) generated by \(Y_1 \cup Z_0 \cup Z_1\) and by all commutators in the variables from \(X = Y_0\cup Y_1\cup Z_0\cup Z_1\). The elements of \(B(Y_0)\) are called proper \(\ast\)-polynomials. The authors discuss unitary \(\ast\)-algebras having polynomial growth of the \(\ast\)-codimensions. In addition, they prove that if \(A\) is a unitary \(\ast\)-algebra with polynomial growth of the codimensions, then \(c^\ast _n(A)\) is a polynomial with rational coefficients and the leading term \(q\) satisfies a certain inequality. Moreover, they construct algebras with graded involution and with superinvolution realizing the smallest and the largest value of \(q\).
scientific article; zbMATH DE number 7357199