Exponents of identities of group rings. (Q650474)

Let \(F\) be a field of characteristic zero, \(G\) a group and \(FG\) the group algebra of \(G\) over \(F\). In the paper quantitative characteristics of the identities of group algebras of finite groups are considered. The authors obtain relations connecting the PI-exponents of identities with involution and Lie identities of a group algebra and its Lie subalgebra of skew-symmetric elements. Graded algebras and graded identities are also considered. The main results of the paper are the following: Theorem 1. Let \(F\) be a field of characteristic zero, let \(G\) be a finite group, and let \(k\) be the maximal dimension of irreducible representations of \(G\) over the algebraic closure of the field \(F\). Assume also that \(k\geq 3\) and \(R=FG\) is the group algebra of \(G\) over \(F\) with some involution *. If \(R^{(-)}\) is the Lie subalgebra of the skew-symmetric elements in \(R\), then \[ \exp^L(R)\leq 2{k+1\over k}\exp^L(R^{(-)})\qquad\text{for }k\neq 4 \] \[ \exp^L(R)\leq 4{k+1\over k}\exp^L(R^{(-)})\qquad\text{for }k=4. \] Theorem 2. Let \(F\) be a field of characteristic zero, let \(G\) be a finite group, and let \(k\) be the maximal dimension of irreducible representations of \(G\) over the algebraic closure of the field \(F\). Assume also that \(k\geq 3\) and \(R=FG\) is the group algebra of \(G\) over \(F\) with some involution *. If \(R^{(-)}\) is the Lie subalgebra of the skew-symmetric elements in \(R\), then \[ \exp^*(R)\leq 4{k\over k-1}\exp^L(R^{(-)})\qquad\text{for }k\neq 4 \] \[ \exp^*(R)\leq 8{k\over k-1}\exp^L(R^{(-)})\qquad\text{for }k=4. \] Theorem 3. Let \(F\) be a field of characteristic zero, and let \(R=FS_n\), where \(n\geq 4\) and \(n\neq 6\) If \(^*\colon R\to R\) is the involution induced from the group \(S_n\) and \(R^{(-)}\) is the Lie subalgebra of skew-symmetric elements in \(R\), then \[ \exp^L(R)=2{k+1\over k}\exp^L(R^{(-)}),\qquad\exp^*(R)=2{k\over k-1}\exp^L(R^{(-)}), \] where \(k\) stands for the maximal dimension of the irreducible representations of \(S_n\).