Lie idempotent algebras

Abstract

Besides the inner product inherited from each symmetric group algebra KS n , on the direct sum KS=⊕ n KS n , a second (convolution) product arises from the Hopf algebra structure on KS. For any subalgebra A of KS for both the inner and the convolution product, it is shown that the Lie idempotent algebra L( A) of A generated by all Lie idempotents in A with respect to the convolution product is also closed under inner products. Its nth homogeneous component L n( A) is thus a subalgebra of KS n , for all n. Provided that A contains a Lie idempotent in KS k , for all k⩽ n, Solomon's descent algebra D n is a subalgebra of L n( A) . In this case, a number of results on the structure of L n( A) is derived including a description of its Jacobson radical, its principal indecomposables and its Cartan invariants. Furthermore, Solomon's epimorphism from D n onto the ring of class functions Cl K ( S n ) of S n extends to an epimorphism of algebras c n : L n( A)→ Cl K(S n) . Additional results are obtained concerning a bialgebra structure of L( A) . In the particular case of A=⊕ n D n , this amounts to a new approach to several well-known results on D n .