01-9   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Manfred Schocker:

Lie Idempotent Algebras

Besides the inner product inherited from each symmetric group algebra $KS_n$, on the direct sum $KS=\bigoplus_n KS_n$, 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 $n$-th 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 $k\le n$, 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 $c_n:\L_n(\A)\lra \Cl_K(S_n)$. Additional results are obtained concerning a bialgebra structure of $\L(\A)$. In the particular case of $\A=\bigoplus_n\D_n$, this amounts to a new approach to several well-known results on $\D_n$.

Bibliographical note: Adv. Math. 175, No.2, 243-270 (2003)


Mail an Jens Burmeister
[Thu Feb 19 18:56:35 2009]
Impressum