by Bonnafé, C.; Pfeiffer, G.

We study different problems related to the Solomon’s descent algebra Σ(W) of a finite Coxeter group (W,S): positive elements, morphisms between descent algebras, Loewy length… One of the main result is that, if W is irreducible and if the longest element is central, then the Loewy length of Σ(W) is equal to $\displaystyle{\left\lceil\frac{|S|}{2}\right\rceil}$.

DOI: 10.1007/s10468-008-9090-9
Online Date: 6/11/2008
Print publication date: 12/1/2008
View article on SpringerLink

by Weyer, Geert Van de

We give a description of the bimodule of double derivations of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials induce double Poisson brackets on S. In case S = ℂ^⊕n
, a criterion for any degree two element to give a double Poisson bracket is deduced. For S = ℂ^⊕n
and S′ = ℂ^⊕m
the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss

_n
(S * T) of the free product S * T is given.

DOI: 10.1007/s10468-008-9088-3
Online Date: 6/6/2008
Print publication date: 10/1/2008
View article on SpringerLink

by Jedwab, Andrea; Montgomery, Susan

We prove that all irreducible representations of the bismash product $H_n = k^{C_n } \# kS_{n – 1} $ have Frobenius–Schur indicator +1, where k is an algebraically closed field of characteristic 0. If n = p, a prime, we find all indicators for $J_n = k^{S_{n – 1} } \# k^{C_n } $. We also study more general bismash products.

DOI: 10.1007/s10468-008-9099-0
Online Date: 6/4/2008
Print publication date: 2/1/2009
View article on SpringerLink

by Coelho, Flávio U.; Tosar, Cecilia

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X
^*, we consider the set $J_{X^*}=\{i \in \mathbb{Z}\, |\, H^i(X^*)\neq 0\}$ and we define the application $l(X^*)=\text{max}J_{X^*}-\text{min}J_{X^*}+1$. We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.

DOI: 10.1007/s10468-008-9105-6
Online Date: 6/3/2008
Print publication date: 2/1/2009
View article on SpringerLink

by Kennedy, Christopher; Winter, David J.

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras.

DOI: 10.1007/s10468-008-9101-x
Online Date: 6/3/2008
Print publication date: 2/1/2009
View article on SpringerLink

by Lin, Weiqiang; Tan, Shaobin

In this paper, we deal with the classification of the irreducible -graded and
²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a -graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight -graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the -graded modules to construct a class of
²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the
²-graded L-modules. We first prove that a
²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial
²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible -graded modules and the irreducible
²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial.

DOI: 10.1007/s10468-008-9107-4
Online Date: 6/3/2008
Print publication date: 12/1/2008
View article on SpringerLink