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Repository of the University of Hasselt containing publications in the fields of statistics, computer science, information strategies and material from the Institute for behavioural sciences.
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1.
Group-Cograded Multiplier Hopf (*-)Algebras - Abd El-Hafez, A.T.; DELVAUX, Lydia; Van Daele, A.
Let G be a group and assume that (Ap)p2G is a family of algebras with identity. We have
a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p; q 2 G, there is given a unital
homomorphism ¢p;q : Apq ! Ap Aq satisfying certain properties.
Consider now the direct sum A of these algebras. It is an algebra, without identity, except
when G is a finite group, but the product is non-degenerate. The maps ¢p;q can be used
to define a coproduct ¢ on A and the conditions imposed on these maps give that (A; ¢)
is a multiplier Hopf algebra....
2.
Deformation theory of abelian categories - Lowen, Wendy; VAN DEN BERGH, Michel
In this paper we develop the basic infinitesimal deformation theory
of abelian categories. This theory yields a natural generalization of the wellknown
deformation theory of algebras developed by Gerstenhaber. As part of
our deformation theory we define a notion of flatness for abelian categories.
We show that various basic properties are preserved under flat deformations
and we construct several equivalences between deformation problems.
3.
The Gelfand-Kirillov conjecture for semi-direct products of Lie algebras - OOMS, Alfons
Let g be an n-dimensional Lie algebra over a field k of characteristic zero and let W be a g-module of dimension at least n. Sufficient conditions are given in order for the semi-direct product g + W to satisfy the Gelfand-Kirillov conjecture. This implies that this conjecture holds for an important class of Frobenius Lie algebras. Special attention is devoted to the case where g = sl(2,k).
4.
Double Poisson algebras - VAN DEN BERGH, Michel
In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and Ginzburg. Our (quasi-) Poisson brackets induce classical ( quasi-) Poisson brackets on representation spaces. As an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure.
5.
Yetter-Drinfel'd modules for group-cograded multiplier Hopf algebras - DELVAUX, Lydia
We give a representation-theoretic and a categorical interpretation of the Drinfel'd double into the framework of group-cograded multiplier Hopf algebras. The Drinfel'd double as constructed by Zunino for a finite-type Hopf group-coalgebra is an example of this construction in the sense that the components of the group-cograded multiplier Hopf algebras are unital and finite-dimensional algebras and the admissible action is related with the adjoint action of the group on itself.
6.
Computing invariants and semi-invariants by means of Frobenius Lie algebras - Ooms, Alfons
Let U(g) be the enveloping algebra of a finite-dimensional Lie algebra g over a field k of characteristic zero, Z(U(g)) its center and Sz(U(g)) its semi-center. A sufficient condition is given in order for Sz(U(g)) to be a polynomial algebra over k. Surprisingly, this condition holds for many Lie algebras, especially among those for which the radical is nilpotent, in which case Sz(U(g)) = Z(U(g)). In particular, it allows the explicit description of Z(U(g)) for more than half of all complex, indecomposable nilpotent Lie algebras of dimension at most 7. (C) 2008 Elsevier Inc. All rights reserved.
7.
Noncommutative resolutions and rational singularities - Stafford, J. T.; Van den Bergh, M.
Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.
8.
The Kontsevich Weight of a Wheel with Spokes Pointing Outward - VAN DEN BERGH, Michel
This is a companion note to "Hochschild cohomology and Atiyah classes" by Damien Calaque and the author. Using elementary methods we compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernoulli numbers. The same result had been obtained earlier by Torossian (unpublished) and also recently by Thomas Willwacher using more advanced methods.
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