by Rump, Wolfgang

The method of differentiation for the category Λ-lat of lattices over an order Λ will be extended to integral almost Abelian categories A instead of Λ-lat. In particular, this yields a differentiation for finitely generated left modules over left Artinian rings.

DOI: 10.1023/B:ALGE.0000042182.98997.7c
Print publication date: 10/1/2004
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by Bichon, Julien

Let A be a compact quantum group, let n∈N
^* and let A
_aut(X

_n
) be the quantum permutation group on n letters. A free wreath product construction A*_w
A
_aut(X

_n
) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.

DOI: 10.1023/B:ALGE.0000042148.97035.ca
Print publication date: 10/1/2004
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by Adamović, Dražen

Let
$$\hat {\mathfrak{g}}$$
be the affine Lie algebra associated to the simple finite-dimensional Lie algebra
$${\mathfrak{g}}$$
. We consider the tensor product of the loop
$$\hat {\mathfrak{g}}$$
-module
$$\overline {V\left( \mu \right)} $$
associated to the irreducible finite-dimensional
$${\mathfrak{g}}$$
-module V(μ) and the irreducible highest weight
$$\hat {\mathfrak{g}}$$
-module L

_k,λ. Then L

_k,λ can be viewed as an irreducible module for the vertex operator algebra M

_k,0. Let A(L

_k,λ) be the corresponding
$$A\left( {M_{k,0} } \right)\left( { = U\left( {\mathfrak{g}} \right)} \right)$$
-bimodule. We prove that if the
$${U\left( {\mathfrak{g}} \right)}$$
-module
$$A\left( {L_{k,0} } \right) \otimes _{U\left( \mathfrak{g} \right)} V\left( \mu \right)$$
is zero, then the
$${\hat {\mathfrak{g}}}$$
-module
$$\left( {L_{k,0} } \right) \otimes _{U\left( {\mathfrak{g}} \right)} V\left( \mu \right)$$
is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.

DOI: 10.1023/B:ALGE.0000042147.02049.38
Print publication date: 10/1/2004
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by Enochs, Edgar; Estrada, S.; García-Rozas, J. R.; Oyonarte, L.

The study of flat covers and cotorsion envelopes has turned out to be very useful since their existence was proved in [3] for the category of R-modules. The problem is even more interesting in categories of sheaves on a topological space, because these categories do not have enough projectives. But the existence of flat covers and cotorsion envelopes allow us to form flat and cotorsion resolutions to compute cohomology.

DOI: 10.1023/B:ALGE.0000042145.72104.cc
Print publication date: 10/1/2004
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by Green, R. M.

We introduce cell modules for the tabular algebras defined in a previous work; these modules are analogous to the representations arising from left Kazhdan–Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type A equipped with its Kazhdan–Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.

DOI: 10.1023/B:ALGE.0000042144.20113.c0
Print publication date: 10/1/2004
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by Rosengren, Hjalmar

We define a natural concept of duality for the
$${\mathfrak{h}}$$
-Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a deformation both of the function algebra F(SL(2)) and of the enveloping algebra U(sl(2)). Matrix elements of the self-duality in the Peter–Weyl basis are 6j-symbols; this leads to a new algebraic interpretation of the hexagon identity or quantum dynamical Yang–Baxter equation for quantum and classical 6j-symbols.

DOI: 10.1023/B:ALGE.0000042118.49950.ca
Print publication date: 10/1/2004
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