by Vonessen, Nikolaus

Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts “algebraically,” i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.

DOI: 10.1007/s10468-007-9052-7
Online Date: 5/24/2007
Print publication date: 10/1/2007
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by Delvaux, L.

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct.

DOI: 10.1007/s10468-007-9053-6
Online Date: 5/23/2007
Print publication date: 12/1/2007
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by Kim, Jeong-Ah; Shin, Dong-Uy

In this paper, we give a new realization of the crystal basis B(∞) using modified Nakajima monomials for the quantum finite algebras. Moreover, as an application, we obtain the image of the Kashiwara embedding Ψ

_ι
from this realization of B(∞).

DOI: 10.1007/s10468-007-9056-3
Online Date: 5/15/2007
Print publication date: 3/1/2008
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by Han, Gang; Sun, Bin-Yong

Let $\mathfrak{g}_{0}$ be a real semisimple Lie algebra. Let $\mathfrak{g}_{0} = \mathfrak{k}_{0} \oplus \mathfrak{p}_{0}$ be the corresponding Cartan decomposition and $\mathfrak{h}_{0} = \mathfrak{t}_{0} \oplus \mathfrak{a}_{0}$ be a maximally compact Cartan subalgebra of $\mathfrak{g}_{0}$. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}$ be the corresponding complexifications. The set $\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}$ consists of all the linear forms on $\mathfrak{t}$ which are the restriction to $\mathfrak{t}$ of the roots in the root system $\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}$ of $\mathfrak{g}$ with respect to $\mathfrak{h}$. The main result of the paper is to prove that $\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}$ is also a (maybe non-reduced) root system and its Weyl group can be identified with a subgroup of the Weyl group of $\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}$. Let $Spin\,\nu :\mathfrak{k} \to End\,S$ be the composition of the isotropy representation $\nu :\mathfrak{k} \to \mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)}$ with the spin representation $Spin:\mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)} \to End\,S$. Finally as an application, we give a nice description of the $\mathfrak{k}$-module structure on S in terms of the restricted root system $\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}$ and its Weyl group.

DOI: 10.1007/s10468-007-9061-6
Online Date: 5/15/2007
Print publication date: 10/1/2007
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by Daele, Alfons Van; Wang, Shuanhong

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism.

DOI: 10.1007/s10468-007-9045-6
Online Date: 5/12/2007
Print publication date: 10/1/2007
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by Dancer, K. A.; Gould, M. D.; Links, J.

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U

_q
[osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U

_q
[osp(m|n)] module. The case W = V is studied as an example.

DOI: 10.1007/s10468-007-9049-2
Online Date: 5/12/2007
Print publication date: 12/1/2007
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by Gräter, Joachim

Let R be a commutative integral domain with field of fractions F and let Q be a finite-dimensional central simple F-algebra. If R is a Prüfer domain then it is still unknown whether or not R can be extended to a Prüfer order in Q in the sense of Alajbegović and Dubrovin (J. Algebra, 135: 165–176, 1990). In this paper we investigate a more general class of rings which we call rings of Prüfer type and we will prove an extension theorem for these rings. Under special assumptions this result also leads to an extension theorem for certain Prüfer domains.

DOI: 10.1007/s10468-007-9046-5
Online Date: 5/2/2007
Print publication date: 8/1/2007
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