by Launois, S.; Lenagan, T. H.

Let $\mathbb{K}$ be a field and q be a nonzero element of $\mathbb{K}$ that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ of quantum matrices. Next, we describe all height one primes of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ whenever ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ has only finitely many height one primes. Finally, we compute the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$. Then, by using the preferred basis of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ and PBW bases, we prove that the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ is isomorphic to the torus ${\left( {\mathbb{K}*} \right)}^{{m + n – 1}} $ when m ≠ n and (m,n) ≠ (1, 3),(3, 1).

DOI: 10.1007/s10468-007-9059-0
Online Date: 4/28/2007
Print publication date: 8/1/2007
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by Liu, Guohua; Zhu, Shenglin

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra $(H,R\,)$ with $R^{\,21}R\in C(H\otimes H\,)$ (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field $k$. We denote by $B$ the vector space generated by the left tensorand of $R^{\,21}R$. Then $B$ is a sub-Hopf algebra of $H$. We proved that when $\dim B$ is odd, $H$ has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when $\dim B$ is even, $H$ is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.

DOI: 10.1007/s10468-006-9021-6
Online Date: 4/27/2007
Print publication date: 12/1/2007
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by Lewis, Mark L.; White, Donald L.

Recall that a finite group G satisfies the one-prime hypothesis if the greatest common divisor for any pair of distinct degrees in cd(G) is either 1 or a prime. In this paper, we classify the nonsolvable groups that satisfy the one-prime hypothesis. As a consequence of our classification, we show that if G is a nonsolvable group satisfying the one-prime hypothesis, then |cd(G)| ≤ 8, and hence, if G is any group satisfying the one-prime hypothesis, then |cd(G)| ≤ 9.

DOI: 10.1007/s10468-007-9057-2
Online Date: 4/17/2007
Print publication date: 8/1/2007
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by Petit, Toukaiddine; Oystaeyen, Freddy

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over $\mathbb{K}$ a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle $\omega\in\mathrm{Z}_{gr}^2(L,\mathbb{K})$, such that A is isomorphic to U

_ω
(L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L.

DOI: 10.1007/s10468-007-9048-3
Online Date: 4/14/2007
Print publication date: 8/1/2007
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by Gómez-Torrecillas, J.; Vercruysse, J.

We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory.

DOI: 10.1007/s10468-007-9050-9
Online Date: 4/14/2007
Print publication date: 6/1/2007
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