Class groups of dihedral extensions

Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p‐ranks of the class groups Cl(K) and Cl(k). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

Central extensions of current groups

 In this paper we study central extensions of the identity component G of the Lie group C ^∞ (M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω¹(M,Y)/dC ^∞ (M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊¹. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ^∞ (M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G. Received: 11 April 2002 / Revised version: 28 August 2002 / Published online: 28 March 2003     

Unitary groups and ramified extensions

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.     

Central extensions of groups of sections

If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra \mathfrakk{\mathfrak{k}} of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.     

Commutative ideal extensions of Abelian groups

No Abstract. May 26, 1999     

On noncommutative serving extensions of groups

The paper studies serving extensions of noncommutative groups in the sense of the definition transferred naturally from the theory of Abelian groups. The basic results are those continuing the Los Theorem on the de composability of Abelian serving extensions of algebraic compact groups in the category of all groups.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 283–291, March, 1972.The author wishes to take this opportunity to thank her scientific adviser Prof. L. Ya. Kulikov for his attention to this work.     

Central extensions of gauge groups revisited

We present an explicit construction for the central extension of the group Map(X, G) where X is a compact manifold and G is a Lie group. If X is a complex curve we obtain a simple construction of the extension by the Picard variety Pic(X). The construction is easily adapted to the extension of Aut(E), the gauge group of automorphisms of a nontrivial vector bundle E.