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On the classification of the real flexible division algebras

Erik Darpö (2006)

Colloquium Mathematicae

We investigate the class of finite-dimensional real flexible division algebras. We classify the commutative division algebras, completing an approach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of quadratic division algebras, and classify the generalised pseudo-octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexible division algebras to the normal...

On the cohomology of vector fields on parallelizable manifolds

Yuly Billig, Karl-Hermann Neeb (2008)

Annales de l’institut Fourier

In the present paper we determine for each parallelizable smooth compact manifold M the second cohomology spaces of the Lie algebra 𝒱 M of smooth vector fields on M with values in the module Ω ¯ M p = Ω M p / d Ω M p - 1 . The case of p = 1 is of particular interest since the gauge algebra of functions on M with values in a finite-dimensional simple Lie algebra has the universal central extension with center Ω ¯ M 1 , generalizing affine Kac-Moody algebras. The second cohomology H 2 ( 𝒱 M , Ω ¯ M 1 ) classifies twists of the semidirect product of 𝒱 M with the...

On the combinatorics of Kac's asymmetry function

R. M. Green (2010)

Commentationes Mathematicae Universitatis Carolinae

We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is...

On the composition structure of the twisted Verma modules for 𝔰𝔩 ( 3 , ℂ )

Libor Křižka, Petr Somberg (2015)

Archivum Mathematicum

We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra 𝔰𝔩 ( 3 , ℂ ) , including the explicit structure of singular vectors for both 𝔰𝔩 ( 3 , ℂ ) and one of its Lie subalgebras 𝔰𝔩 ( 2 , ℂ ) , and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as D -modules on the Schubert cells in the full flag manifold for SL ( 3 , ℂ ) .

On the CR-structure of certain linear group orbits in infinite dimensions

Wilhelm Kaup (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits  M explicitly and show as main result that every continuous CR-function on  M has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

On the definition of the dual Lie coalgebra of a Lie algebra.

Bertin Diarra (1995)

Publicacions Matemàtiques

Let L be a Lie algebra over a field K. The dual Lie coalgebra Lº of L has been defined by W. Michaelis to be the sum of all good subspaces V of the dual space L* of L: V is good if tm(V) ⊂ V ⊗ V, where m is the multiplication of L. We show that Lº = tm-1(L* ⊗ L*) as in the associative case.

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