50 research outputs found
The centroid of extended affine and root graded Lie algebras
We develop general results on centroids of Lie algebras and apply them to
determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody
Lie algebras, and Lie algebras graded by finite root systems.Comment: 35 page
Universal central extensions of direct limits of Lie superalgebras
We show that the universal central extension of a direct limit of perfect Lie
superalgebras L_i is (isomorphic to) the direct limit of the universal central
extensions of L_i. As an application we describe the universal central
extensions of some infinite rank Lie superalgebras
A survey of equivariant map algebras with open problems
This paper presents an overview of the current state of knowledge in the
field of equivariant map algebras and discusses some open problems in this
area.Comment: 18 pages. v2: Minor correction
\'Etale Descent of Derivations
We study \'etale descent of derivations of algebras with values in a module.
The algebras under consideration are twisted forms of algebras over rings, and
apply to all classes of algebras, notably associative and Lie algebras, such as
the multiloop algebras that appear in the construction of extended affine Lie
algebras.Comment: 15 pages, to appear in Transformation Group
On conjugacy of Cartan subalgebras in non-fgc Lie tori
We establish the conjugacy of Cartan subalgebras for generic Lie tori "of
type A". This is the only conjugacy problem of Lie tori related to Extended
Affine Lie Algebras that remained open.Comment: 28 pages, to be published in Transformation Group
Irreducible finite-dimensional representations of equivariant map algebras
Suppose a finite group acts on a scheme X and a finite-dimensional Lie
algebra g. The corresponding equivariant map algebra is the Lie algebra M of
equivariant regular maps from X to g. We classify the irreducible
finite-dimensional representations of these algebras. In particular, we show
that all such representations are tensor products of evaluation representations
and one-dimensional representations, and we establish conditions ensuring that
they are all evaluation representations. For example, this is always the case
if M is perfect.
Our results can be applied to multiloop algebras, current algebras, the
Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the
known classifications of irreducible finite-dimensional representations of
these algebras. Moreover, we obtain previously unknown classifications of
irreducible finite-dimensional representations of other types of equivariant
map algebras, such as the generalized Onsager algebra.Comment: 25 pages; v2: results generalized to schemes and arbitrary
finite-dimensional g; v3: change of notation, minor typos corrected, some
explanations added; v4: minor typos corrected and references update