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