Erhard Neher

The alternative torus and the structure of elliptic quasi-simple Lie algebras of type ₂

We present the complete classification of the tame irreducible elliptic quasi-simple Lie algebras of type A2, and in particular, specialize on the case where the coordinates are not associative. Here the coordinates are Cayley-Dickson algebras over Laurent polynomial rings in v > 3 variables, which we call alternative tori. In giving our classification we need to present much information on these alternative tori and the Lie algebras coordinatized by them.

Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids

Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R, the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R, and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R. This result is then used to classify Lie algebras graded by R: we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowys intersection matrix algebras arising from multiply-affinized Cartan matrices.

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 particu- lar, 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 classifica- tions of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.

An Introduction to Universal Central Extensions of Lie Superalgebras

We provide an introduction to the theory of universal central extensions of Lie superalgebras. In particular, we show that a Lie superalgebra has a universal central extension if and only if it is perfect. We also consider the question of lifting automorphisms and derivations to the universal central extension, and describe the universal central extension of a semidirect product.

Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey

This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.

Derivations and invariant forms of Jordan and alternative tori

Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types A 1 and A 2 . In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.

Tits-Kantor-Koecher Superalgebras of Jordan Superpairs Covered by Grids

Abstract In this paper we describe the Tits-Kantor-Koecher superalgebras associated to Jordan superpairs covered by grids,extending results from Neher (Neher,E. (1996). Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid. Amer. J. Math. 118; 439–491) to the supercase. These Lie superalgebras together with their central coverings are precisely the Lie superalgebras graded by a 3-graded root system.

Reflection systems and partial root systems

Abstract We develop a general theory of reflection systems and, more specifically, partial root systems which provide a unifying framework for finite root systems, Kac–Moody root systems, extended affine root systems and various generalizations thereof. Nilpotent and prenilpotent subsets are studied in this setting, based on commutator sets and the descending central series. We show that our notion of a prenilpotent pair coincides, for Kac–Moody root systems, with the one defined by Tits in terms of positive systems and the Weyl group.

EXTENSIONS AND BLOCK DECOMPOSITIONS FOR FINITE-DIMENSIONAL REPRESENTATIONS OF EQUIVARIANT MAP ALGEBRAS

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. The irreducible finite-dimensional representations of these algebras were classified in [NSS12], where it was shown that they are all tensor products of evaluation representations and one-dimensional representations.In the current paper, we describe the extensions between irreducible finite-dimensional representations of an equivariant map algebra in the case that X is an affine scheme of finite type and g is reductive. This allows us to also describe explicitly the blocks of the category of finite-dimensional representations in terms of spectral characters, whose definition we extend to this general setting. Applying our results to the case of generalized current algebras (the case where the group acting is trivial), we recover known results but with very different proofs. For (twisted) loop algebras, we recover known results on block decompositions (again with very different proofs) and new explicit formulas for extensions. Finally, specializing our results to the case of (twisted) multiloop algebras and generalized Onsager algebras yields previously unknown results on both extensions and block decompositions.

Isoparametric triple systems of FKM-type II

This paper continues the investigation of isoparametric triple systems of FKM-type started in [9]. We classify all such triple systems which are congruent to an isoparametric triple system of algebra type. We also consider the question to what extent the Clifford sphere of an FKM-triple is determined by the triple product. As an application the automorphism group of an FKM-triple is described.