A presentation for reduced extended affine weyl groups

In 1985 K. Saito [Sal] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5J, we gave a presentation called “a presentation by conjugation” for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will give a presentation wh.ich we call a “generalized present.ation by conjugation” for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to “a presentation by conjugation”. Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have “a presentation by conjugation”. In [ST] another presentation for EAWGs of nullity 2 is given.     

The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.     

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

The degeneracy of extended affine Lie algebras

We construct degenerate extended affine Lie algebras from a given nondegenerate extended affine Lie algebra and show that all degenerate extended affine Lie algebras are obtained in this way. Received: 21 January 1997     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure has the residue of each point a dual affine plane, then is either an extended dual affine plane or a special Laguerre plane. This research was partially supported by NSF Grant MCS-8102361.     

Weyl modules for the twisted loop algebras

The notion of a Weyl module, previously defined for the untwisted affine algebras, is extended here to the twisted affine algebras. We describe an identification of the Weyl modules for the twisted affine algebras with suitably chosen Weyl modules for the untwisted affine algebras. This identification allows us to use known results in the untwisted case to compute the dimensions and characters of the Weyl modules for the twisted algebras.     

Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

Supplement to multiplier theorems

A multiplier theorem in (J. Combinatorial Theory Ser. A, in press) is extended to cyclic group divisible difference sets (GDDSs) of small size. A multiplier theorem for abelian difference sets in (Proc. Amer. Math. Soc.68 (1978), 375–379) is extended to abelian GDDSs. A remark on the existence of cyclic affine planes is made based on a previously proved multiplier theorem.     

A matrix-splitting method for symmetric affine second-order cone complementarity problems

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. The purpose of this paper is to propose an iterative method for the symmetric affine SOCCP that is based on the idea of matrix splitting. Matrix-splitting methods have originally been developed for the solution of the system of linear equations and have subsequently been extended to the LCP and the affine variational inequality problem. In this paper, we first give conditions under which the matrix-splitting method converges to a solution of the affine SOCCP. We then present, as a particular realization of the matrix-splitting method, the block successive overrelaxation (SOR) method for the affine SOCCP involving a positive definite matrix, and propose an efficient method for solving subproblems. Finally, we report some numerical results with the proposed algorithm, where promising results are obtained especially for problems with sparse matrices.     

Affine PBW bases and MV polytopes in rank 2

Mirkovi?–Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. Our main tool is a new characterization of rank-2 affine MV polytopes.     

Properly Discontinuous Groups of Affine Transformations: A Survey

In this survey article we discuss the structure of properly discontinuous groups of affine transformations and in particular of affine crystallographic groups. One of the main open questions is Auslander's conjecture claiming that every affine crystallographic group is virtually solvable.     

Closed Conjugacy Classes,Closed Orbits and Structure of Algebraic Groups

In this paper we prove that if an affine algebraic group (in characteristic zero) has all its conjugacy classes closed, then it is nilpotent. A classical result (called sometimes the Kostant-Rosenlicht Theorem) guarantees that if an affine algebraic group G is unipotent, then all its orbits on affine varieties are closed. We prove the converse of that theorem in arbitrary characteristics.     

Nonreduced extended affine root systems of nullity 3

In 1985 K. Saito [S] introduced the concept of an extended affine root system (EARS). His study of these root systems was motivated by his interest in singularities. Later in [AABGP], this notion played an important role in the study of extended affine Lie algebras. Saito classified the EARS's of nullity ≤ 2 which have the further property that the quotient modulo a "marking" is reduced. In [AABGP], a construction was given of all EARS's, and this was used to give a classification of EARS's of reduced type. However, when the EARS was not reduced, the isomorphism problem for the construction is quite difficult and the classification was only done for EARS'S with nullity ≤ 2. The present paper extends this classification to nullity 3.