Arturo Pianzola

Extended affine Lie algebras and their root systems

Covering extended affine Lie algebras and their root systems, this work is intended for graduate students, research mathematicians, and mathematical physicists interested in Lie theory.

Realization of graded-simple algebras as loop algebras

Abstract Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a ℤ n -graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields. 2000 Mathematics Subject Classification: 16W50, 17B70; 17B65, 17B67.

MULTILOOP REALIZATION OF EXTENDED AFFINE LIE ALGEBRAS AND LIE TORI

An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient conditions for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs.

AUTOMORPHISMS OF TOROIDAL LIE ALGEBRAS AND THEIR CENTRAL QUOTIENTS

We describe the structure of the group of automorphisms of Lie algebras of the form when R has trivial Picard group. We also look at the group of automorphisms of central quotients of the universal covering algebra of as well as conjugacy questions.

On infinite root systems

We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.

Conjugacy theorems for loop reductive group schemes and Lie algebras

The conjugacy of split Cartan subalgebras in the finite-dimensional simple case (Chevalley) and in the symmetrizable Kac–Moody case (Peterson–Kac) are fundamental results of the theory of Lie algebras. Among the Kac–Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras—extended affine Lie algebras—that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson–Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat–Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest.

Developments and Trends in Infinite-Dimensional Lie Theory

Preface.- Part A: Infinite-Dimensional Lie (Super-)Algebras.- Isotopy for Extended Affine Lie Algebras and Lie Tori.- Remarks on the Isotriviality of Multiloop Algebras.- Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras - A Survey.- Tensor Representations of Classical Locally Finite Lie Algebras.- Lie Algebras, Vertex Algebras, and Automorphic Forms.- Kac-Moody Superalgebras and Integrability.- Part B: Geometry of Infinite-Dimensional Lie (Transformation) Groups.- Jordan Structures and Non-Associative Geometry.- Direct Limits of Infinite-Dimensional Lie Groups.- Lie Groups of Bundle Automorphisms and Their Extensions.- Gerbes and Lie Groups.- Part C: Representation Theory of Infinite-Dimensional Lie Groups Functional Analytic Background for a Theory of Infinite- Dimensional Reductive Lie Groups.- Heat Kernel Measures and Critical Limits.- Coadjoint Orbits and the Beginnings of a Geometric Representation Theory.- Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces.- Index.

Descent constructions for central extensions of infinite dimensional Lie algebras

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives information about the structure of the group of automorphisms of such algebras.

Automorphisms and twisted loop algebras of finite dimensional simple Lie superalgebras

We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this andcohomology considerations, we list all different isomorphism classes of the corresponding twisted loop superalgebras. 2000 MSC: Primary 17B67. Secondary 17B40, 20G15.

Affine Kac-Moody Lie algebras as torsors over the punctured line

Abstract We interpret and develop a theory of loop algebras as torsors (principal homogeneous spaces) over Spec ( k [ t , t −1 ]). As an application, we recover Kacs realization of affine Kac-Moody Lie algebras.