Showing papers on "Cartan matrix published in 1989"

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

Abstract: We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras “continuum Lie algebras.” The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.

On generators and relations for certain involutory subalgebras of kac-moody lie algebras ∗

Abstract: We find generators and relations for those subalgebras of Kac-Moody Lie algebras that are the fixed point algebras of certain involutions. Specifically the involution must involve the Cartan involution which interchanges the positive and negative generators. We go on to apply these results to the G.I.M. algebras, which were introduced as natural generalizations of Kac-Moody algebras by P. Slodowy. We show such algebras are isomorphic to subalgebras of Kac-Moody algebras. From this we are able to derive someinteresting interrelations between certain Kac-Moody algebras.

Killing and the Coxeter transformation of Kac-Moody algebras

Abstract: On etudie le spectre de la transformation de Coxeter associee a une algebre de Kac-Moody arbitraire et plus particulierement a une algebre affine

The greatest mathematical paper of all time

Abstract: Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.

Representations of Affine Kac-Moody Algebras

Abstract: In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary semisim-ple finite-dimensional Lie algebra. A semisimple Lie algebra is a direct sum of simple ideals which are pairwise orthogonal with respect to the Killing form. It follows that the Cartan matrix of a semisimple Lie algebra decomposes to a block diagonal form, each block representing a simple ideal. Similarly, the Dynkin diagram is a disconnected union of Dynkin diagrams of simple Lie algebras. Next we shall study certain infinite-dimensional Lie algebras which have many similarities with the simple finite-dimensional Lie algebras. In particular, they can be described in terms of generalized Cartan matrices. These algebras were independently introduced in Kac [1968] and Moody [1968].

Quantum Lie Superalgebras and Supergroups

Abstract: The quantum inverse problem method allows for a systematic treatment of a variety of exactly solvable models of quantum field theory in 1 + 1 dimensions and vertex type lattice models in statistical physics [cf. e.g. 1,2]. Moreover, it enables one to construct new integrable models and leads also to new mathematical structures [3,4] known as quantum Lie groups and algebras, quadratic algebras, quantum Hopf algebras. Recently these concepts entered into numerous publications on the operator solution to the quantum Liouville equation, rational conformal field theory, representation of braid groups, knot and link invariants, quantum topological field theory etc. Unfortunately, quantum groups and algebras appear as an artifact, and the efforts to give them a physical meaning or interpretation are far from being convincing [cf. e.g. 5].

On the Representations of the Classical Groups

Abstract: This thesis deals with some aspects of both the ordinary and modular representations of the classical groups. The main theme of the thesis is a conjecture proposed by J.L. Alperin about the number of modular characters of a finite group. A new proof of this conjecture for a class of groups containing the classical groups is presented and as a consequence of the conjecture being proved true the following have been calculated • the p-rank of the Cartan matrix of an arbitrary parabolic subgroup of the general linear group. • the number of ordinary irreducible characters of an arbitrary parabolic subgroup of the general linear group up to dimension three. • the number of ordinary irreducible characters of an arbitrary parabolic subgroup of the special linear group of dimension three.