Showing papers on "Representation theory published in 1986"

Representation Theory of Semisimple Groups: An Overview Based on Examples

Abstract: In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Cohomology of Infinite-Dimensional Lie Algebras

Abstract: 1. General Theory.- 1. Lie algebras.- 2. Modules.- 3. Cohomology and homology.- 4. Principal algebraic interpretations of cohomology.- 5. Main computational methods.- 6. Lie superalgebras.- 2. Computations.- 1. Computations for finite-dimensional Lie algebras.- 2. Computations for Lie algebras of formal vector fields. General results.- 3. Computations for Lie algebras of formal vector fields on the line.- 4. Computations for Lie algebras of smooth vector fields.- 5. Computations for current algebras.- 6. Computations for Lie superalgebras.- 3. Applications.- 1. Characteristic classes of foliations.- 2. Combinatorial identities.- 3. Invariant differential operators.- 4. Cohomology of Lie algebras and cohomology of Lie groups.- 5. Cohomology operations in cobordism theory..- References.

Noncommutative Harmonic Analysis

Abstract: Some basic concepts of Lie group representation theory The Heisenberg group The unitary group Compact Lie groups Harmonic analysis on spheres Induced representations, systems of imprimitivity, and semidirect products Nilpotent Lie groups Harmonic analysis on cones $\mathrm {SL}(2,R)$ $\mathrm {SL}(2, \mathbf C)$, and more general Lorentz groups Groups of conformal transformations The symplectic group and the metaplectic group Spinors Semisimple Lie groups.

Differential algebraic groups

Abstract: The lectures will attempt to describe the general theory of differential algebraic groups that has been developed in recent years in analogy with and as a generalization of the older theory of algebraic 0rou9s. Limitations of time "make it necessary to omit a number of topics and to give broad descriptions instead of proofs. I have qiven Professor Tuan at least one copy of each of several references, and I hope that these can be consulted by anyone interested in any of the details. The main reference is the manuscript of my forth­coming book [8]; two earlier works that may prove helpful on occasion are my 1973 book [6] and my paoer [7]. Also included are reprints of five papers by P.J. Cassidy [1­5], one paper by J. Kovacic [9], and one paper by W.Y. Sit [10], all bearing on results in the theory that I shall not have time to describe. These lectures are intended as a selective survey of the subject.

The Yangians, Bethe Ansatz and combinatorics

Abstract: An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed. The connection between the quantum inverse transform method and the representation theory of a symmetric group is considered. A new approach to the completeness problem of Bethe vectors is also given.

Representation theory of finite-dimensional algebras

Abstract: Methods and results from the representation theory of nite di- mensional algebras have led to many interactions with other areas of math- ematics. The aim of this workshop was, in addition to stimulating progress in the representation theory of algebras, to further develop such interactions with commutative algebra, algebraic geometry, group representation theory, Lie-algebras and quantum groups, but also with the new theory of cluster algebras.

Invariant complex structures on reductive Lie groups.

Abstract: An invariant complex structure on a real even dimensional Lie group G0 is a complex structure on the underlying manifold in the usual sense with the additional property that left multiplication (but not necessarily right multiplication) in the group is holomorphic with respect to that structure. If g0 denotes the Lie algebra of G0 and g its complexification, then such structures are in one-to-one correspondence with complex subalgebras q c g such that g = q iq where is complex conjugation in g. H. C. Wang [Wa54] was one of the first to describe invariant complex structures on compact semisimple Lie groups and more generally on compact homogeneous manifolds with finite fundamental groups. In the non-compact case, Morimoto [Mo 56] showed that there always exist invariant complex structures on any even dimensional reductive Lie group. Using exhaustive Lie algebra calculations, Sasaki [Sa 81], [Sa 82] gave a detailed classification of all invariant complex structures on the groups GL(2, M), 17(2), and SL(3, AP). The complex structures found in the above works, except for two very special structures on SL(3, M), come with natural parameter spaces and are regulär in the following sense: there exists a Cartan subalgebra heg, stable under , such that

The Plancherel theorem for general semisimple groups

Abstract: © Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Invariant quadratic forms on finite dimensional lie algebras

Abstract: Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [ L, L ] ∩ R (with the radical R ) in the orthogonal L ⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

Varieties of polynomial growth of lie algebras over a field of characteristic zero

Abstract: The purpose of this note is to prove that condition (i) is not only necessary but also sufficient for the polynomial growth of the variety V. In addition, we prove some results concerning techniques of applying the theory of representations of the symmetric group to the study of varieties of Lie algebras or other linear algebras. Writing elements we will omit parentheses placed in the left-normal way. We say that an element w contain8 a skewsy~etric array consisting of m variables if an equality of the form

New S function series and non-compact Lie groups

Abstract: A number of new infinite series of S functions are described in terms of their generating functions and S function content. Applications to the character theory of non-compact Lie groups are noted.