INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY

1974 ◽  
Vol 6 (2) ◽  
pp. 237-238
Author(s):  
W. Ledermann
Keyword(s):  
Representation Theory ◽  
Lie Algebras

Representations of affine Lie algebras and soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 391-391
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Gordon Equation ◽  
Vries Equation ◽  
Affine Lie Algebras ◽  
Loop Groups ◽  
Sine Gordon Equation ◽  
Kyoto School ◽  
Hidden Symmetries ◽  
Soliton Equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

scholarly journals Quiver Hecke Algebras and 2-Lie Algebras

2012 ◽  
Vol 19 (02) ◽  
pp. 359-410 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Hecke Algebras ◽  
Geometric Description ◽  
Monoidal Categories ◽  
Quiver Varieties ◽  
Basic Properties

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

scholarly journals ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES

2001 ◽  
Vol 03 (04) ◽  
pp. 533-548 ◽  
Author(s):  
NAIHUAN JING ◽  
KAILASH C. MISRA ◽  
CARLA D. SAVAGE
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Mathematical Physics ◽  
Infinite Dimensional ◽  
Algebra Representation ◽  
The Sixties ◽  
Lie Algebra Representation ◽  
New Interpretation

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

Multi-Sided Braid Type Subfactors

2001 ◽  
Vol 53 (3) ◽  
pp. 546-564 ◽  
Author(s):  
Juliana Erlijman
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Relative Commutants

AbstractWe generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II1 factor R in [E1]—which arise by considering unitary braid representations with certain properties—to multisided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types A, B, C, D. We also compute the (first) relative commutants.

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