The Nagano-Yagi-Goldmann theorem states that on the torus T, every affine (or projective) structure is invariant or is constructed on the basis of some Goldmann rings [N-Y]. It shows the interest to study the invariant affine structure on the torus T or on abelian Lie groups. Recently, the works of Kim [K] and Dekempe-Ongenae [D-O] precise the number of non equivalent invariant affine structure...

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley–Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that M1,M2 ∈ L =⇒ M1M2 − M2M1 ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L1 = ML...

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if Lie sub(super)algebra determined by it not finite dimensional or but (super)algebra any submatrix A, obtained striking out row and column intersecting on main diagonal, sum (super)algebras. said to be affine, all its matrices are affine. list superalgebras over complex numbers ...

We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian subalgebras of simple Lie algebras.

A complete classification of the WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of SU(2), and level 1 of all simple algebras. In this paper we solve the classification problem for SU(3) modular invariant partition functions. Our approach will also be applicable to other affine Lie algebras, and we include some p...

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang–Baxter equation. The method is based on an affine realization of certain seaweed algebras and their quantum analogues. We also propose a method of ω-affinization, which enables us to quantize rational r-matrices of sl(3).

The degenerate affine and affine BMW algebras arise naturally in the context of SchurWeyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we...

In this paper, we give a finite number of defining relations satisfied by a finite number of generators for the elliptic Lie algebras and superalgebras gR with rank ≥ 2. Here the R’s denote the reduced and non-reduced elliptic root systems with rank ≥ 2. We also show that if L is an extended affine Lie algebra (EALA) whose non-isotropic roots form the R, then there exists a natural homomorphism...

Recent interests in quantum groups are stimulated by their marvelous relations with quantum Yang-Baxter equations, conformal field theory, invariants of links and knots, and q-hypergeometric series. Besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q-deformations or quantum analogues of known st...

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of these, the principal and homogeneous realizations, are of particular interest. The principal vertex operator construction for the a...