Roots of unity: Representations of quantum groups

Abstract

Representations of Quantum Groups U ε (g n ), g n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n − 1 quantum groups for ε a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.

Tullio Regge: An Eclectic Genius

“…Then, from the Weyl representations equation (7.15), the products A q , B q , C q of the unitary operators A, B, C lie in the center of the algebra. This is typical of quantum algebras [27].…”

Nilpotent representations of classical quantum groups at roots of unity

“…(Indeed, the module as below is similar to the maximal cyclic representation as in [4].). Theorem 1.1 (Schnizer module [6]). For any a = (a i,j ) 1≤i≤j≤n ∈ (C × ) N , b = (b i,j ) 1≤i≤j≤n ∈ C N , λ = (λ 1 , · · · , λ n ) ∈ C n , we obtain a U ε (sl(n + 1, C))-module structure on V : Φ λ,a,b : U ε (sl(n + 1, C)) −→ End(V ).…”

“…For type A, such modules are constructed very explicitly in [4], which is called maximal cyclic representations. For any simple Lie algebra, Schnizer introduced an alternative construction of such modules in [5], [6], which we also call a maximal cyclic representation or "Schnizer module".…”

“…(Indeed, the module as below is similar to the maximal cyclic representation as in [4].). Theorem 1.1 (Schnizer module [6]). For any a = (a i,j…”

Representations of Uq(sl(N)) at roots of unity

“…In [10][11][12][13][14][15][16][17][18], explicit expressions for representations with periodic (or cyclic) actions of the generators are given.…”