UniversalR-matrix for quantized (super)algebras

Abstract

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universal .R-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties of ^-exponential functions. We don't use quantum Weyl group.

Факторизация $R$-матрицы, $Q$-оператор и разделение переменных в случае $XXX$-спиновой цепочки с группой симметрии $SL(2,\mathbb{C})$

“…в котором использованы операторы, входящие в матрицу монодромии (10). Соотно-шение для матричных элементов, стоящих на пересечении первой строки и второго столбца, имеет вид…”

“…В рамках этого метода Q-оператор строит-ся как след специальной матрицы монодромии по вспомогательному пространству бесконечномерного представления q-осцилляторной алгебры. При этом использу-ется явный вид универсальной R-матрицы [10]. Абстрактное представление через генераторы для универсальной R-матрицы является достаточно сложным.…”

A q-superdimension formula for irreps of type I quantum superalgebras

“…Associated with g one can define the quantum superalgebra U q (g) ͑q is assumed not a root of unity͒ which has the structure of a Z 2 -graded quasi-triangular Hopf algebra. 3 We will not give the full defining relations of U q (g) here and refer to Ref. 4 for details.…”

Eigenvalues of Casimir invariants for type I quantum superalgebras

“…Associated with g one can define the quantum superalgebra U q (g) (q is assumed not a root of unity) which has the structure of a Z 2 -graded quasi-triangular Hopf algebra [3]. We will not give the full defining relations of U q (g) here and refer to [4] for details.…”

“…Likewise, their Z 2 -graded counterparts quantum superalgebras [2,3,4] play a similar role in relation to supersymmetric solvable models. There has been significant interest in the study of these models, particularly those describing systems of correlated electrons which are of importance in condensed matter physics.…”