scholarly journals Representations of Quantum Affine Algebras in their R-Matrix Realization

2020 ◽  
Author(s):  
Naihuan Jing ◽  
◽  
Ming Liu ◽  
Alexander Molev ◽  
◽  
...  
Keyword(s):  
Irreducible Representations ◽  
Quantum Affine Algebras ◽  
R Matrix ◽  
Finite Dimensional ◽  
Gauss Decomposition ◽  
Affine Algebras

We use the isomorphisms between the R-matrix and Drinfeld presentations of the quantum affine algebras in types B, C and D produced in our previous work to describe finite-dimensional irreducible representations in the R-matrix realization.We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the R-matrix and Drinfeld presentations of the Yangians.

scholarly journals GENERALIZED GAUSS DECOMPOSITION OF TRIGONOMETRIC R-MATRICES

1995 ◽  
Vol 10 (19) ◽  
pp. 1375-1392 ◽  
Author(s):  
S.M. KHOROSHKIN ◽  
A.A. STOLIN ◽  
V.N. TOLSTOY
Keyword(s):  
Tensor Product ◽  
Bilinear Form ◽  
Central Charge ◽  
Irreducible Representations ◽  
Verma Modules ◽  
R Matrix ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Gauss Decomposition ◽  
Affine Algebras

The general formula for the universal R-matrix for quantized nontwisted affine algebras, obtained by the first and third authors, is applied to zero central charge, highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonometric R-matrix in tensor product of these modules. In particular, [Formula: see text] current realization of the universal R-matrix is presented. It gives a new universal presentation for the trigonometric R-matrix with a parameter in tensor product of Uq(sl2)-Verma modules. Detailed analysis of a scalar factor arising in finite-dimensional representations of the universal R-matrix for any Uq(ĝ) is given. We interpret this scalar factor as a multiplicative bilinear form on highest weight polynomials of irreducible representations and express this form in terms of infinite q-shifted factorials.

scholarly journals On finite-dimensional representations of two-parameter quantum affine algebras

2016 ◽  
Vol 15 (03) ◽  
pp. 1650054 ◽  
Author(s):  
Naihuan Jing ◽  
Honglian Zhang
Keyword(s):  
Irreducible Representations ◽  
Quantum Affine Algebras ◽  
Finite Dimensional ◽  
Affine Algebras ◽  
One To One ◽  
Two Parameter

We introduce the notion of Drinfeld polynomials of two-parameter quantum affine algebras and establish a one-to-one correspondence between finite-dimensional irreducible representations and sets of [Formula: see text]-tuples of pairs of polynomials with certain conditions.

scholarly journals Generalized Green Functions and Unipotent Classes for Finite Reductive Groups, I

2006 ◽  
Vol 184 ◽  
pp. 155-198 ◽  
Author(s):  
David Hernandez ◽  
Hiraku Nakajima
Keyword(s):  
Weight Module ◽  
Green Functions ◽  
Reductive Groups ◽  
Quantum Affine Algebras ◽  
Finite Dimensional ◽  
Affine Algebras

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

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