Twisted Quantum Affine Algebras Research Articles

R-matrix Presentation of Quantum Affine Algebra in Type A(2)2n−1

In this paper, we give an RTT presentation of the twisted quantum affine algebra of type A 2−1 (2) and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first such presentation for twisted quantum affine algebras with nontrivial central elements.

$$Q{\widetilde{Q}}$$-Systems for Twisted Quantum Affine Algebras

We establish the \(Q {\widetilde{Q}}\)-systems for the twisted quantum affine algebras that were conjectured in Frenkel and Hernandez (Commun Math Phys 362(2):361–414, 2018). We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations.

Twisted quantum affinizations and their vertex representations

In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.

Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in [FG].

Two-parameter twisted quantum affine algebras

We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.

Drinfeld Realization of Quantum Twisted Affine Algebras via Braid Group

The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig’s braid group action.

Addendum to “Drinfeld Realization of Twisted Quantum Affine Algebras”

We provide necessary details to several arguments that appeared in our previous paper “Drinfeld Realization of Twisted Quantum Affine Algebras,” Commun. Algebra 35 (2007) 3683–3698.

Kirillov-Reshetikhin Conjecture: The General Case

We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and we get explicit formulas for the character of their tensor products (the untwisted case was treated in [16, 33, 34]). The proof provides several new developments for the representation theory of twisted quantum affine algebras, in particular on the twisted Frenkel–Reshetikhin q-characters that we define (expected in [11, 12]) and on the parameterization of simple finite dimensional representations without Drinfeld presentation. We also prove the twisted T-system [30]. As an application, we get a proof of explicit (q)-characters formulas conjectured in various papers. We prove an isomorphism of Grothendieck rings between a twisted quantum affine algebra and the corresponding simply-laced quantum affine algebra.

Drinfeld Realization of Twisted Quantum Affine Algebras

The quantum affine algebra has two realizations, the usual Drinfeld–Jimbo definition and a new Drinfeld realization given by Drinfeld. In this article, we use the adjoint action to prove that these two realizations are isomorphic for the twisted quantum affine algebra.

Construction of Perfect Crystals Conjecturally Corresponding to Kirillov-Reshetikhin Modules over Twisted Quantum Affine Algebras

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.

Universal [formula omitted]-matrix for quantum affine algebras Uq( A2(2)) and [formula omitted] with Drinfeld comultiplication

We derive an integral formula for the universal R-matrix for the twisted quantum affine algebra U q ( A 2 (2)) and quantum affine superalgebra U q( osp(1 | 2)) with Drinfeld comultiplication.

Difference L operators and a Casorati determinant solution to the T-system for twisted quantum affine algebras

We propose factorized difference operators L(u) associated with the twisted quantum affine algebras Uq(A(2)2n), Uq(A(2)2n−1), Uq(D(2)n+1), Uq(D(3)4). These operators are shown to be annihilated by a screening operator. Based on the solutions of the difference equation L(u)w(u) = 0, we also construct a Casorati determinant solution to the T-system for Uq(A(2)2n), Uq(A(2)2n−1).

Drinfeld basis of the twisted quantum affine algebraUq(A(2)2) from the Gauss decomposition of anL-operator

We establish the algebraic isomorphism between the Drinfeld basis of the twisted quantum affine algebra Uq(A(2)2) and the twisted Reshetihkin and Semenov-Tian-Shansky algebra by using the Gauss decomposition technique of Ding and Frenkel.

Izergin–Korepin model with a boundary

The Izergin-Korepin model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the twisted quantum affine algebra U-q[((2))(2)]. We give the bosonization of the vacuum state with zero particle content. Excitation states are given by the action of the vertex operators on the vacuum state. We derive the boundary S-matrix. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy. (C) 2001 Elsevier Science B.V. All rights reserved.

Q-Wedge Modules for Quantized Enveloping Algebras of Classical Type

We use the fusion construction in twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a by-product we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the q-wedge products.

Vertex operators for twisted quantum affine algebras

We construct explicitly the q q -vertex operators (intertwining operators) for the level one modules V ( Λ i ) V(\Lambda _i) of the classical quantum affine algebras of twisted types using interacting bosons, where i = 0 , 1 i=0, 1 for A 2 n − 1 ( 2 ) A_{2n-1}^{(2)} ( n ≥ 3 n\geq 3 ), i = 0 i=0 for D 4 ( 3 ) D_4^{(3)} , i = 0 , n i=0, n for D n + 1 ( 2 ) D_{n+1}^{(2)} ( n ≥ 2 n\geq 2 ), and i = n i=n for A 2 n ( 2 ) A_{2n}^{(2)} ( n ≥ 1 n\geq 1 ). A perfect crystal graph for D 4 ( 3 ) D_4^{(3)} is constructed as a by-product.

Twisted Quantum Affine Algebras

We give a highest weight classification of the finite-dimensional irreducible representations of twisted quantum affine algebras. As in the untwisted case, such representations are in one-to-one correspondence with n-tuples of monic polynomials in one variable. But whereas in the untwisted case $n$ is the rank of the underlying finite-dimensional complex simple Lie algebra ?, in the twisted case n is the rank of the subalgebra of ? fixed by the diagram automorphism. The way in which such an n-tuple determines a representation is also more complicated than in the untwisted case.

Functional relations and analytic Bethe ansatz for twisted quantum affine algebras

Functional relations are proposed for transfer matrices of solvable vertex models associated with the twisted quantum affine algebras $U_q(X^{(\kappa)}_n)$ where $X^{(\kappa)}_n = A^{(2)}_n, D^{(2)}_n, E^{(2)}_6$ and $D^{(3)}_4$. Their solutions are obtained for $A^{(2)}_n$ and conjectured for $D^{(3)}_4$ in the dressed vacuum form in the analytic Bethe ansatz.