Quantum Affine Algebra Of Type Research Articles

Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras II: Types $C_r$, $F_4$, and $G_2$

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type C_r , F_4 , and G_2 at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.

Generalized Young walls and crystal bases for quantum affine algebra of type $A$

We give a new realization of the crystal of using generalized Young walls, a modified notion of Young walls of type . Moreover, by the fact that the irreducible highest weight crystal lies in the crystal , we construct the crystal using generalized Young walls.

Paths and Tableaux Descriptions of Jacobi-Trudi Determinant Associated with Quantum Affine Algebra of Type Cn

We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type C_n. Like the D_n case studied by the authors recently, applying the Gessel-Viennot path method with an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.

Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D n

We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type D n . Unlike the A n and B n cases, a simple application of the Gessel-Viennot path method does not yield an expression of the determinant by a positive sum over a set of tuples of paths. However, applying an additional involution and a deformation of paths, we obtain an expression by a positive sum over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.

Modified Nakajima monomials and the crystal [formula omitted

In this paper, we give a new realization of the crystal B ( ∞ ) using modified Nakajima monomials. For the quantum affine algebras of type A n ( 1 ) , we give an explicit characterization of the monomials in B ( ∞ ) , and construct a natural crystal isomorphism between monomial realization and path realization. As a corollary, we obtain a new realization of the crystal B ( ∞ ) for the quantum classical algebras of type A n .

Monomials of q and q, t-characters for non simply-laced quantum affinizations

Nakajima [23][24] introduced the morphism of q, t-characters for finite dimensional representation of simply-laced quantum affine algebras: it is a t-deformation of the Frenkel-Reshetikhin’s morphism of q-characters (sum of monomials in infinite variables). In [15] we generalized the construction of q, t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of [15]: the monomials of q and q, t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated in [24]) and the coefficients are non negative. In particular those q, t-characters can be considered as t-deformations of q-characters. In the proof we show that for quantum affine algebras of type A, B, C and quantum toroidal algebras of type A(1) the l-weight spaces of fundamental representations are of dimension 1. Eventually we show and use a generalization of a result of [13][10][22]: for general quantum affinizations we prove that the l-weights of a l-highest weight simple module are lower than the highest l-weight in the sense of monomials.

Box ball system associated with antisymmetric tensor crystals

A new box ball system associated with an antisymmetric tensor crystal of the quantum affine algebra of type A is considered. This includes the so-called colored box ball system with capacity 1 as the simplest case. Infinite number of conserved quantities are constructed and the scattering rule of two olitons are given explicitly.

Crystal bases of the Fock space representations and string functions

Let U q( g) be the quantum affine algebra of type A n (1), A 2 n−1 (2), A 2 n (2), B n (1), D n (1), and D n+1 (2), and let F(Λ) be the Fock space representation for a level 1 dominant integral weight Λ. Using the crystal basis of F(Λ) and its characterization in terms of abacus, we construct an explicit bijection between the set of weight vectors in F(Λ) λ−mδ ( m⩾0) for a maximal weight λ and the set of certain ordered sequences of partitions. As a corollary, we obtain the string function of the basic representation V( Λ).

An Integral PBW Basis of the Quantum Affine Algebra of Type $A_2^{(2)}$

We construct an integral PBW basis and an integral crystal basis of the quantum affine algebra of type A_2^{(2)} .

Trace Construction of a Basis for the Solution Space¶of sl N qKZ Equation

The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type A_{N-1} is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix element of the composition of intertwining operators is explicitly calculated. The integral formula for the trace is presented.

Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type $\mathfrak{gl}_N$

A representation of the quantum toroidal algebra of type \mathfrak{sl}_N is defined on every integrable irreducible highest weight module of the quantum affme algebra of type \mathfrak{gl}_N . The q -version of the level-rank duality giving the reciprocal decomposition of the q -Fock space with respect to mutually commutative actions of U'q(\mathfrak{gl}_N) of level L and U'_q(\mathfrak{sl}_L) of level N is described.

Algèbres affines quantiques aux racines de l’unité et K-théorie équivariante

We show that the algebra homomorphism [Display omitted] constructed by Ginzburg and Vasserot between the quantum affine algebra of type g l n and the equivariant K-theory group of the Steinberg variety (of incomplete flags), specializes to a surjective homomorphism [Display omitted] . In particular, this shows that the parameterization of irreducible U ε res ( s l ^ n ) -modules and the multiplicity formulas of [6], [8] are still valid when ε is a root of unity.

Q-Deformed Fock spaces and modular representations of spin symmetric groups

We use the Fock space representation of the quantum affine algebra of type $A^{(2)}_{2n}$ to obtain a description of the global crystal basis of its basic level 1 module. We formulate a conjecture relating this basis to decomposition matrices of spin symmetric groups in characteristic $2n+1$.

Twisted vertex representations of quantum affine algebras

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 structures. Quantum groups were first defined by Drinfeld [2] and Jimbo [9] (also see [4]) as a q-deformation of the universal enveloping algebras of the KacMoody algebras in the work of trigonometric solutions of Yang-Baxter equations. In the same spirit it was shown in [13], [14], that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable Kac-Moody algebras and those of the corresponding quantum groups, where both spaces have the same dimension in the case of generic q (i.e. q is not a root of unity). Moreover, one can be very explicit in the case of quantum gl(n) to write down the irreducible highest weight representations. Quantum affine algebras are the quantum groups associated to affine Lie algebras. Following Drinfeld's realization [-3] of q-analog of loop algebras, the vertex representation of untwisted simply laced quantum affine algebras was constructed in Frenkel-Jing [6], which is a q-deformation of Frenkel-Kac [7] and Segal [15] construction in the theory of affine Lie algebras. Subsequently, the same was done for the quantum affine algebra of type B in [1]. In the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the Dynkin diagram, which generalizes certain important cases in the ordinary twisted vertex operator calculus [-5,