We re-examine the level-one irreducible highest weight representations of the quantum affine superalgebra $U_q(\hat{sl}(2|1))$ and derive the characters and supercharacters associated with these representations. We calculate the exchange relations of the vertex operators and find that these vertex operators satisfy the graded Faddeev-Zamolodchikov algebra. We give an integral expression of the correlation functions of the q-deformed supersymmetric t-J model and derive the difference equations which they satisfy.
In this paper, we propose an elliptic algebra $A_{q,p;\hatπ}(\hat{gl_2})$ which is based on the relations $RLL=LLR^{*}$, where $R$ and $R^{*}$ are the dynamical R-maxtrices of $A^{(1)}_{1}$ type face model with the elliptic moduli shifted by the center of the algebra. From the Ding-Frenkel correspondence, we find that its corresponding (Drinfeld) current algebra at level one is the algebra of screening currents for q-deformed Virasoro algebra.We realize the elliptic algebra at level one by Miki's construction from the bosonization for the type I and type II vertex operators.We also show that t…
We introduce the elliptic superalgebra $U_{q,p}(\hat{sl}(M|N))$ as one parameter deformation of the quantum superalgebra $U_q(\hat{sl}(M|N))$. For an arbitrary level $k \neq 1$ we give the bosonization of the elliptic superalgebra $U_{q,p}(\hat{sl}(1|2))$ and the screening currents that commute with $U_{q,p}(\hat{sl}(1|2))$ modulo total difference.
On the level-1 Fock space modules of the algebra $U_q(\hat{sl_n})$ we define a level-0 action $U_0$ of the $U_q(\hat{sl_n})$, and an action of an abelian algebra of conserved Hamiltonians commuting with the $U_0$. An irreducible decomposition of the Fock space with respect to the level-0 action is derived by constructing a base of the Fock space in terms of the Non-symmetric Macdonald Polynomials.
We consider an unified model, called ancestor model, associated with twisted trigonometric $R$ matrix which model leads to several descendant integrable lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its …
An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic algebra $A_{q,p}(\hat{sl}_2)$.
We determine the exchange relations of the level-one q-vertex operators of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$. We study in details the level-one irreducible highest weight representations of $U_q[\hat{gl(2|2)}]$, and compute the characters and supercharacters associated with these irreducible modules.
We decompose the level-1 irreducible highest weight modules of the quantum affine algebra $U_q(\hat{sl}_n)$ with respect to the level-0 $U'_q (\hat{sl}_n)$--action defined in q-alg/9702024. The decomposition is parameterized by the skew Young diagrams of the border strip type.
We study the level-one irreducible highest weight representations of $U_q[\hat{gl(1|1)}]$ and associated q-vertex operators. We obtain the exchange relations satisfied by the vertex operators. The characters and supercharacters associated with these irreducible representations are calculated.
The $\hatκ$-deformed extended Galilei Hopf group algebra, ${\rm Fun}_{\hatκ}(\tilde G_{(m)})$, is introduced. It provides an example of a cocycle bicrossproduct structure, and is shown to be the contraction limit of a pseudoextension of the $κ$-Poincaré group algebra. The possibility of obtaining another deformed extended Galilei groups is discussed, including one obtained from a non-standard Poincaré deformation.
We study the level-one irreducible highest weight representations of the quantum affine superalgebra $U_q[\hat{sl(N|1)}]$, and calculate their characters and supercharacters. We obtain bosonized q-vertex operators acting on the irreducible $U_q[\hat{sl(N|1)}]$-modules and derive the exchange relations satisfied by the vertex operators. We give the bosonization of the multi-component super $t-J$ model by using the bosonized vertex operators.
An algebra homomorphism $ψ$ from the nonstandard q-deformed (cyclically symmetric) algebra $U_q(so_3)$ to the extension ${\hat U}_q(sl_2)$ of the Hopf algebra $U_q(sl_2)$ is constructed. Not all irreducible representations of $U_q(sl_2)$ can be extended to representations of ${\hat U}_q(sl_2)$. Composing the homomorphism $ψ$ with irreducible representations of ${\hat U}_q(sl_2)$ we obtain representations of $U_q(so_3)$. Not all of these representations of $U_q(so_3)$ are irreducible. Reducible representations of $U_q(so_3)$ are decomposed into irreducible components. In this way we obtain all …
The method of vector coherent states is generalized to study representations of the affine Lie algebra $\hat{sl}(2)$. A large class of highest weight irreps is explicitly constructed, which contains the integrable highest weight irreps as special cases.
We study the highest weight and continuous tensor product representations of q-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of the q-deformed algebra $sl_q(2,\Cb)$ is calculated in detail.
We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of W_{q,t}[g] to the representation ring of the quantum affine algebra U_q(\hat g), as discovered recently by Frenkel and Reshetikhin, is further elucidated in some examples.
The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra $U_q[\hat{sl(2|1)}]$. We give the bosonization of the boundary states. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy.
We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using Lusztig $a$-values.
In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares …
We find certain functional identities for the Gauss q-power function of a sum of q-commuting variables. Then we use these identities to obtain two-parameter twists of the quantum affine algebra U_q (\hat{sl}_2) and of the Yangian Y(sl_2). We determine the corresponding deformed trigonometric and rational quantum R-matrices, which then are used in the computation of deformed XXX and XXZ Hamiltonians.
We extend to the sl(N) case the results that we previously obtained on the construction of W_{q,p} algebras from the elliptic algebra A_{q,p}(\hat{sl}(2)_c). The elliptic algebra A_{q,p}(\hat{sl}(N)_c) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N-1)/2 integers, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^1/2)^{NM} = q^{-c-N} for M in Z. It becomes Abelian when in addition p=q^{Nh} where h is a non-zero integer. The Poisson structur…
Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an …
We bosonize certain components of level $\ell$ $U_q(\hat{sl}_2)$-intertwiners of $(\ell + 1)$-dimensions. For $\ell = 2$, these intertwiners, after certain modification by bosonic vertex operators, are added to the algebra $U_q(\hat{sl}_2)$ at level 2 to construct all irreducible highest weight representations of level 1 for the quantum affine algebra $U_q(\hat{sp}_4)$.
We compute the braiding for the `principal gradation' of $U_q(\hat{{\it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $τ$ in the deformation parameter $q=e^{2πiτ}$. We also examine the convergence using probability, assuming a uniform …
The $U_q(\hat{gl}(N|N))$-analog of the half-infinite $t-J$ model with a boundary is considered by using the vertex operator approach. We find explicit bosonic formula of the boundary state in the integrable highest-weight module over the quantum superalgebra $U_q(\hat{gl}(N|N))$.
An algebra homomorphism $ψ$ from the q-deformed algebra $U_q({\rm iso}_2)$ with generating elements $I$, $T_1$, $T_2$ and defining relations $[I,T_2]_q=T_1$, $[T_1,I]_q=T_2$, $[T_2,T_1]_q=0$ (where $[A,B]_q=q^{1/2}AB-q^{-1/2}BA$) to the extension ${\hat U}_q({\rm m}_2)$ of the Hopf algebra $U_q({\rm m}_2)$ is constructed. The algebra $U_q({\rm iso}_2)$ at $q=1$ leads to the Lie algebra ${\rm iso}_2 \sim {\rm m}_2$ of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra $U_q({\rm m}_2)$ is treated as a Hopf $q$-deformation of the universal enveloping algebra of ${\rm iso}_2$ and…
The generators $(J_{\pm}, J_0)$ of the algebra $U_q(sl(2))$ is our starting point. An invertible nonlinear map involving, apart from q, a second arbitrary complex parameter h, defines a triplet $({\hat X},{\hat Y},{\hat H})$. The latter set forms a closed algebra under commutation relations. The nonlinear algebra $U_{q,h}(sl(2))$, thus generated, has two different limits. For $q \to 1$, the Jordanian h-deformation $U_{h}(sl(2))$ is obtained. For $h \to 0$, the q-deformed algebra $U_{q}(sl(2))$ is reproduced. From the nonlinear map, the irreducible representations of the doubly-deformed algebra…
A level-one representation of the quantum affine superalgebra $\U_q(\hat{\frak{sl}}(M+1|N+1))$ and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields. Character formulas of level-one irreducible highest weight modules of $\U_q(\hat{\frak{sl}}(2|1))$ are conjectured.
We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$. The unitary ones are classified as well. Similar results are obtained for the sin-algebras.
(1) Utilizing a Braid group action on a completion of U_q(\hat{sl_{n+1}}), an algebra homomorphism from the toroidal algebra U_q(sl_{n+1,tor}) (n\ge 2) with fixed parameter to a completion of U_q(\hat{gl_{n+1}}) is obtained. (2) The toroidal actions by Saito induces a level 0 U'_q(\hat{sl_{n+1}}) action on level 1 integrable highest weight modules of U_q(\hat{sl_{n+1}}). Another level 0 U'_q(\hat{sl_{n+1}}) action is defined by Jimbo, et al., in the case n=1. Using the fact that the intertwiners of U_q(\hat{sl_{n+1}}) modules are intertwiners of toroidal modules for an appropriate comultiplica…
Recently Varagnolo and Vasserot established that the q-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra $U_q(sl_n,tor)$ (n > 2) with the level (0,1). In the present article we propose a more detailed proof of this fact then the one given by Varagnolo and Vasserot. The proof is based on certain non-trivial properties of Cherednik's commuting difference operators. The quantum toroidal action on the Fock space depends on a certain parameter. We find that with a specific choice of this parameter the action on the Fock spaces gives ri…
We consider principal subspace W(Λ) of integrable highest weight module L(Λ) for quantum affine algebra $U_q(\hat{\mathfrak{sl}}_{n+1})$. We introduce quantum analogues of the quasi-particles associated with the principal subspaces for $\hat{\mathfrak{sl}}_{n+1}$ and discover certain relations among them. By using these relations we find, for certain highest weight Λ, combinatorial bases of principal subspace W(Λ) in terms of monomials of quantum quasi-particles.
Using the fusion product of the representations of the Lie algebra $\mathfrak{sl}_2$ we construct a set of the integrable highest weight $\hat{\mathfrak{sl}_2}$-modules $L^D$, depending on the vector $D\in\mathbb{N}^{k+1}$. In a special cases of $D$ our modules are isomorphic to the irreducible $\hat{\mathfrak{sl}_2}$-modules $L_{i,k}$. We construct a basis of the $L^D$ and study the decomposition of $L^D$ on the irreducible components. We also write a formulas for the characters of $L^D$.
We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(cΛ_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(cΛ_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.
New deformed affine algebras A_{\hbar,η}(\hat{g}) are defined for any simply-laced classical Lie algebra g, which are generalizations of the algebra A_{\hbar,η}(\hat{sl_2}) recently proposed by Khoroshkin, Lebedev and Pakuliak (KLP). Unlike the work of KLP, we associate to the new algebras the structure of an infinite Hopf family of algebras in contrast to the one containing only finite number of algebras introduced by KLP. Bosonic representation for A_{\hbar,η}(\hat{g}) at level 1 is obtained, and it is shown that, by repeated application of Drinfeld-like comultiplications, a realization of A…
We show that there are precisely two, up to conjugation, anti-involutions sigma_{\pm} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension \hat{D}^{\pm} of the Lie subalgebra of this algebra fixed by - sigma_{\pm}, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the truncated polynomial algebra C[u] / (u^{m+1}) and its…
In this paper, we re-examine certain integrable modules of Chari-Presslely for an (untwisted) affine Lie algebra $\hat{\g}$ by exploiting basic formal variable techniques. We define and study two categories ${\mathcal{E}}$ and ${\mathcal{C}}$ of $\hat{\g}$-modules using generating functions, where ${\mathcal{E}}$ includes evaluation modules and ${\mathcal{C}}$ unifies highest weight modules, evaluation modules and their tensor product modules, and we classify integrable irreducible $\hat{\g}$-modules in categories ${\mathcal{E}}$ and ${\mathcal{C}}$.
In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined starting from a semi-simple algebraic group, so that the derived category of $A$-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let $\hat{\g}$ be the affine Lie algebra corresponding to the Langlands dual Lie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we show that the category of $\hat{\g}$ modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a q…
We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight represe…
A classification of finite dimensional irreducible representations of the nonstandard $q$-deformation $U'_q(so_n)$ of the universal enveloping algebra $U(so(n, C))$ of the Lie algebra $so(n, C)$ (which does not coincides with the Drinfeld--Jimbo quantized universal enveloping algebra $U_q(so_n)$) is given for the case when $q$ is not a root of unity. It is shown that such representations are exhausted by representations of the classical and nonclassical types. Examples of the algebras $U'_q(so_3)$ and $U'_q(so_4)$ are considered in detail. The notions of weights, highest weights, highest weigh…
Bosonized q-vertex operators related to the 4-dimensional evaluation modules of the quantum affine superalgebra $U_q[\hat{sl(2|1)}]$ are constructed for arbitrary level $k=α$, where $α\neq 0, -1$ is a complex parameter appearing in the 4-dimensional evaluation representations. They are intertwiners among the level-$α$ highest weight Fock-Wakimoto modules. Screen currents which commute with the action of $U_q[\hat{sl(2|1)}]$ up to total differences are presented. Integral formulae for N-point functions of type I and type II q-vertex operators are proposed.
A decomposition of the level-one $q$-deformed Fock representations of $\uqn$ is given. It is found that the action of $\upqn$ on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra $\widehat{H}_N$ in the limit $N \rightarrow \infty$. The $q$-deformed Fock space is shown to be isomorphic as a $\upqn$-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation of $\upqn$ and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose the $q$-wedging operators, which a…
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational ve…
For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only integers or only half-integers as in highest weights of the latter) are given explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$ not equal to a root of unity, that representation operators indeed satisfy relevant (trilinear) relations and define finite dimensional irreduc…
We consider $\hat{sl_2}$ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra $U(sl_2\otimes\C[t])$. The first one is generated by $sl_2\otimes t^N$, and the second one is generated by $e\otimes P(t), f\otimes R(t)$ where $P(t), R(t)$ are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and $q$-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion pro…
Let $V(Λ_i)$ (resp., $V(-Λ_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(Λ_i)\otimes V(-Λ_j)$ is filtered by submodules $F_n=U_q(\hat{sl}_{2})(v_i\otimes \bar{v}_{n-i})$, $n\ge 0, n\equiv i-j\bmod 2$, where $v_i\in V(Λ_i)$ is the highest vector and $\bar{v}_{n-i}\in V(-Λ_j)$ is an extremal vector. We show that $F_n/F_{n+2}$ is isomorphic to the level 0 extremal weight module $V(n(Λ_1-Λ_0))$. Using this we give a functional realization of the completion of $V(Λ_i)\otimes V(-Λ_j)$ by the filtration $(F_n)_{n\geq0}$. The subsp…
We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for $U_q(\hat{\mathfrak{sl}}_e)$, generalising the LLT algorithm.
We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of $K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the integer level representation of the Ding-Iohara quantum algebra naturally produces the currents of the deformed $\mathcal{W}$ algebra. Then w…
Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_…
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov, and the Kac-Moody algebra A_l^(1) with the twisted algebra A_2l^(2). In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford supe…
The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one subspace V_1 is isomorphic to the irreducible highest weight \hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable \hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.
We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}. As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest-weights iΛ_1+(k-i)Λ_2, where 1 \leq i \leq k and Λ_1 and Λ_2 are fundamental weights of \hat{\goth{sl}(3)}.
Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
本文用两类三角超对称 R 矩阵构造对合差分算子，证明分次 R 矩阵满足结合 Yang-Baxter 方程，并借助 Polychronakos 冻结技巧得到 q 形变、可含各向异性的新型长程自旋链。
We give a simple characterization of the highest weight vertices in the crystal graph of the level l Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words classically used in the decomposition of tensor products of fundamental $\mathfrak{sl}_{n}$-modules. This yields a combinatorial decomposition of the Fock spaces in their irreducible components and the branching law for the restriction of the irreducible highest weight $\mathfrak{sl}_{\infty}$-modules to $\hat{\mathfrak{sl}_{e}}$.
Local Weyl modules over two-dimensional currents with values in $gl_r$ are deformed into spaces with bases related to parking functions. Using this construction we 1) propose a simple proof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; 2) describe the limits of Weyl modules in terms of semi-infinite forms and find the limits of characters; 3) propose a lower bound and state a conjecture for dimensions of Weyl modules with arbitrary highest weights. Also we express characters of deformed Weyl modules in terms of $ρ$-parking functions and …
We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.
We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra $\hat{\mathfrak{gl}}_N$ by the evaluation map.
In this paper we study general quantum affinizations $\U_q(\hat{\Glie})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Pressley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), t…
Consider any representation $ϕ$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $Λ(g)$. We show that the representation $ϕ$ extends canonically to the representation $\tildeϕ$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncomm…
