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We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types E 6 (1) , E 7 (1) and E 8 (1). In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks. [ABSTRACT FROM AUTHOR]

In Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1-25), four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formulas for the refined BPS invariant. We give an algebraic proof of these identities using the formula for the universal R-matrix of the quantum affine algebra developed by Ito (Hiroshima Math. J. 40 (2010), 133-183), which yields various product presentations of the universal R-matrix by choosing various convex orders on an affine root system. By the uniqueness of the universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities, including the ones proposed in Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1-25), which turn out to correspond to convex orders of multiple row type. [ABSTRACT FROM AUTHOR]

We continue the study of prime simple modules for quantum affine algebras from the perspective of q-fatorization graphs. In this paper we establish several properties related to simple modules whose q-factorization graphs are afforded by trees. The two most important of them are proved for type A. The first completes the classification of the prime simple modules with three q-factors by giving a precise criterion for the primality of a 3-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose q-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general. [ABSTRACT FROM AUTHOR]

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal B (λ) for the quantum affine algebras of types A 2 n (2) , D n + 1 (2) , A 2 n − 1 (2) , D n (1) , B n (1) and C n (1) . We show that the crystal consisting of reduced Young walls is isomorphic to the crystal B (λ). Moreover, we provide a new realization of the crystal B (∞) in terms of reduced virtual Young walls and reduced extended Young walls. [ABSTRACT FROM AUTHOR]

Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of any untwisted quantum affine algebra U q (g ̂) has a cluster algebra structure. This makes it possible to compute q -characters of real prime simple modules associated to cluster variables as a result of a sequence of cluster mutations. In this paper, for minimal affinizations of type G2, we describe a cluster algebra algorithm to compute their q -characters and prove the geometric q -character formula conjectured by Hernandez and Leclerc. [ABSTRACT FROM AUTHOR]

The (q, t)-Cartan matrix specialized at t = 1 , usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type. In this paper, we study the (q, t)-Cartan matrix specialized at q = 1 , called the t-quantized Cartan matrix, and investigate the relations with (ii ′ ) its corresponding unipotent quantum coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type. [ABSTRACT FROM AUTHOR]

Let U q − be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on U q − induced from a diagram automorphism. Let U _ q − be the quantum algebra obtained from σ , and B ˜ (resp. B _ ˜) the canonical signed basis of U q − (resp. U _ q −). Assume that U q − is simply-laced of finite or affine type. In our previous papers [10] , [11] , we have proved by an elementary method, that there exists a natural bijection B ˜ σ ≃ B _ ˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases. [ABSTRACT FROM AUTHOR]

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra g , which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra U q ( g ′ ^) of the simply laced Lie algebra g ′ corresponding to g . Our construction is motivated by the analysis of the second classical limit of the deformed -algebra of g , which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of U q ( g ′ ^) . We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of U q (L g ^) , where L g ^ is the (twisted) affine Kac–Moody algebra Langlands dual to g ^ . We discuss the analogous structures in the Gaudin model which appears in the limit q → 1 . Finally, we describe a conjectural construction of the simple g -crystals in terms of the folded q-characters. [ABSTRACT FROM AUTHOR]

Let U ′ q (g) be a quantum affine algebra of untwisted affine A D E type, and C 0 g the Hernandez-Leclerc category of finite-dimensional U ′ q (g)-modules. For a suitable infinite sequence ˆ w 0 = ⋯ s i - 1 s i 0 s i 1 ⋯ of simple reflections, we introduce subcategories C [a,b] g of C 0 g for all a ⩽ b ∈ Z ⊔ {± ∞}. Associated with a certain chain C of intervals in [a,b], we construct a real simple commuting family M (C) in C [a,b] g, which consists of Kirillov-Reshetikhin modules. The category C [a,b]g provides a monoidal categorification of the cluster algebra K (C[a,b]g), whose set of initial cluster variables is [M(C)]. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on C - g by Hernandez-Leclerc since it is C [-∞,0] g, and is also applicable to C 0 g since it is C [-∞,∞]g. [ABSTRACT FROM AUTHOR]

We apply the theory of ϕ -coordinated modules, developed by H.-S. Li, to the Etingof–Kazhdan quantum affine vertex algebra associated with the trigonometric R-matrix of type A. We prove, for a certain associate ϕ of the one-dimensional additive formal group, that any (irreducible) ϕ -coordinated module for the level c ∈ C quantum affine vertex algebra is naturally equipped with a structure of (irreducible) restricted level c module for the quantum affine algebra in type A and vice versa. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra. [ABSTRACT FROM AUTHOR]

We construct a family of PBWD (Poincaré–Birkhoff–Witt–Drinfeld) bases for the quantum loop algebras U v (L sl n) , U v 1 , v 2 (L sl n) , U v (L sl (m | n)) in the new Drinfeld realizations. In the 2-parameter case, this proves (Hu et al. in Commun Math Phys 278(2):453–486, 2008, Theorem 3.11) (stated in loc. cit. without a proof), while in the super case it proves a conjecture of Zhang (Math. Z. 278(3–4):663–703, 2014). The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin and Odesskii (Anal. i Prilozhen 23(3):45–54, 1989; Anal i Prilozhen 31(3):57–70, 1997; Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000). NATO Sci Ser II Math Phys Chem, vol 35, Kluwer Academic Publishers, Dordrecht, pp 123–137, 2001). Our approach is similar to that of Enriquez (J Lie Theory 13(1):21–64, 2003) in the formal setting, but the key novelty is an explicit shuffle algebra realization of the corresponding algebras, which is of independent interest. This also allows us to strengthen the above results by constructing a family of PBWD bases for the RTT forms of those quantum loop algebras as well as for the Lusztig form of U v (L sl n) . The rational counterparts provide shuffle algebra realizations of type A (super) Yangians and their Drinfeld–Gavarini dual subalgebras. [ABSTRACT FROM AUTHOR]

Let U'q(g) be a quantum affine algebra of untwisted affine ADE type and let C0g be Hernandez-Leclerc's category. For a duality datum D in C0g, we denote by FD the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor FD sending simple modules to simple modules. Moreover, under the same condition, the functor FD has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in C0g, and show that all simple modules in C0g can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property. [ABSTRACT FROM AUTHOR]

Let g0 be a simple Lie algebra of type ADE and let U′q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g0) on the quantum Grothendieck ring Kt(g) of Hernandez-Leclerc's category C0g. Focused on the case of type AN-1, we construct a family of monoidal autofunctors {fi}i∈Z on a localization TN of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A∞. Under an isomorphism between the Grothendieck ring K(TN) of TN and the quantum Grothendieck ring Kt(A(1)N-1), the functors {fi}1≤i≤N-1 recover the action of the braid group B(AN-1). We investigate further properties of these functors. [ABSTRACT FROM AUTHOR]

Let U q − be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra g , and U _ q − the algebra corresponding to the orbit algebra of g obtained from a diagram automorphism σ on g. Let B σ be the set of σ -fixed elements in the canonical basis of U q − , and B _ the canonical basis of U _ q −. Lusztig proved that there exists a canonical bijection B σ ≃ B _ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where g is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence for PBW-bases, by using a new type of PBW-bases of U q − obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima. [ABSTRACT FROM AUTHOR]

The work is devoted to a probably new connection between deformed Virasoro algebra and quantum affine algebra sl 2 . We give an explicit realization of Virasoro current via the vertex operators of the level 1 integrable representations of quantum affine algebra sl 2 . The same is done for a twisted version of deformed Virasoro algebra. [ABSTRACT FROM AUTHOR]

Let Δ be the set of all quadruples (ν , μ , d , e) of integers that satisfy 0 ≤ D − d 2 ≤ ν ≤ μ ≤ D − d ≤ D , e + d + D is even , | e | ≤ 2 ν − D + d , d ∈ { e + D − 2 ν , min ⁡ { D − μ , e + D − 2 ν + 2 (N − 2 D) } }. In [6] , it is shown for the Terwilliger algebra T of the Grassmann scheme J q (N , D) , N ≥ 2 D , that the isomorphism classes of irreducible T -modules W are determined by their endpoint ν , dual endpoint μ , diameter d , and auxiliary parameter e , that come from the Leonard system attached to W , and it is claimed without proof that the quadruples (ν , μ , d , e) belong to Δ, if d ≥ 1. Let Λ be the set of triples (α , β , ρ) of non-negative integers that satisfy 0 ≤ α ≤ D − ρ 2 , 0 ≤ β ≤ N − D − ρ 2 , 0 ≤ α + β ≤ D − ρ. We construct a mapping from Λ to Δ which is bijective if N > 2 D and 2 : 1 if N = 2 D. We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T -modules, by embedding the standard module of J q (N , D) in a bigger space that allows a U q ( s l ˆ 2) -module structure [9]. As a byproduct we have the following: for a fixed ρ , 0 ≤ ρ ≤ D , set N ′ = N − 2 ρ , D ′ = D − ρ , and Λ ρ = { (α , β) | (α , β , ρ) ∈ Λ }. Then Λ ρ is precisely the set that parameterizes the isomorphism classes of irreducible T -modules for the Johnson scheme J (N ′ , D ′) [3]. [ABSTRACT FROM AUTHOR]

We introduce and investigate new invariants of pairs of modules M and N over quantum affine algebras Uq'(g) by analyzing their associated R-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable Uq'(g)-modules to become a monoidal categorification of a cluster algebra. [ABSTRACT FROM AUTHOR]

We introduce a sequence of q-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For 𝔤 = 𝔰 𝔩 2 ̂ , we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category 𝒪 of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into q-characters of simple modules in this category 𝒪 . [ABSTRACT FROM AUTHOR]

We consider the trigonometric classical r-matrix for gl N and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit q → 1 to elements of the Bethe subalgebra for the XXZ model. [ABSTRACT FROM AUTHOR]

We show how affine PBW bases can be used to construct affine MV polytopes, and that the resulting objects agree with the affine MV polytopes recently constructed using either preprojective algebras or KLR algebras. To do this we first generalize work of Beck-Chari-Pressley and Beck-Nakajima to define affine PBW bases for arbitrary convex orders on positive roots. Our results describe how affine PBW bases for different convex orders are related, answering a question posed by Beck and Nakajima. [ABSTRACT FROM AUTHOR]

In this paper we prove that every Kirillov–Reshetikhin module of type G 2 ( 1 ) and D 4 ( 3 ) has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al. [ABSTRACT FROM AUTHOR]

We construct a Young wall model for higher level -type adjoint crystals. The Young walls and reduced Young walls are defined in connection with affine energy function. We prove that the affine crystal consisting of reduced Young walls provides a realization of highest weight crystals B(λ) and B(∞). [ABSTRACT FROM AUTHOR]

It is known that the incidence algebra of a subspace lattice over a finite field with q elements is a homomorphic image of the quantum algebra U q 1 / 2 ( sl 2 ) . In this paper, we extend this situation. For a fixed proper subspace (which is an object of the subspace lattice), we define naturally a new algebra H which contains the incidence algebra as a proper subalgebra, and show how it is related to the quantum affine algebra U q 1 / 2 ( sl ˆ 2 ) . We show that there is an algebra homomorphism from U q 1 / 2 ( sl ˆ 2 ) to H , and that H is generated by its image together with the center. Moreover, we show that any irreducible H -module is also irreducible as a U q 1 / 2 ( sl ˆ 2 ) -module and is isomorphic to the tensor product of two evaluation modules. We also obtain a small set of generators of the center of H . [ABSTRACT FROM AUTHOR]

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 $$V_{c,1}$$ , $$c\ge 1$$ , associated with the principal subspaces $$W(c\Lambda _0)$$ of the integrable highest weight $$U_q (\widehat{\mathfrak {sl}}_2)$$ -modules $$L(c\Lambda _0)$$ . Using quantum integrability, we derive combinatorial bases for $$V_{c,1}$$ and compute the corresponding character formulae. [ABSTRACT FROM AUTHOR]

Let be an untwisted affine Kac-Moody Lie algebra. The top of every irreducible highest weight integrable -module is the finite-dimensional irreducible -module, where the action of the simple Lie algebra is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level Frenkel-Jing operators corresponding to Drinfeld realization of the quantum affine algebra . By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator , we obtain an infinite-dimensional vector space . Next, we introduce an associative algebra , a certain quantum analogue of the universal enveloping algebra , and construct some infinite-dimensional -modules corresponding to the finite-dimensional irreducible -modules . We show that the space carries a structure of an -module and, furthermore, we prove that the -module is isomorphic to the -module . [ABSTRACT FROM AUTHOR]

Let $$U'_q(\mathfrak {g})$$ be a twisted affine quantum group of type $$A_{N}^{(2)}$$ or $$D_{N}^{(2)}$$ and let $$\mathfrak {g}_{0}$$ be the finite-dimensional simple Lie algebra of type $$A_{N}$$ or $$D_{N}$$ . For a Dynkin quiver of type $$\mathfrak {g}_{0}$$ , we define a full subcategory $${\mathcal C}_{Q}^{(2)}$$ of the category of finite-dimensional integrable $$U'_q(\mathfrak {g})$$ -modules, a twisted version of the category $${\mathcal C}^{(1)}_{Q}$$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor $${\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}$$ , where $$\mathrm{Rep}(R)$$ is the category of finite-dimensional modules over the quiver Hecke algebra R of type $$\mathfrak {g}_{0}$$ with nilpotent actions of the generators $$x_k$$ . We show that $${\mathcal F}_{Q}^{(2)}$$ sends any simple object to a simple object and induces a ring isomorphism . [ABSTRACT FROM AUTHOR]

We consider two different methods of associating vertex algebraic structures with the level 1 principal subspaces for U q ( sl ˆ 2 ) . In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B.L. Feigin and A.V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal q _ -vertex algebras, investigate their properties and construct the nonlocal q _ -vertex algebra and its module, generated by Frenkel–Jing operator and Koyama's operator respectively. By finding the combinatorial bases of their suitably defined subspaces, we establish a connection with the sum sides of the Rogers–Ramanujan identities. Finally, we discuss further applications to quantum quasi-particle relations. [ABSTRACT FROM AUTHOR]

We describe a cluster algebra algorithm for calculating the q-characters of Kirillov-Reshetikhin modules for any untwisted quantum affine algebra Uq .bg/. This yields a geometric q-character formula for tensor products of Kirillov-Reshetikhin modules. When g is of type A;D;E, this formula extends Nakajima's formula for q-characters of standard modules in terms of homology of graded quiver varieties. [ABSTRACT FROM AUTHOR]

We introduce semi-infinite Lakshmibai–Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of the previous result due to Naito and Sagaki (which uses Littelmann's Lakshmibai–Seshadri paths), in which the level-zero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight. [ABSTRACT FROM AUTHOR]