Quantum Affine Algebras Research Articles

Toroidal Grothendieck rings and cluster algebras

We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories \({\mathscr {C}}_1\) of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.

PBW theoretic approach to the module category of quantum affine algebras

Let $U_{q}'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}_{\mathfrak{g}}^{0}$ be Hernandez-Leclerc’s category. For a duality datum $\mathcal{D}$ in $\mathcal{C}_{\mathfrak{g}}^{0}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Moreover, under the same condition, the functor $\mathcal{F}_{\mathcal{D}}$ has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in $\mathcal{C}_{\mathfrak{g}}^{0}$, and show that all simple modules in $\mathcal{C}_{\mathfrak{g}}^{0}$ 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.

On the quantum affine vertex algebra associated with trigonometric R-matrix

We apply the theory of \(\phi \)-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 \(\phi \) of the one-dimensional additive formal group, that any (irreducible) \(\phi \)-coordinated module for the level \(c\in {\mathbb {C}}\) 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.

Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17], the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17], our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras

We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points HilbN(C2∕Γ) on a simple singularity C2∕Γ, where Γ is a finite subgroup of SL(2). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at ζ=exp(2π −1 2(h∨+1)) are always 1, which is a special case of an earlier conjecture by Kuniba. Here h∨ is the dual Coxeter number. We also prove the claim, which was not known for E7, E8 before.

Q-data and Representation Theory of Untwisted Quantum Affine Algebras

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $\Lambda(V,W)$ and $\Lambda^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.

Another admissible quantum affine algebra of type [formula omitted] with quantum Weyl group

The article is a continuation of Hu and Zhuang (2021), we construct another admissible quantum affine algebra Uq(sl̂2) of affine type A1(1) with different defining structural constants and variant q-Serre relations, its present formulae of the quantum root vectors are more involved than those in Hu and Zhuang (2021). We prove that as Hopf algebras, Uq(sl̂2) is neither isomorphic to the standard quantum affine algebra Uq(sl̂2) nor to the one Uq(sl̂2) constructed in Hu and Zhuang (2021). The new quantum affine algebra has also the quantum Weyl group as its automorphism subgroup, by which its quantum root vectors are well-characterized, and leads to a description of the Poincaré-Birkhoff–Witt basis in terms of the Chevalley generators.

Higher Level q-Oscillator Representations for $$U_q(C_n^{(1)}),U_q(C^{(2)}(n+1))$$ and $$U_q(B^{(1)}(0,n))$$

We introduce higher level $q$-oscillator representations for the quantum affine (super)algebras of type $C_n^{(1)},C^{(2)}(n+1)$ and $B^{(1)}(0,n)$. These representations are constructed by applying the fusion procedure to the level one $q$-oscillator representations which were obtained through the studies of the tetrahedron equation. We prove that these higher level $q$-oscillator representations are irreducible. For type $C_n^{(1)}$ and $C^{(2)}(n+1)$, we compute their characters explicitly in terms of Schur polynomials.

Correlation Functions of Quantum Toroidal $\mathfrak{gl}_1$ Algebra

In this paper, we study the correlation functions of the quantum toroidal $\mathfrak{gl}_1$ algebra. The first key properties we establish are similar to those of the correlation functions of quantum affine algebras $U_q\mathfrak{n}_+$ as established by Enriquez in (Eneiquez, 2000), while the proof of the remaining key ``vanishing property" relies on a certain ``Master Equality'' of formal series, which constitutes the main technical result of this paper.

Cluster realization of Weyl groups and q-characters of quantum affine algebras

We consider an infinite quiver $Q(\mathfrak{g})$ and a family of periodic quivers $Q_m(\mathfrak{g})$ for a finite dimensional simple Lie algebra $\mathfrak{g}$ and $m \in \mathbb{Z}_{>1}$. The quiver $Q(\mathfrak{g})$ is essentially same as what introduced by Hernandez and Leclerc for the quantum affine algebra. We construct the Weyl group $W(\mathfrak{g})$ as a subgroup of the cluster modular group for $Q_m(\mathfrak{g})$, in a similar way as what studied by the author, Ishibashi and Oya, and study its applications to the $q$-characters of quantum non-twisted affine algebras $U_q(\hat{\mathfrak{g}})$ introduced by Frenkel and Reshetikhin, and to the lattice $\mathfrak{g}$-Toda field theory. In particular, when $q$ is a root of unity, we prove that the $q$-character is invariant under the Weyl group action. We also show that the $A$-variables for $Q(\mathfrak{g})$ correspond to the $\tau$-function for the lattice $\mathfrak{g}$-Toda field equation.

Representations of Quantum Affine Algebras in their R-Matrix Realization

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.

A quantum cluster algebra approach to representations of simply laced quantum affine algebras

We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t-analogs of q-characters of simple modules of quantum affine algebras.

Diagram automorphisms and canonical bases for quantum affine algebras

Let Uq− 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 Bσ be the set of σ-fixed elements in the canonical basis of Uq−, and B_ the canonical basis of U_q−. Lusztig proved that there exists a canonical bijection Bσ≃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 Uq− obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.

Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

Drinfeld-type presentations of loop algebras

Let g \mathfrak {g} be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ \mu be a diagram automorphism of g \mathfrak {g} , and let L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) be the loop algebra of g \mathfrak {g} associated to μ \mu . In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] of L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) . The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] as special examples. As an application, when g \mathfrak {g} is of simply-laced-type, we prove that the classical limit of the μ \mu -twisted quantum affinization of the quantum Kac-Moody algebra associated to g \mathfrak {g} introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] .

Quantum geometry and \u03b8-angle in five-dimensional super Yang-Mills

Five-dimensional Sp(N) supersymmetric Yang-Mills admits a ℤ2 version of a theta angle θ. In this note, we derive a double quantization of the Seiberg-Witten geometry of mathcal{N} = 1 Sp(1) gauge theory at θ = π, on the manifold S1× ℝ4. Crucially, ℝ4 is placed on the Ω-background, which provides the two parameters to quantize the geometry. Physically, we are counting instantons in the presence of a 1/2-BPS fundamental Wilson loop, both of which are wrapping S1. Mathematically, this amounts to proving the regularity of a qq-character for the spin-1/2 representation of the quantum affine algebra {U}_qleft(hat{A_1}right) , with a certain twist due to the θ-angle. We motivate these results from two distinct string theory pictures. First, in a (p, q)-web setup in type IIB, where the loop is characterized by a D3 brane. Second, in a type I′ string setup, where the loop is characterized by a D4 brane subject to an orientifold projection. We comment on the generalizations to the higher rank case Sp(N) when N > 1, and the SU(N) theory at Chern-Simons level κ when N > 2.

Vertex representations of quantum N-toroidal algebras for type C

Quantum [Formula: see text]-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum [Formula: see text]-toroidal algebra for type [Formula: see text]. In particular, we also obtain a level-one module of the quantum toroidal algebra for type [Formula: see text] as a special case.

Quantum Grothendieck rings as quantum cluster algebras.

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type A, we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type A1, we identify remarkable relations in this quantum Grothendieck ring.

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A.