Quantum Toroidal Research Articles

Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group

AbstractWe first construct an action of the extended double affine braid group $$\mathcal {\ddot{B}}$$ B ¨ on the quantum toroidal algebra $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $$\mathcal {\ddot{B}}$$ B ¨ we produce automorphisms and anti-involutions of $$U_{q}(\mathfrak {g}_{\textrm{tor}})$$ U q ( g tor ) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and $$k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$$ k 0 a 0 ⋯ k n a n up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of $$U_{q}(\mathfrak {sl}_{n+1,\textrm{tor}})$$ U q ( sl n + 1 , tor ) .

Representations of quantum toroidal superalgebras and plane s-partitions

We construct Fock and MacMahon modules for the quantum toroidal superalgebra [Formula: see text] associated with the Lie superalgebra [Formula: see text] and parity [Formula: see text]. The bases of the Fock and MacMahon modules are labeled by super-analogs of partitions and plane partitions with various boundary conditions, while the action of generators of [Formula: see text] is given by Pieri-type formulas. We study the corresponding characters.

Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

Commutative subalgebra of a shuffle algebra associated with quantum toroidal [formula omitted

We define and study the shuffle algebra Shm|n of the quantum toroidal algebra Em|n associated to Lie superalgebra glm|n. We show that Shm|n contains a family of commutative subalgebras Bm|n(s) depending on parameters s=(s1,…,sm+n), ∏isi=1, given by appropriate regularity conditions. We show that Bm|n(s) is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the s-weighted R-matrix computed on the degree zero part of Em|n modules of levels ±1.

On pentagon identity in Ding-Iohara-Miki algebra

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general “master pentagon identity” for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We prove this curious identity and discuss its implications.

Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras in the spirit of Feigin and Odesskii (Funktsional. Anal. Prilozhen. 31(3):57–70, 1997), thus generalizing its rational version of Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) and the type A construction of Finkelberg and Tsymbaliuk (Arnold Math. J. 5(2–3):197–283, 2019). As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of Di Francesco and Kedem (Commun. Math. Phys. 369(3):867–928, 2019).

Gauge/Bethe correspondence from quiver BPS algebras

We study the Gauge/Bethe correspondence for two-dimensional mathcal{N} = (2, 2) supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the R-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional mathcal{N} = 2 gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature.

On Hopf algebraic structures of quantum toroidal algebras

ABSTRACT We define an algebra using a simplified set of generators for the quantum toroidal algebra and show that there exists an epimorphism from to . We derive a closed formula of the comultiplication on the generators of that extends that of the quantum affine algebra . As a consequence, we show that is a Hopf algebra for n = 1, 2 and give conjectural formulas in the general case. We further show that is isomorphic to a double algebra.

Correlation Functions of Quantum Toroidal gln Algebra

In this paper, we study the correlation functions of the quantum toroidal \mathfrak{gl}_n algebra. Their first key properties are established in analogy to those of the correlation functions of quantum affine algebras U_q\mathfrak{n}_+. The core of the paper is the proof of the vanishing of those correlation functions at length 3 wheel conditions, which is done separately for n = 1, n = 2, n≥ 3 cases with different ``Master Equalities" of formal series (the n = 1 case has been previously discussed by the author in (Cui, 2021)). Another important contribution is the discovery of cubic Serre relations for the quantum toroidal.

Deformations of $$\mathcal {W}$$ algebras via quantum toroidal algebras

We study the uniform description of deformed \(\mathcal {W}\) algebras of type \(\textsf {A}\) including the supersymmetric case in terms of the quantum toroidal \({\mathfrak {g}}{\mathfrak {l}}_1\) algebra \({{\mathcal {E}}}\). In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra \(\mathcal {K}\) over \({{\mathcal {E}}}\) which gives a uniform construction of basic deformed \(\mathcal {W}\) currents and screening operators in types \(\textsf {B},\textsf {C},\textsf {D}\) including twisted and supersymmetric cases. We show that a completion of algebra \(\mathcal {K}\) contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except \(\textsf {D}^{(2)}_{\ell +1}\). We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

Quantum W1+∞ subalgebras of BCD type and symmetric polynomials

The infinite affine Lie algebras of type ABCD, also called gl̂(∞), ô(∞), and sp̂(∞), are equivalent to subalgebras of the quantum W1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (Â∞), a Majorana fermion (B̂∞ and D̂∞), or a symplectic boson (Ĉ∞). Explicit formulas for the action of the quantum W1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B̂∞, a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.

Quantum toroidal and shuffle algebras

In this paper, we prove that the quantum toroidal algebra Uq,q‾(gl¨n) is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R−matrix of the quantum toroidal algebra.

New quantum toroidal algebras from 5D mathcal{N} = 1 instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D mathcal{N} = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S1× ℂ2/ℤp where the action of ℤp is determined by two integer parameters (ν1, ν2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of mathfrak{gl} (p). We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of mathfrak{gl} (p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν1, ν2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.

Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

We introduce and study the quantum toroidal algebra $\mathcal{E}_{m|n}(q_1,q_2,q_3)$ associated with the superalgebra $\mathfrak{gl}_{m|n}$ with $m\neq n$, where the parameters satisfy $q_1q_2q_3=1$. We give an evaluation map. The evaluation map is a surjective homomorphism of algebras $\mathcal{E}_{m|n}(q_1,q_2,q_3) \to \widetilde{U}_q\,\widehat{\mathfrak{gl}}_{m|n}$ to the quantum affine algebra associated with the superalgebra $\mathfrak{gl}_{m|n}$ at level $c$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^{m-n}=c^2$. We also give a bosonic realization of level one $\mathcal{E}_{m|n}(q_1,q_2,q_3)$-modules.

Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sln, denoted by Uq1,q2,q3(n) and Yh1,h2,h3(n), respectively. Our motivation arises from the milestone work [11], where a similar relation between the quantum loop algebra Uq(Lg) and the Yangian Yh(g) has been established by constructing an isomorphism of C[[ħ]]-algebras Φ:Uˆexp⁡(ħ)(Lg)⟶∼Yˆħ(g) (with ˆ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of h=0. The same construction can be applied to the toroidal setting with qi=exp⁡(ħi) for i=1,2,3 (see [11,22]). In the current paper, we are interested in the more general relation: q1=ωmneh1/m,q2=eh2/m,q3=ωmn−1eh3/m, where m,n≥1 and ωmn is an mn-th root of 1. Assuming ωmnm is a primitive n-th root of unity, we construct a homomorphism Φm,nωmn between the completions of the formal versions of Uq1,q2,q3(m) and Yh1/mn,h2/mn,h3/mn(mn).

(q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody {U}_q{left(widehat{mathfrak{g}}right)}_k to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_1right) and {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_nright) when the screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations.Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type An. The correlation functions of these operators satisfy the (q, t)-Knizhnik-Zamolodchikov (KZ) equation, which features the ℛ-matrix. The matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly.We also present an important application of the DIM formalism to the study of 6d gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic (q, t)-KZ equations.

Extremal loop weight modules and tensor products for quantum toroidal algebras

ABSTRACTWe define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin–Jimbo–Miwa–Mukhin [7] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal mathfrak{g}{mathfrak{l}}_1 ). Webs of DIM representations are in correspondence with (p, q)-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the mathcal{N}=1 quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.

Classical limits of quantum toroidal and affine Yangian algebras

In this short article, we compute the classical limits of the quantum toroidal and affine Yangian algebras of sln by generalizing our arguments for gl1 from [7] (an alternative proof for n>2 is given in [10]). We also discuss some consequences of these results.

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

We construct by fusion product new families of irreducible representations of the quantum affinization $\mathcal {U}_{q}(\hat {sl}_{\infty })$ . The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $\mathcal {U}_{q}(sl_{n+1}^{tor})$ : we prove the conjectural link between $\mathcal {U}_{q}(\hat {sl}_{\infty })$ and $\mathcal {U}_{q}(sl_{n+1}^{tor})$ stated in Hernandez (J. Algebra 329, 147–162, 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv: 1305.3481 ).