Toroidal Algebras 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 ) .

Schur-Weyl duality for toroidal algebras of type A

We state and prove an analog of the Schur-Weyl duality for a quotient of the classical 2-toroidal Lie algebra of type A. We then provide a method to extend this duality to the m-toroidal case, m > 2.

Tau-functions beyond the group elements

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a group element, i.e. satisfying the property Δ(X)=X⊗X, then their generating functions obey bilinear Hirota equations and hence are named τ-functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of τ-function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call generalized τ-functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain.

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.

More on affine Dynkin quiver Yangians

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to Seiberg duality of the quivers, and we investigate the relations of the dual quiver Yangians. We also mention the construction of the twisted quiver Yangians. It is conjectured that the truncations of the (twisted) quiver Yangians can give rise to certain \U0001d4b2-algebras. Incidentally, we give the screening currents of the \U0001d4b2-algebras in terms of the free field realization in the case of generalized conifolds. Moreover, we discuss the toroidal and elliptic algebras for any general quivers.

(q,t)-deformed (skew) Hurwitz τ-functions

We follow the general recipe for constructing commutative families of W-operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to a (q,t)-deformation of the earlier studied models. As before, a key role is played by an appropriate deformation of the cut-and-join rotation operator. We outline its expression both in terms of generators of the quantum toroidal algebra and in terms of the Macdonald difference operators.

Shifted Quantum Groups and Matter Multiplets in Supersymmetric Gauge Theories

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $$\mathfrak {gl}(1)$$ and quantum affine $$\mathfrak {sl}(2)$$ algebras (resp. denoted $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ and $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ ). It defines several new representations, including finite dimensional highest $$\ell $$ -weight representations for the toroidal algebra, and a vertex representation of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ acting on Hall–Littlewood polynomials. It also explores the relations between representations of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ and $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ in the limit $$q_1\rightarrow \infty $$ ( $$q_2$$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $${{\mathcal {N}}}=1$$ and 3d $${{\mathcal {N}}}=2$$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

Elliptic quantum toroidal algebra $$U_{q,t,p}({\mathfrak {gl}}_{1,tor})$$ and affine quiver gauge theories

Elliptic quantum toroidal algebra $$U_{q,t,p}({\mathfrak {gl}}_{1,tor})$$ and affine quiver gauge theories

A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES

AbstractWe categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

Quantum N-toroidal Algebras and Extended Quantized GIM Algebras of N-fold Affinization

We introduce the notion of quantum N-toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of N-fold affinization. We show that the quantum N-toroidal algebras are quotients of the extended quantized GIM algebras of N-fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras.

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).

The R-matrix of the quantum toroidal algebra

We consider the R-matrix of the quantum toroidal algebra of type gl1, both abstractly and in Fock space representations. We provide a survey of a certain point of view on this object which involves the elliptic Hall and shuffle algebras, and we show how to obtain certain explicit formulas.

Computing the R-matrix of the quantum toroidal algebra

We consider the problem of the R-matrix of the quantum toroidal algebra Uq,t(gl..1) in the Fock representation. Using the connection between the R-matrix R(u) (u being the spectral parameter) and the theory of Macdonald operators, we obtain explicit formulas for R(u) in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator, which completely describe the functional dependence of R(u) on the spectral parameter u. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with R(u) and thus gives another approach to the study of the poles of the R-matrix as a function of u.

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.

Quiver Hall–Littlewood functions and Kostka–Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

Quantum Toroidal Comodule Algebra of Type A<sub>n−1</sub> and Integrals of Motion

We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform description of deformed $W$ algebras associated with Lie (super)algebras of types BCD. We show that $\mathcal{K}_n$ possesses a family of commutative subalgebras.

Affine Laumon spaces and a conjecture of Kuznetsov

We prove a conjecture of Kuznetsov stating that the equivariant K-theory of affine Laumon spaces is the universal Verma module for the quantum affine algebra U_q(gl_n^). We do so by reinterpreting the action of the quantum toroidal algebra U_q(gl_n^^) on the K-theory from [14] in terms of the shuffle algebra studied in [12], which constructs an embedding of U_q(gl_n^) into U_q(gl_n^^)

Shifted quiver quantum toroidal algebra and subcrystal representations

Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations.In this paper, we define the shifted QQTA and study a class of their representations. We define 1d and 2d subcrystals of the original 3d crystal by removing a few arrows from the original quiver diagram and show how the shifted QQTA acts on them. We construct the 2d crystal representations from the 1d crystal representations by utilizing a generalized coproduct acting on different shifted QQTAs. We provide a detailed derivation of subcrystal representations of ℂ3, ℂ3/ℤn(n ≥ 2), conifold, suspended pinch point, and ℂ3/(ℤ2× ℤ2).

A note on quiver quantum toroidal algebra

Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian mathfrak{gl} 1. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations.In this paper, we propose a q-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge C. When it is trivial (C = 1), QQTA has a representation acting on the three-dimensional crystals, like Li-Yamazaki’s quiver Yangian. While we focus on the toric Calabi-Yau threefolds without compact 4-cycles, our analysis can likely be generalized to all toric Calabi-Yau threefolds.