Toroidal Algebras Research Articles

A not-so-simple Lie bracket expansion

Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form. Lie algebras and quantum groups are not topics that you are apt to hear undergraduates math majors discussing in their spare time. However, there are a surprising number of nontrivial questions in this area that are undergraduate appropriate. In this paper, we will give a brief overview of the broad mathematical setting, and then discuss an accessible problem that involves expanding a nested set of brackets. Lie algebras, their universal enveloping algebras and quantum groups are a fundamental part of representation theory that have many applications within mathematics and mathematical physics. Lie algebras and Lie groups were originally discovered by Sophus Lie in the late nineteenth century [Borel 2001]. Given a Lie algebra, we associate a unique associative algebra called the universal enveloping algebra. In 1985, Jimbo and Drinfeld discovered q-analogues of these universal enveloping algebras called “quantum groups”, which have been a recent area of study (see [Lusztig 1993]). In order to find the quantum analogue of a Lie algebra it is often desirable to understand the defining relationships of the Lie algebra inside of its universal enveloping algebra. The motivation for this project came from a paper by Lum in which he gives a nice presentation of a toroidal Lie algebra that could be useful in understanding this Lie algebra’s quantum group [Lum 1998]. All of these relations utilize a nested set of brackets called t.k/. For simplicity, we have modified t.k/ by a scalar. In this paper we seek to understand the expansion of this object.

The Fermion Representation of Quantum Toroidal Algebra on 3D Young Diagrams

We develop an equivalence between the diagonal slices and the perpendicular slices of 3D Young diagrams via Maya diagrams. Furthermore, we construct the fermion representation of quantum toroidal algebra on the 3D Young diagrams perpendicularly sliced.

Twisted Quantum Toroidal Algebras $${T_q^-(\mathfrak g)}$$ T q - ( g )

We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.

Fermion Representation of Quantum Toroidal Algebra

We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.

Characters of representations of the quantum toroidal algebra : Plane partitions with “stands”

An expression for the generating function of plane partitions a i,j subject to the constraints a m,n = 0 and a i,j ⩾ k j , 1 ⩽ j ⩽ n, which is the character of an irreducible representation of the quantum toroidal algebra Open image in new window , is obtained.

Quantum extremal loop weight modules and monomial crystals

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A_1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.

The composition Hall algebra of a weighted projective line

Abstract In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of the quantized enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types D4(1, 1), E6(1, 1), E7(1, 1) and E8(1, 1). In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

The quantum toroidal algebra $$\widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}$$ : Calculation of characters of some representations as generating functions of plane partitions

The generating function of plane partitions {ai,j} subject to the constraint am,n = 0 is expressed and calculated as the character of an irreducible representation of the quantum toroidal algebra \(\widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}\) in the case K = q1mq2n.

Toroidal vertex algebras and their modules

We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal Lie algebras.

Two-parameter quantum vertex representations via finite groups and the McKay correspondence

We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of SL2(C) via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of ADE types.

Automorphism Group of a Class of Gradation Shifting Toroidal Lie Algebras

Let [Formula: see text] be the ring of Laurent polynomials in commuting variables. As a generalization of the toroidal Lie algebra, the gradation shifting toroidal Lie algebra [Formula: see text] is isomorphic to the corresponding (centerless) toroidal Lie algebra so(n, ℂ) ⨂ A of type B or D as a vector space, with the Lie bracket twisted by n fixed elements E1,…,En from A. In this paper, we study the automorphisms of the gradation shifting toroidal algebra [Formula: see text], which is proved to be closely related to a class of subgroups of GL(n,ℤ), called the linear groups over semilattices. We use the linear group over a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra [Formula: see text], which extends our earlier work.

Irreducible modules for extended affine Lie algebras

We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.

The algebra [formula omitted] and applications

In this note we consider the algebra U q ( sl ˆ ∞ ) and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras U q ( sl n + 1 tor ) , more precisely predictions of character formulae for representations of U q ( sl n + 1 tor ) . In this context, we state a general positivity conjecture for representations of U q ( sl ˆ ∞ ) viewed as representations of U q ( sl n + 1 tor ) , that we prove for Kirillov–Reshetikhin modules.

Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory.

K-THEORY OF QUIVER VARIETIES, q-FOCK SPACE AND NONSYMMETRIC MACDONALD POLYNOMIALS

We have two constructions of the level-(0, 1) irreducible representation of the quantum toroidal algebra of type A. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type ˆ A. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using the nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the q uiver varieties.

Quantum toroidal algebras and their representations

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac–Moody algebras by using the Drinfeld quantum affinization process. They are quantum group analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to which they are related via Schur–Weyl duality. In this review paper, we give a glimpse of some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate the theory with several examples. We also announce new results and explain possible further developments, in particular on finite-dimensional representations at roots of unity.

On the toroidal Leibniz algebras

Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras ˙ g⊗C(t ±1 1 ,...,t ±1 ν ) in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.

Complex multiplication, rationality and mirror symmetry for Abelian varieties

We show that complex multiplication on Abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an Abelian variety of CM-type to be of CM-type. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.

Drinfeld coproduct, quantum fusion tensor category and applications

The class of quantum affinizations includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a coproduct (the Drinfeld coproduct) which does not produce tensor products of modules in the usual way because it is defined in a completion. In this paper we propose a new process to produce quantum fusion modules from it : for all quantum affinizations, we construct by deformation and renormalization a new (non semi-simple) tensor category Mod. For quantum affine algebras this process is new and different from the usual tensor product. For general quantum affinizations, for example for toroidal algebras, so far, no process to produce fusion modules was known. We derive several applications from it : we construct the fusion of (finitely many) arbitrary l-highest weight modules, and prove that it is always cyclic. We establish exact sequences involving fusion of Kirillov-Reshetikhin modules related to new T-systems. Eventually for a large class of quantum affinizations we prove that the subcategory of finite length modules of Mod is stable under the new monoidal bifunctor.