Double-loop algebras and the Fock space

Varagnolo, Michela; Vasserot, Éric

doi:10.1007/s002220050242

Cited by 62 publications

(68 citation statements)

References 9 publications

(6 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is called a double loop algebra, or toroidal algebra, and has been studied by various people; see for example [22,48,49,56] and the references therein. A first step to a geometric approach to the toroidal algebra using quiver varieties for the affine Dynkin graph of type A was given by M. Varagnolo and E. Vasserot [57].…”

Section: Res M (P ) = L(w) ⊕ L(0)mentioning

confidence: 99%

Quiver varieties and finite dimensional representations of quantum affine algebras

Nakajima

2000

J. Amer. Math. Soc.

336

371

View full text Add to dashboard Cite

We study finite dimensional representations of the quantum affine algebra U q ( g ^ ) {\mathbf {U}}_q(\widehat {\mathfrak {g}}) using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

show abstract

Section: Res M (P ) = L(w) ⊕ L(0)mentioning

confidence: 99%

Quiver varieties and finite dimensional representations of quantum affine algebras

Nakajima

2000

J. Amer. Math. Soc.

336

371

View full text Add to dashboard Cite

show abstract

“…All other matrix elements of R W W (z) and R W W * (z) vanish. The matrix elements of R W * W (z) and R W * W * (z) are obtained from (26) and (27) via…”

Section: The Modelmentioning

confidence: 99%

An integrable -model: corner transfer matrices and Young skew diagrams

Gade¹

2004

Nuclear Physics B

View full text Add to dashboard Cite

The path space of an inhomogeneous vertex model constructed from the vector representation of Uq gl(2|2) and its dual is studied for various choices of composite vertices and assignments of gl(2|2)-weights. At q = 0, the corner transfer matrix Hamiltonian acts trigonally on the space of half-infinite configurations subject to a particular boundary condition. A weight-preserving oneto-one correspondence between the half-infinite configurations and the weight states of a level-one module of Uq sl(2|2) /H with grade −n is found for n ≥ −3 if the grade −n is identified with the diagonal element of the CTM Hamiltonian. In each case, the module can be decomposed into two irreducible level-one modules, one of them including infinitely many weight states at fixed grade. Based on a mapping of the path space onto pairs of border stripes, the character of the reducible module is decomposed in terms of skew Schur functions. Relying on an explicit verification for simple border stripes, a correspondence between the paths and level-zero modules of Uq sl(2|2) constructed from an infinite-dimensional Uq gl(2|2) -module is conjectured.

show abstract

“…The following formulas taken from [Varagnolo and Vasserot 1998] help one to understand better the bracket on sl n (A). It is generally difficult to compute explicitly the bracket with respect to the decomposition sl n (A) ∼ = sl n (A)⊕HC 1 (A), but on can get some nice formulas by using a different splitting of sl n (A).…”

Section: Matrix Lie Algebras Over Ringsmentioning

confidence: 99%

“…For m 1 , m 2 ∈ sl n , a 1 , a 2 ∈ A, and ( · , · ) the Killing form on sl n , set Varagnolo and Vasserot 1998]. The Lie algebra sl n (A) is isomorphic to the vector space sl n ⊗ A ⊕ A, A endowed with the bracket…”

Section: Matrix Lie Algebras Over Ringsmentioning

confidence: 99%

Double affine Lie algebras and finite groups

Guay¹,

Hernandez²,

Loktev³

2009

Pacific J. Math.

View full text Add to dashboard Cite

We begin to study the Lie theoretical analogues of symplectic reflection algebras for a finite cyclic group ; we call these algebras "cyclic double affine Lie algebras". We focus on type A: In the finite (respectively affine, double affine) case, we prove that these structures are finite (respectively affine, toroidal) type Lie algebras, but the gradings differ. The case that is essentially new is sl n (‫ [ރ‬u, v] ). We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra A 1 instead of the polynomial ring ‫ [ރ‬u, v], and, more generally, a rank one rational Cherednik algebra. We study quasifinite highest weight representations of these Lie algebras.

show abstract

Double-loop algebras and the Fock space

Cited by 62 publications

References 9 publications

Quiver varieties and finite dimensional representations of quantum affine algebras

Quiver varieties and finite dimensional representations of quantum affine algebras

An integrable -model: corner transfer matrices and Young skew diagrams

Double affine Lie algebras and finite groups

Contact Info

Product

Resources

About