Affine Lie Algebra Of Type Research Articles

Combinatorial bases of principal subspaces for the affine Lie algebra of type [formula omitted

We consider principal subspaces WL(kΛ0) and WN(kΛ0) of standard module L(kΛ0) and generalized Verma module N(kΛ0) at level k≥1 for the affine Lie algebra of type B2(1). By using the theory of vertex operator algebras, we find combinatorial bases of principal subspaces in terms of quasi-particles. From quasi-particle bases, we obtain character formulas for WL(kΛ0) and WN(kΛ0).

Combinatorial bases of modules for affine Lie algebra B 2(1)

Abstract We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

Combinatorial Bases of Feigin–Stoyanovsky's Type Subspaces of Level 2 Standard Modules for

Let be an affine Lie algebra of type and L(Λ) its standard module with a highest weight vector v Λ. For a given ℤ-gradation , we define Feigin–Stoyanovsky's type subspace as By using vertex operator relations for standard modules we reduce the Ponicaré–Brikhoff–Witt spanning set of W(Λ) to a basis and prove its linear independence by using Dong–Lepowsky intertwining operators.

Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types [formula omitted

Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions ( q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.

Modules for the core of extended affine Lie algebras of typeA1with coordinates in rank 2 quantum tori

We construct a Lie algebra L from rank 3 quantum tori and show that it is isomorphic to the core of extended affine Lie algebras of type A 1 . Then we construct two classes - which turn out to be exhaustive - of irreducible ℤ-graded highest weight L-modules and give necessary and sufficient conditions for these modules to have finite-dimensional homogeneous subspaces. As a consequence, we also determine all the irreducible ℤ-graded L-modules with nonzero center and finite-dimensional homogeneous subspaces.

Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A

Let L(− 1 (l+1),0) be the simple vertex operator algebra associated to an affine Lie algebra of typeA (1) with the lowest admissible halfinteger level − 1 (l + 1), for even l. We study the category of weak modules for that vertex operator algebra which are in category O as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra [formula omitted

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type A 1 ( 1 ) . A realization of the basic representation that is of “ D 2 ( 2 ) ”-type plays the central role.

Vertex operator algebra analogue of embedding of [formula omitted] into [formula omitted

Let L B ( − 5 2 , 0 ) (resp. L F ( − 5 2 , 0 ) ) be the simple vertex operator algebra associated with affine Lie algebra of type B 4 ( 1 ) (resp. F 4 ( 1 ) ) with the lowest admissible half-integer level − 5 2 . We show that L B ( − 5 2 , 0 ) is a vertex subalgebra of L F ( − 5 2 , 0 ) with the same conformal vector, and that L F ( − 5 2 , 0 ) is isomorphic to the extension of L B ( − 5 2 , 0 ) by its only irreducible module other than itself. We also study the representation theory of L F ( − 5 2 , 0 ) , and determine the decompositions of irreducible weak L F ( − 5 2 , 0 ) -modules from the category O into direct sums of irreducible weak L B ( − 5 2 , 0 ) -modules.

The sixth Painlevé equation arising fromD(1)4hierarchy

The sixth Painleve equation arises from a Drinfel'd-Sokolov hierarchy associated with the affine Lie algebra of type D_4 by similarity reduction.

Vertex operator algebras associated to type B affine Lie algebras on admissible half-integer levels

Let L ( n − l + 1 2 , 0 ) be the vertex operator algebra associated to an affine Lie algebra of type B l ( 1 ) at level n − l + 1 2 , for a positive integer n. We classify irreducible L ( n − l + 1 2 , 0 ) -modules and show that every L ( n − l + 1 2 , 0 ) -module is completely reducible. In the special case n = 1 , we study a category of weak L ( − l + 3 2 , 0 ) -modules which are in the category O as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.

Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395–412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A 2 ℓ ( 2 ) and of type D ℓ + 1 ( 2 ) . We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.

Weight multiplicities for affine Lie algebras of type [formula omitted] and Kostka numbers

In this paper, we prove the conjecture on the polynomial behavior of weight multiplicities for affine Lie algebras of type A r ( 1 ) , which was originally due to Benkart and Kass. More precisely, we prove that the degree of the weight multiplicity function m λ ( μ ) is equal to the depth associated with dominant integral weights λ and μ, and compute the leading coefficient of m λ ( μ ) explicitly in terms of a Kostka number. As applications, we verify other conjectures on the leading coefficient of m λ ( μ ) , and the polynomial behavior of weight multiplicities for classical Lie algebras of type A r .

Rectangular Schur functions and the basic representation of affine Lie algebras

An expression is given for the plethysm p 2 ∘ S □ , where p 2 is the power sum of degree two and S □ is the Schur function indexed by a rectangular partition. The formula can be well understood from the viewpoint of the basic representation of the affine Lie algebra of type A 2 ( 2 ) .

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A n (1) , we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.

Virasoro Algebra, Dedekind ?-function and Specialized Macdonald Identities

We motivate and prove a series of identities which form a generalization of the Euler pentagonal number theorem, and are closely related to specialized Macdonald identities for powers of the Dedekind η-function. More precisely, we show that what we call “denominator formula” for the Virasoro algebra has “higher analogue” for all cs,t-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as fusion and modular invariance that require all the irreducible modules of the series. In particular, in the case of c2,2k+1-minimal models we give a new proof of a family of specialized Macdonald identities associated with twisted affine Lie algebras of type A(2) 2k k ≥ 2 (i.e., BCk affine root system) which involve (2k2 - k)-th powers of the Dedekind η-function.

Similarity reduction of the modified Yajima–Oikawa equation

We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A_2^{(1)}. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom two, and admits a group of B\"acklund transformations isomorphic to the affine Weyl group of type A_2^{(1)}. We show that the system is equivalent to a two-parameter family of the fifth Painlev\'e equation.

Classification of Quantum Tori with Involution

AbstractQuantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type A1, C and BC. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type C.

Derivations and invariant forms of Jordan and alternative tori

Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types A 1 and A 2 . In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.

Classification of division Zn-graded alternative algebras

The octonion torus (or Cayley torus) appears as a coordinate algebra of extended affine Lie algebras of type A 2 and F 4 . A generalized version of the octonion torus, called division Z n -graded alternative algebras , is classified in this paper. Using the result, we can complete the classification of division ( A 2 , Z n ) -graded Lie algebras, up to central extensions, which are a generalization of the cores of extended affine Lie algebras of type A 2 .

Combinatorial R Matrices for a Family of Crystals: B(1)n, D(1)n, A(2)2n, and D(2)n + 1 Cases

For coherent families of crystals of affine Lie algebras of type B(1)n, D(1)n, A(2)2n, and D(2)n+1 we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux. This is a continuation of previous work by the authors (2000, in “Physical Combinatorics” (M. Kashiwara and T. Miwa, Eds.), Birkhäuser, Boston).