Classification of Finite Irreducible Modules over the Lie Conformal Superalgebra CK 6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK ₆, for which E(1, 6) is the annihilation superalgebra.     

A BGG-Type Resolution for Tensor Modules over General Linear Superalgebra

We construct a Bernstein–Gelfand–Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector. Ngau Lam was partially supported by an NSC-grant 96-2115-M-006-008-MY3 of the ROC.     

Finite-Dimensional Half-Integer Weight Modules over Queer Lie Superalgebras

We give a new interpretation of representation theory of the finite-dimensional half-integer weight modules over the queer Lie superalgebra \({\mathfrak{q}(n)}\). It is given in terms of the Brundan’s work on finite-dimensional integer weight \({\mathfrak{q}(n)}\)-modules by means of Lusztig’s canonical basis. Using this viewpoint we compute the characters of the finite-dimensional half-integer weight irreducible modules. For a large class of irreducible modules whose highest weights are of special types (i.e., totally connected or totally disconnected) we derive closed-form character formulas that are reminiscent of the Kac–Wakimoto character formula for basic Lie superalgebras.     

Infinite-Dimensional Lie Superalgebras and Hook Schur Functions

 Making use of a Howe duality involving the infinite-dimensional Lie superalgebra and the finite-dimensional group GL _l of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of in terms of hook Schur functions. We use the reduction procedure of to to derive a character formula for a certain class of level 1 highest weight irreducible representations of, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra . These modules turn out to form the complete set of integrable -modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible -modules may be written as a sum of products of hook Schur functions. Received: 6 March 2002 / Accepted: 15 January 2003 Published online: 14 March 2003 RID="*" ID="*" Partially supported by NSC-grant 91-2115-M-002-007 of the R.O.C. RID="**" ID="**" Partially supported by NSC-grant 90-2115-M-006-015 of the R.O.C. Communicated by M. Aizenman     

Equivalence of Blocks for the General Linear Lie Superalgebra

We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category ${\mathcal{O}}$ for a general linear Lie superalgebra to an integral block of ${\mathcal{O}}$ for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of ${\mathcal{O}}$ .     

A Level-One Representation of the Quantum Affine Superalgebra \UqMN

A level-one representation of the quantum affine superalgebra? and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields. Character formulas of level-one irreducible highest weight modules of are conjectured. Received: 30 September 1996 / Accepted: 4 February 1997     

An admissible level $$\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) $$-model: modular transformations and the Verlinde formula

The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebra \(\widehat{{\mathfrak {osp}}} (1|2)\) at level \(-\frac{5}{4}\) are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.     

INHOMOGENEOUS DIFFERENTIAL REALIZATION,BOSON-FERMION REALIZATION OF LIE SUPERALGEBRA B(0,1) AND ITS REPRESENTATIONS

The differential realization of Lie superalgebra B(0,1) on the space of inhomogeneous polynomials,and the corresponding inhomogeneous Boson-Fermion realization are studiel.A new kind of indecomposable and irreducible representations of Lie superalgebra B(0,1) is studied on the universal enveloping algebra of Heisenberg-Weyl superalgebra,and on its subspaces and quotient spaces.All the finite dimensional irreducible representations are naturally obtained as special cases.     

One-Parameter Indecomposable and Irreducible Representations of gl(2 | 1) Superalgebra

Using inhomogeneous boson–fermion realization, one-parameter indecomposable and irreducible representations of the gl(2 | 1) superalgebra are studied on subspace and quotient spaces of the universal enveloping algebra of Heisenberg–weyl superalgebra. All the finite-dimensional irreducible representations of one-parameter of the gl(2 | 1) superalgebra are naturally obtained as special cases. The parameter has relation to the Hubbard interaction parameter U in the Hubbard model for correlated electrons.     

Orthosymplectic representations of Lie superalgebras

We investigate when an irreducible finite-dimensional representation of a Lie superalgebra is orthosymplectic. Then we turn to basic classical Lie superalgebras and give the conditions for orthosymplecticity in terms of Kac-Dynkin labels.     

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).     

Finite-dimensional irreducible representations of the quantum superalgebraU q [gl(n/1)]

It is shown that every finite-dimensional irreducible module over the general linear Lie superalgebragl(n/1) can be deformed to an irreducible module ofU _q [gl(n/1)], aq-analogue of the universal enveloping algebra ofgl(n/1). The results are extended also to all Kac modules, which in the atypical cases remain indecomposible. Within each module expressions for the transformations of the Gel'fand-Zetlin basis under the action of the algebra generators are written down. An analogoue of the Poincaré-Birkhoff-Witt theorem is formulated.     

Algebraic Bethe ansatz for gl(2, 1) invariant 36-vertex models

Four-dimensional irreducible representations of the superalgebra gl(2, 1) carry a freee parameter. We compute the spectra of the corresponding transfer matrices by means of the nested algebraic Bethe ansatz together with a generalized fusion procedure.     

and as Vertex Operator Extensions¶of Dual Affine Algebras

We discover a realisation of the affine Lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with “dual” levels (and an auxiliary level-1 algebra). The duality relation between the levels is . We construct the representation of on a sum of tensor products of , , and modules and decompose it into a direct sum over the spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs. Conversely, we show how the representations are constructed from representations. Received: 29 July 1999 / Accepted: 6 February 2000     

New Weyl groups forA 1 (1) and characters of singular highest weight modules

We consider singular Verma modules overA ₁ ⁽¹⁾ , i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group).     

Freely Generated Vertex Algebras and Non–Linear Lie Conformal Algebras

We introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.Acknowledgement. We would like to thank M. Artin, B. Bakalov, A. Dandrea and P. Etingof for useful discussions. This research was conducted by A. De Sole for the Clay Mathematics Institute. The paper was partially supported by the NSF grant DMS0201017.