Fusion products and toroidal algebras

Abstract

We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(\lambda)$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Feigin and Loktev. Moreover, we prove a conjecture by Chari, Fourier and Sagaki on truncated Weyl modules for certain classes of dominant integral weights and show that they are realizable as fusion products. In the last section we consider the case $\mathfrak{g}=\mathfrak{sl}_2$ and compute a PBW type basis for truncated Weyl modules of the associated current algebra

On truncated Weyl modules

“…For instance, it follows from the results of [16] for N = 2, g = sl n , and λ any multiple of a fundamental weight. Other cases for general N , with restrictions on λ or g were proved in [23,28]. Our main result (Theorem 2.3.2) extends the list of known cases for which the conjecture holds.…”

“…Motivated by this conjecture, it is usual to simplify notation and write V 1 * · · · * V k instead of V 1 a 1 * · · · * V k a k . This conjecture has been proved in some special cases (see [7,12,13,17,23, 26] and references therein). In all these special cases, each V j is a quotient of a graded local Weyl module and the cyclic generator v j is a highest-weight generator.…”

“…We let W (λ) and W N (λ) denote the graded Weyl module for g [t] = g ⊗ C[t] and g[t] N = g ⊗ C [t] t N C[t] , respectively, and refer to W N (λ) as a truncated Weyl module. One motivation for studying the truncated Weyl modules comes from a conjecture stated in [6] related to Schur positivity which, as seen in [16,23,28], can be rephrased as a conjecture on the realization of truncated Weyl modules as fusion products of certain irreducible modules. In particular, a positive answer for this conjecture leads to a way of computing the characters of truncated Weyl modules, which is still an open problem in general.…”

Representations of Lie superalgebras with Fusion Flags

“…This space is called the fusion product and is denoted by V t (λ 1 ) z 1 * · · · * V t (λ k ) * z k , where V t (λ) z is a non-graded t[t]-module. It was proved in [5,17,20] that certain (truncated) Weyl modules can be realized as fusion products of finite-dimensional irreducible t-modules. We say that a representation V has a fusion flag if there exists a sequence…”

Nonsymmetric Macdonald polynomials and PBW filtration: Towards the proof of the Cherednik–Orr conjecture

“…To give the precise formulation of the conjecture, recall that the Demazure modules W λ are cyclic modules for the current algebra g ⊗ C[t] with w λ being the cyclic vector. The PBW filtration on the universal enveloping algebra of the current algebra induces the increasing filtration F s on the Demazure module (the PBW filtration on the representations of affine algebras is studied in [8,22]). Each space of this filtration is invariant with respect to the Cartan subalgebra and the associated graded space W gr λ is bi-graded by the operators d and by the PBW-grading operator D. We have the PBW character ch PBW W λ = k,s≥0 q k p s ch{v ∈ F s /F s−1 , dv = kv}.…”