Mathematics

We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

The K -type formulas of the space of K -finite solutions to the Heisenberg ultrahyperbolic equation ??s f=0 for the non-linear group SL ? (3,R) are classified. This completes a previous study of Kable for the linear group SL(m,R) in the case of m=3 , as well as generalizes our earlier results on a certain second order differential operator. As a by-product we also show several properties of certain sequences { P j (x;y) } ??j=0 and { Q j (x;y) } ??j=0 of tridiagonal determinants, whose generating functions are given by local Heun functions. In particular, it is shown that these sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palindromic property. We further show that classical sequences of Cayley continuants { Cay j (x;y) } ??j=0 and Krawtchouk polynomials { K j (x;y) } ??j=0 also admit this property. In the end a new proof of Sylvester's formula for certain tridiagonal determinant Sylv(x;n) is provided from a representation theory point of view.

Given a Lie superalgebra g with a subalgebra g ≥0 , and a finite-dimensional irreducible g ≥0 -module F , the induced g -module M(F)=U(g) ⊗ U( g ≥0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra g=E(5,10) with the subalgebra g ≥0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F) . Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as 'exceptional' de Rham complexes for E(5,10) .

In this paper we classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. In both the cases we classify modules, when a part of center acts non-trivially and trivially on modules.

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus LT which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial LT -action and where the centre of the underlying Lie algebra acts trivially.

In this paper, we give a new approach to classify all simple Harish-Chandra modules for the N=1 Ramond algebra based on the so called A-cover theory developed in \cite{BF}

In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra k ˆ and its contact subalgebra k + . Based on these results, we give a classification of simple Harish-Chandra modules for k ˆ and k + .

We classify all simple strong Harish-Chandra modules for the Lie superalgebra W(m,n) . We show that every such module is either strongly cuspidal or a module of the highest weight type. We construct tensor modules for W(m,n) , which are parametrized by simple finite-dimensional gl(m,n) -modules and show that every simple strongly cuspidal W(m,n) -module is a quotient of a tensor module. Finally, we realize modules of the highest weight type as simple quotients of the generalized Verma modules induced from tensor modules for W(m−1,n) .

We classify n -hereditary monomial algebras in three natural contexts: First, we give a classification of the n -hereditary truncated path algebras. We show that they are exactly the n -representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n -hereditary quadratic monomial algebras. In the case n=2 , we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating A 3 ??k A 3 , where A 3 is the Dynkin quiver of type A with bipartite orientation. In the case n?? , we show that the only n -representation finite algebras are the n -representation-finite Nakayama algebras with quadratic relations.

The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to sl 3 and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arXiv:1005.0185, arXiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral k are always rational in category O , whilst they always admit nonsemisimple relaxed highest-weight modules unless k+ 3 2 ∈ Z ≥0 .