We prove that the quiver problem is NP complete.

In this paper we classify degenerate Verma modules over the linearly compact Lie superalgebra E(4,4). This completes the description of Verma modules over the exceptional linearly compact Lie superalgebras. As in the other cases all degenerate modules and morphisms between them give rise to infinite bilateral complexes which may be viewed as a generalization of de Rham complexes.

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big N=4 superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal W-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big N=4 superconformal algebra.

We find modular transformations of normalized characters for the following W-algebras: (a) W_k^min(𝔤), where 𝔤=D_n (n≥ 4), or E_6, E_7, E_8, and k is a negative integer ≥ -2 , or ≥ -h^∨/6-1 , respectively; (b) quantum Hamiltonian reduction of the 𝔤̂ -module L(k Λ _0) , where 𝔤 is a simple Lie algebra, f is its non-zero nilpotent element, and k is a principal admissible level with the denominator u>θ (x) , where 2x is the Dynkin characteristic of f, and θ is the highest root of 𝔤 . We prove that these vertex algebras are modular invariant. A conformal vertex algebra V is called modular invariant if its character tr_V q^L_0-c/24 converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of V is important since, in particular, conjecturally it implies that V is simple, and that V is rational, provided that it is lisse.

Simplicity of universal minimal quantum affine W-algebras is studied. As an application, we find the values of the center, for which the vacuum module over a superconformal algebra is irreducible.

The theory of admissible modules over symmetrizable anisotropic Kac-Moody superalgebras, introduced by Kac and Wakimoto in late 80's, is a well-developed subject with many applications, including representation theory of vertex algebras. Recently this theory was developed in a more general setup by Gorelik and Serganova. In the present paper we develop in this more general setup the theory of admissible modules over arbitrary symmetrizable Kac-Moody superalgebras.

We prove that the span of normalized characters of subprincipal admissible modules over an affine Lie algebra of subprincipal admissible level k is SL_2(𝐙)-invariant and find the explicit modular transformation formula.

We discuss the theory of Poisson vertex algebras and their generalizations in relation to integrability of Hamiltonian PDE. In particular, we discuss the theory of affine classical W-algebras and apply it to construct a large class of integrable Hamiltonian PDE. We also discuss non commutative Hamiltonian PDE in the framework of double PVA, and differential-difference Hamiltonian equations in the framework of multiplicative PVA.

Using spectral flow, we provide a proof of [9, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of minimal W-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [9]). When 𝔤=spo(2|2n), F(4), D(2,1;m/n), it is also proven that the unitarity of extremal (=massless) representations of the minimal W-algebra W^k_min(𝔤) in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.

We study the embeddings of the exceptional infinite-dimensional Lie superalgebra E(1, 6) in the exceptional Lie superalgebras E(5, 10) and E(4, 4). These questions arose in recent works on enhanced symmetries in some supersymmetric theories by N. Garner, S. Raghavendran, I. Saberi and B. Williams.

We develop the notion of a (pro-) conformal pseudo operad and apply it to the construction of the basic cohomology complex of a vertex algebra. The paper heavily uses the ideas and constructions of the work of Tamarkin [Tam02].

The theory of triples of Poisson brackets and related integrable systems, based on a classical R-matrix R in End_F(g), where g is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel-Ragnisco and Li-Parmentier [OR89,LP89]. In the present paper we develop an "affine" analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson lambda-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.

We prove that any unitary highest weight module over a universal minimal quantum affine $W$-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine $W$-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal $W$-algebra is unitary.

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4), for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

We give a review of the B-type Kadomtsev–Petviashvili (BKP) hierarchy and find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada–Kotera hierarchy). The name comes from the fact that, for n=3, the simplest equation of the hierarchy is the famous Sawada–Kotera equation.

Let g be a simple finite dimensional complex Lie algebra and let g be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a simple highest weight g-module are either bounded or are given by a linear function of the weight. We explain and generalize this observation using Kazhdan-Lusztig theory, by computing values at q = 1 of certain (parabolic) affine inverse KazhdanLusztig polynomials. In particular, we obtain explicit character formulas for some g-modules of negative integer level k when g is of type D-n, E-6, E-7, E-8 and k >= -2, -3, -4, -6 respectively, as conjectured by Kac and Wakimoto. The calculation relies on the explicit description of the canonical basis in the cell quotient of the anti -spherical module over the affine Hecke algebra corresponding to the subregular cell. We also present an explicit description of the corresponding objects in the derived category of equivariant coherent sheaves on the Springer resolution, they correspond to irreducible objects in the heart of a certaint-structure related to the so called non -commutative Springer resolution.

To a positive-definite even lattice Q, one can associate the lattice vertex algebra VQ, and any automorphism σ of Q lifts to an automorphism of VQ. In this paper, we investigate the orbifold vertex algebra V , which consists of the elements of VQ fixed under σ, in the case when σ has prime order. We describe explicitly the irreducible V -modules, compute their characters, and determine the modular transformations of characters. As an application, we find the asymptotic and quantum dimensions of all irreducible V -modules. We consider in detail the cases when the order of σ is 2 or 3, as well as the case of permutation orbifolds.

We begin a systematic study of unitary representations of minimal W-algebras. In particular, we classify unitary minimal W-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the KP hierarchy: by a generalized Jacobi-Trudy formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its n-reductions. In particular, for n=3 we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.

Foundations of the theory of vertex algebras are extended to the non-Archimedean setting.