Related papers: Coset Vertex Operator Algebras and $\W$-Algebras

The positive part $U^+_q$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation by two generators $X,Y$ that satisfy the $q$-Serre relations. The $q$-Onsager algebra $\mathcal O_q$ has a presentation by two generators $A,B$ that satisfy…

When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain…

We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $\Lambda$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all…

In this paper, we first introduce the notion of twisted $\mathcal O$-operators on a Hom-Lie-Yamaguti algebra by a given $(2,3)$-cocycle with coefficients in a representation. We show that a twisted $\mathcal O$-operator induces a…

Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial…

By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided…

Let $\mathcal{T}_{+}(E)$ be the tensor algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. In earlier work, we showed that the completely contractive representations of $\mathcal{T}_{+}(E)$, whose restrictions to $M$ are…

In this paper we give an explicit formula for level 1 vertex operators related to $U_q(\widehat{sl}(n))$ as operators on the Fock spaces. We derive also their commutation relations. As an applications we culculate the one point functions of…

Recently a one-parameter family of deformed $ \mathcal{L} w_{1+\infty}$ soft symmetry algebras, denoted $ \mathcal{L}_{\Lambda}w_{1+\infty}$, acting on tree-level gravitational theories in AdS$_4$ has been discovered. Here we show that all…

In this article, we construct three new holomorphic vertex operator algebras of central charge $24$ using the $\mathbb{Z}_2$-orbifold construction associated to inner automorphisms. Their weight one subspaces has the Lie algebra structures…

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The…

In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the…

We classify all homogeneous odd (i.e., parity-reversing) Rota--Baxter operators of weight zero on the modified Witt-type Lie superalgebra $W = \langle L_m, G_n \rangle_{m,n\in\Z}$. Our classification shows that nontrivial such operators are…

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$ over $\mathbb{C}$, $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $\mathfrak{h}$ whose eigenvalues are…

We construct W-algebra generalizations of the ^sl(2) algebra -- W-algebras W^{(2)}_n generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky--Polyakov algebra. We define…

Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie…

Given a finite-dimensional reductive Lie algebra $\mathfrak{g}$ equipped with a nondegenerate, invariant, symmetric bilinear form $B$, let $V^k(\mathfrak{g},B)$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ and $B$…

In [8], the affine vertex algebra $L_k(\mathfrak{sl}_2)$ is realized as a subalgebra of the vertex algebra $Vir_c \otimes \Pi(0)$, where $Vir_c$ is a simple Virasoro vertex algebra and $\Pi(0)$ is a half-lattice vertex algebra. Moreover,…

The paper deals with weighted spaces $L_p^w(G)$ on a locally compact group G. If w is a positive measurable function on G then we define the space $L_p^w(G)$, $p\ge1$, as $L_p^w(G)=\{f:fw\in L_p(G)\}$. We consider weights such that these…

We show that a square zero, degree one element in W(V), the Weyl algebra on a vector space V, is equivalent to providing V with the structure of an algebra over the properad Cobar(coFrob), the properad arising from the cobar construction…