A construction of some ideals in affine vertex algebras.
A family of regular vertex operator algebras with two generators
For every m ∈ ℂ ∖ 0, −2 and every nonnegative integer k we define the vertex operator (super)algebra D m,k having two generators and rank . If m is a positive integer then D m,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D m,k is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.
Algèbres et équations non-linéaires
Combinatorial bases of modules for affine Lie algebra B 2(1)
We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases...
Drinfeld-Sokolov hierarchies on truncated current Lie algebras
In this paper we construct on truncated current Lie algebras integrable hierarchies of partial differential equations, which generalize the Drinfeld-Sokolov hierarchies defined on Kac-Moody Lie algebras.
Formal calculus and umbral calculus.
Invariant theory and the algebra with negative integral central charge
The vertex algebra with central charge may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer , it was conjectured in the physics literature that should have a minimal strong generating set consisting of elements. Using a free field realization of due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of ,...
Nonassociativity in VOA theory and finite group theory
We discuss some examples of nonassociative algebras which occur in VOA (vertex operator algebra) theory and finite group theory. Methods of VOA theory and finite group theory provide a lot of nonassociative algebras to study. Ideas from nonassociative algebra theory could be useful to group theorists and VOA theorists.
On Griess algebras.
On solutions of the KZ and qKZ equations at level zero
-Wakimoto modules and integral formulae of solutions of the quantum Knizhnik-Zamolodchikov equations.
Quasi-particle fermionic formulas for (k, 3)-admissible configurations
We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.
Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra
A Lie algebra is called a generalized Heisenberg algebra of degree n if its centre coincides with its derived algebra and is n-dimensional. In this paper we define for each positive integer n a generalized Heisenberg algebra 𝓗ₙ. We show that 𝓗ₙ and 𝓗 ₁ⁿ, the Lie algebra which is the direct product of n copies of 𝓗 ₁, contain isomorphic copies of each other. We show that 𝓗ₙ is an indecomposable Lie algebra. We prove that 𝓗ₙ and 𝓗 ₁ⁿ are not quotients of each other when n ≥ 2, but 𝓗 ₁ is a...
Sewn sphere cohomologies for vertex algebras
We define sewn elliptic cohomologies for vertex algebras by sewing procedure for coboundary operators.
The 3-state Potts model and Rogers-Ramanujan series
We explain the appearance of Rogers-Ramanujan series inside the tensor product of two basic A 2(2) -modules, previously discovered by the first author in [Feingold A.J., Some applications of vertex operators to Kac-Moody algebras, In: Vertex Operators in Mathematics and Physics, Berkeley, November 10–17, 1983, Math. Sci. Res. Inst. Publ., 3, Springer, New York, 1985, 185–206]. The key new ingredients are (5,6)Virasoro minimal models and twisted modules for the Zamolodchikov W 3-algebra.
The triplet vertex operator superalgebras: twisted sector.
The spectrum of coupled random matrices.
The Virasoro algebra and some exceptional Lie and finite groups.
Tree algebras: An algebraic axiomatization of intertwining vertex operators
We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over . We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over .
Vertex algebras and algebraic curves