scholarly journals A SMASH PRODUCT CONSTRUCTION OF NONLOCAL VERTEX ALGEBRAS

2007 ◽  
Vol 09 (05) ◽  
pp. 605-637 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Lie Algebras ◽  
Suitable Condition ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Smash Product ◽  
Module Algebra ◽  
Primitive Elements ◽  
Product Construction

A notion of vertex bialgebra and a notion of nonlocal vertex module-algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra V satisfying a suitable condition, a canonical bialgebra B(V) is constructed such that primitive elements of B(V) are essentially pseudo-derivations and group-like elements are essentially pseudo-endomorphisms. As an application, vertex algebras associated with the Heisenberg Lie algebras as well as those associated with the nondegenerate even lattices are reconstructed through smash products, and furthermore, a different approach to the construction of modules for the lattice vertex algebras is given.

Strongly graded vertex algebras generated by vertex Lie algebras

2019 ◽  
Vol 21 (08) ◽  
pp. 1850069
Author(s):  
Yufeng Pei ◽  
Jinwei Yang
Keyword(s):  
Lie Algebras ◽  
Category Theory ◽  
Tensor Category ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Intertwining Operators ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators

We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

scholarly journals VERTEX ALGEBRAS AND VERTEX POISSON ALGEBRAS

2004 ◽  
Vol 06 (01) ◽  
pp. 61-110 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Poisson Algebra ◽  
Vertex Algebra ◽  
General Construction ◽  
Vertex Algebras ◽  
Poisson Algebras ◽  
Formal Deformation ◽  
Good Filtration

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex Poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex Poisson algebra are revisited and certain general construction theorems of vertex Poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex Poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each ℕ-graded vertex algebra V=∐n∈ℕV(n) with [Formula: see text], a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

scholarly journals Vertex Lie algebras and cyclotomic coinvariants

2017 ◽  
Vol 19 (02) ◽  
pp. 1650015 ◽  
Author(s):  
Benoît Vicedo ◽  
Charles Young
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bethe Ansatz ◽  
Riemann Sphere ◽  
Marked Point ◽  
Vertex Algebra ◽  
The Other ◽  
Gaudin Models ◽  
Equivariant Functions ◽  
Marked Points

Given a vertex Lie algebra [Formula: see text] equipped with an action by automorphisms of a cyclic group [Formula: see text], we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over “local” Lie algebras [Formula: see text] assigned to marked points [Formula: see text], by the action of a “global” Lie algebra [Formula: see text] of [Formula: see text]-equivariant functions. On the other hand, the universal enveloping vertex algebra [Formula: see text] of [Formula: see text] is itself a vertex Lie algebra with an induced action of [Formula: see text]. This gives “big” analogs of the Lie algebras above. From these we construct the space of “big” cyclotomic coinvariants, i.e. coinvariants with respect to [Formula: see text]. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary, we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [B. Vicedo and C. Young, Cyclotomic Gaudin models: Construction and Bethe ansatz, preprint (2014); arXiv:1409.6937]. At the origin, which is fixed by [Formula: see text], one must assign a module over the stable subalgebra [Formula: see text] of [Formula: see text]. This module becomes a [Formula: see text]-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

Semisimplicity of Locally Finite Graded R#H-Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 109-122
Author(s):  
Thomas Guédénon
Keyword(s):  
Abelian Group ◽  
Smash Product ◽  
Module Algebra ◽  
Locally Finite ◽  
Finite Dimensional

Let k be a field, Γ an abelian group with a bicharacter, R a colour algebra over k (i.e., a Γ-graded associative k-algebra with identity), H a Hopf colour k-algebra acting on R in such a way that R is a graded H-module algebra and the associated smash product R#H is a colour algebra. The aim of this paper is to study the semisimplicity of the category of H-locally finite Γ-graded R#H-modules. From our main result we deduce that if H is finite-dimensional and R is left graded-noetherian and graded-semisimple, then the colour algebra R#H is graded-semisimple if either H is graded-semisimple or if H is colour-cocommutative and R is colour-commutative and projective in the category of graded R#H-modules.

scholarly journals Vertex Algebras, Lie Algebras, and Superstrings

1998 ◽  
Vol 200 (2) ◽  
pp. 363-403 ◽  
Author(s):  
Nils R Scheithauer
Keyword(s):  
Lie Algebras ◽  
Vertex Algebras

Primitive elements of free lie algebras

1970 ◽  
Vol 9 (4) ◽  
pp. 275-284 ◽  
Author(s):  
G. P. Kukin
Keyword(s):  
Lie Algebras ◽  
Primitive Elements ◽  
Free Lie Algebras
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