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.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Moduli Space ◽  
Deformation Theory ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Exterior Degree ◽  
Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

DIFFEOMORPHISM GROUPS, QUANTIZATION, AND SU(∞)

1989 ◽  
Vol 04 (19) ◽  
pp. 5235-5248 ◽  
Author(s):  
JENS HOPPE
Keyword(s):  
Lie Algebras ◽  
Symplectic Manifold ◽  
Poisson Algebra ◽  
General Construction ◽  
Diffeomorphism Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras ◽  
Detailed Presentation

A method of relating infinite dimensional Lie algebras to basis dependent sequences of finite dimensional Lie algebras is introduced. Diff A S2 and Diff A T2 are shown to be N → ∞ limits of SU(N), with a limiting procedure, and " SU (∞)'s", that differ from the usual Kac Moody ones. After a detailed presentation of these two 'principal' examples, the general construction is outlined, which seems to work for the Poisson algebra of more or less any homogenous symplectic manifold.

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 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.

scholarly journals Capable n-Lie algebras and the classification of nilpotent n-Lie algebras with s(A)=3

2016 ◽  
Vol 110 ◽  
pp. 25-29 ◽  
Author(s):  
Hamid Darabi ◽  
Farshid Saeedi ◽  
Mehdi Eshrati
Keyword(s):  
Lie Algebras

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

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