This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras can be formulated in terms of a single multiplication and they behave like associative algebras with respect to it. Chapter 1 is the introduction. In chapter 2 we discuss many examples of vertex Lie algebras and we show that vertex Lie algebras form a full subcategory of the category of "local" Lie algebras. In chapter 3 we introduce associative, commutative, and Poisson vertex algebras and vertex algebra modules and we show that graded associative vertex algebras form a full subcategory of the category of "local" associative algebras. In chapter 4 we give a systematic presentation of the vertex algebra identities, proving in particular the equivalence of various axiom systems, and we use filtrations to prove results about generating subspaces with the PBW-property, with and without repeats. In chapter 5 we explain three constructions of the enveloping vertex algebra of a vertex Lie algebra and prove the PBW-theorem. In chapter 6 we prove the Zhu correspondence between N-graded vertex algebra modules and modules over the Zhu algebra.
%0 Journal Article
%1 rosellen_course_2006
%A Rosellen, Markus
%D 2006
%J math/0607270
%K Quantum algebras {Algebra,Vertex}
%T A Course in Vertex Algebra
%U http://arxiv.org/abs/math/0607270
%X This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras can be formulated in terms of a single multiplication and they behave like associative algebras with respect to it. Chapter 1 is the introduction. In chapter 2 we discuss many examples of vertex Lie algebras and we show that vertex Lie algebras form a full subcategory of the category of "local" Lie algebras. In chapter 3 we introduce associative, commutative, and Poisson vertex algebras and vertex algebra modules and we show that graded associative vertex algebras form a full subcategory of the category of "local" associative algebras. In chapter 4 we give a systematic presentation of the vertex algebra identities, proving in particular the equivalence of various axiom systems, and we use filtrations to prove results about generating subspaces with the PBW-property, with and without repeats. In chapter 5 we explain three constructions of the enveloping vertex algebra of a vertex Lie algebra and prove the PBW-theorem. In chapter 6 we prove the Zhu correspondence between N-graded vertex algebra modules and modules over the Zhu algebra.
@article{rosellen_course_2006,
abstract = {This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras can be formulated in terms of a single multiplication and they behave like associative algebras with respect to it. Chapter 1 is the introduction. In chapter 2 we discuss many examples of vertex Lie algebras and we show that vertex Lie algebras form a full subcategory of the category of "local" Lie algebras. In chapter 3 we introduce associative, commutative, and Poisson vertex algebras and vertex algebra modules and we show that graded associative vertex algebras form a full subcategory of the category of "local" associative algebras. In chapter 4 we give a systematic presentation of the vertex algebra identities, proving in particular the equivalence of various axiom systems, and we use filtrations to prove results about generating subspaces with the {PBW-property,} with and without repeats. In chapter 5 we explain three constructions of the enveloping vertex algebra of a vertex Lie algebra and prove the {PBW-theorem.} In chapter 6 we prove the Zhu correspondence between N-graded vertex algebra modules and modules over the Zhu algebra.},
added-at = {2009-05-11T21:36:02.000+0200},
author = {Rosellen, Markus},
biburl = {https://www.bibsonomy.org/bibtex/216d0672ded02ea16f377092b8bd52ad2/tbraden},
interhash = {503d2fae3e9ff6a2aaccd09687f87f63},
intrahash = {16d0672ded02ea16f377092b8bd52ad2},
journal = {math/0607270},
keywords = {Quantum algebras {Algebra,Vertex}},
month = {July},
timestamp = {2009-05-11T21:36:02.000+0200},
title = {A Course in Vertex Algebra},
url = {http://arxiv.org/abs/math/0607270},
year = 2006
}