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VERTEX ALGEBRAS AND VERTEX POISSON ALGEBRAS

HAISHENG LI

Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA

Department of Mathematics, Harbin Normal University, Harbin, China

https://doi.org/10.1142/S0219199704001264Cited by:55 (Source: Crossref)

Abstract

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 , 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.

Keywords:

AMSC: 17B69, 81R10

References

B. Bakalov and V. G. Kac, Field algebras , arXiv:math.QA/0204282 . Google Scholar
S. Bates and A. Weinstein, Berkeley Mathematics Lecture Notes 8, (1997). Google Scholar
F. Bayenet al., Ann. Phys. 111(1), 61 (1978). Crossref, Web of Science, Google Scholar
A. A. Beilinson and V. Drinfeld, Chiral algebras, unpublished manuscript . Google Scholar
R. E. Borcherds, Proc. Natl. Acad. Sci. USA 83, 3068 (1986), DOI: 10.1073/pnas.83.10.3068. Crossref, Web of Science, Google Scholar
R. E. Borcherds, Progress in Math., Vertex algebras, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996) 160, eds. M. Kashiwaraet al. (Birkhäuser, Boston, 1998) pp. 35–77. Google Scholar
N. Chriss and V. Ginzburg , Representation Theory and Complex Geometry ( Birkhaüser , 1997 ) . Google Scholar
J. Dixmier , Graduate Studies in Mathematics , Enveloping Algebras II ( American Mathematical Society , 1996 ) . Google Scholar
C. Dong and J. Lepowsky , Progress in Math. , Generalized Vertex Algebras and Relative Vertex Operators 112 ( Birkhaüser , 1993 ) . Google Scholar
C. Dong, H.-S. Li and G. Mason, Math. Ann. 310, 571 (1998), DOI: 10.1007/s002080050161. Crossref, Web of Science, Google Scholar
C. Dong, H.-S. Li and G. Mason, Vertex Lie algebra, Poisson algebras and vertex operator algebras, Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proc. of an International Conference297, Contemporary Math., eds. S. Bermanet al. (Amer. Math. Soc., 2002) pp. 69–96. Google Scholar
B. Enriquez and E. Frenkel, Duke Math. J. 92, 459 (1998), DOI: 10.1215/S0012-7094-98-09214-6. Crossref, Web of Science, Google Scholar
P. Etingof and D. Kazhdan, Selecta Math. (N.S.) 6, 105 (2000). Crossref, Web of Science, Google Scholar
Alex J. Feingold, Igor B. Frenkel and John F. X. Ries, Contemporary Math. 121, (1991). Google Scholar
E. Frenkel and D. Ben-Zvi , Mathematical Surveys and Monographs , Vertex Algebras and Algebraic Curves 88 ( Amer. Math. Soc. , 2001 ) . Google Scholar
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993; preprint, 1989 . Google Scholar
I. Frenkel , J. Lepowsky and A. Meurman , Pure and Appl. Math. , Vertex Operator Algebras and the Monster 134 ( Academic Press , Boston , 1988 ) . Google Scholar
I. B. Frenkel and Y. Zhu, Duke Math. J. 66, 123 (1992), DOI: 10.1215/S0012-7094-92-06604-X. Crossref, Web of Science, Google Scholar
M. Gerstenhaber, Ann. of Math. 84, 1 (1966), DOI: 10.2307/1970528. Crossref, Web of Science, Google Scholar
V. G. Kac , University Lecture Series , Vertex Algebras for Beginners 10 ( Amer. Math. Soc. , 1997 ) . Google Scholar
M. Karel and H.-S. Li, J. Algebra 217, 393 (1999), DOI: 10.1006/jabr.1998.7838. Crossref, Web of Science, Google Scholar
M. Kontsevich, Deformation quantization of Poisson manifolds, I , arXiv:q-alg/9709040 . Google Scholar
J. Lepowsky and H.-S. Li , progress in Math. , Introduction to Vertex Operator Algebras and Their Representations 227 ( Birkhäuser , 2003 ) . Google Scholar
H.-S. Li, Representation theory and tensor product theory for vertex operator algebras, Ph.D. thesis, Rutgers University, 1994 . Google Scholar
H.-S. Li, J. Pure Appl. Algebra 109, 143 (1996), DOI: 10.1016/0022-4049(95)00079-8. Crossref, Web of Science, Google Scholar
H.-S. Li, J. Alg. 212, 495 (1999), DOI: 10.1006/jabr.1998.7654. Crossref, Web of Science, Google Scholar
H.-S. Li, Commun. Contemp. Math. 5, 281 (2003), DOI: 10.1142/S0219199703000987. Link, Web of Science, Google Scholar
B.-H. Lian, Commun. Math. Phys. 163, 307 (1994), DOI: 10.1007/BF02102011. Crossref, Web of Science, Google Scholar
A. Meurman and M. Primc, Mem. Amer. Math. Soc. 652, (1999). Google Scholar
M. Primc, J. Pure Appl. Algebra 135, 253 (1999), DOI: 10.1016/S0022-4049(97)00144-8. Crossref, Web of Science, Google Scholar
Y.-C. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale University, 1990 . Google Scholar
Y.-C. Zhu, J. Amer. Math. Soc. 9, 237 (1996), DOI: 10.1090/S0894-0347-96-00182-8. Crossref, Web of Science, Google Scholar