Abstract
This work was motivated by two seemingly unrelated problems:
-
1.
Lie’s problem of classification of “local continuous transformation groups of a finite-dimensional manifold”.
-
2.
The problem of classification of operator product expansions (OPE) of chiral fields in 2-dimensional conformal field theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
D. Alexseevski, D. Leites, I. Shchepochkina, Examples of simple Lie superalgebras of vector fields, C.R. Acad. Bul. Sci. 33 (1980), 1187–1190.
A.A. Belavin, A.M. Polyakov, A.M. Zamolodchikov, Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys. 34 (1984), 763–774.
R.J. Blattner, Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc. 144 (1969), 457–474.
R.J. Blattner, A theorem of Cartan and Guillemin, J. Diff. Geom. 5 (1970), 295–305.
A. Cappelli, C.A. Trugenberger, G.R. Zemba, Stable hierarchal quantum Hall fluids as Wi+∞ minimal models, Nucl. Phys. B 448 (1995), 470–511.
E. Cartan, Les groupes des transformations continués, infinis, simples, Ann. Sci. Ecole Norm. Sup. 26 (1909), 93–161.
S.-J. Cheng, Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras, J. Algebra 173 (1995), 1–43.
S.-J. Cheng, V.G. Kac, A new N = 6 superconformal algebra, Commun. Math. Phys. 186 (1997), 219–231.
S.-J. Cheng, V.G. Kac, Generalized Spencer cohomology and filtered deformations of ℤ-graded Lie superalgebras, Adv. Theor. Math. Phys. 2 (1998), 1141–1182.
S.-J. Cheng, V.G. Kac, Structure of some ℤ-graded Lie superalgebras of vector fields, Transformation Groups 4 (1999), 219–272.
A. D’Andrea, V.G. Kac, Structure theory of finite conformal algebras, Selecta Mathematica 4 (1998), 377–418.
E.S. Fradkin, V. Ya. Linetsky, Classification of superconformal and quasisuperconformal algebras in two dimensions, Phys. Lett. B 291 (1992), 71–76.
V.W. Guillemin, A Jordan-Hölder decomposition for a certain class of infinite dimensional Lie algebras, J. Diff. Geom. 2 (1968), 313–345.
V.W. Guillemin, Infinite-dimensional primitive Lie algebras, J. Diff. Geom. 4 (1970), 257–282.
V.W. Guillemin, S. Sternberg, An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc. 70 (1964), 16–47.
V.G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. USSR-Izvestija 2 (1968), 1271–1311.
V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96.
V.G. Kac, Modular invariance in mathematics and physics, in “Mathematics into the 21st Century”, AMS Centennial Publ. (1992), 337–350.
V.G. Kac, Vertex Algebras for Beginners, University Lecture Series 10, AMS, Providence, RI, 1996. Second edition 1998.
V.G. Kac, The idea of locality, in “Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras” (H.D. Doebner, et al., eds.), World Sci., Singapore (1997), 16–22.
V.G. Kac, Superconformal algebras and transitive group actions on quadrics, Comm. Math. Phys. 186 (1997), 233–252.
V.G. Kac, Classification of infinite-dimensional simple linearly compact Lie superalgebras, Adv. Math. 139 (1998), 1–55.
V.G. Kac, J. van der Leur, On classification of superconformal algebras, in “Strings 88” (S.J. Gates, et al., eds.), World Sci. (1989), 77–106.
V.G. Kac, A.O. Radul, Representation theory of W1+∞, Transformation Groups 1 (1996), 41–70.
V.G. Kac, A.N. Rudakov, Irreducible representations of linearly compact Lie superalgebras, in preparation.
Yu. Kochetkoff, Déformations de superalgébres de Buttin et quantification, CR. Acad. Sci. Paris 299, ser I, 14 (1984), 643–645.
S. Lie, Theorie der transformations Gruppen, Math. Ann. 16 (1880), 441–528.
O. Mathieu, Classification of simple graded Lie algebras of finite growth, Invent. Math. 108 (1992), 445–519.
V.I. Ogievetskii, E.S. Sokachev, The simplest group of Einstein supergravity, Sov. J. Nucl. Phys. 31 (1980), 140–164.
L. Okun, Physics of Elementary Particles, Nauka, 1988 (in Russian).
E. Poletaeva, Semi-infinite cohomology and superconformal algebras, preprint.
P. Ramond, J.H. Schwarz, Classification of dual model gauge algebras, Phys. Lett. B. 64 (1976), 75–77.
A.N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type, Math. USSR-Izvestija 8 (1974), 836–866.
J.A. Schouten, W. van der Kulk, Pfaff’s Problem and its Generalizations, Clarendon Press, 1949.
I. Shchepochkina, New exceptional simple Lie superalgebras, C.R. Bui. Sci. 36:3 (1983), 313–314.
I. Shchepochkina, The five exceptional simple Lie superalgebras of vector fields, preprint.
I.M. Singer, S. Sternberg, On the infinite groups of Lie and Cartan I, J. Analyse Math. 15 (1965), 1–114.
S. Sternberg, Featured review of “Classification of infinite-dimensional simple linearly compact Lie superalgebras” by Victor Kac, Math. Reviews 99m: 17006, 1999.
B.Y. Weisfeiler, Infinite-dimensional filtered Lie algebras and their connection with graded Lie algebras, Funct. Anal. Appl. 2 (1968), 88–89.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Birkhäuser, Springer Basel AG
About this chapter
Cite this chapter
Kac, V.G. (2010). Classification of Infinite-Dimensional Simple Groups of Supersymmetries and Quantum Field Theory. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0422-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0422-2_6
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0421-5
Online ISBN: 978-3-0346-0422-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)