Affine Lie algebras: the realization (case k=1)
In this chapter we describe in detail a “concrete” construction of all “nontwisted” affine Lie algebras. It turns out that such an algebra g can be realized entirely in terms of an “underlying” simple finite dimensional Lie algebra ġ. Namely, its derived algebra [g, g] is the universal central extension (the center being 1-dimensional) of the Lie algebra of polynomial maps from ℂX into ġ. In fact, the “Fourier transform” of the latter algebra appears in the quantum field theory, and is called a current algebra.
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Bibliographical notes and comments
- Kac, V. G. [1968 B] Simple irreducible graded Lie algebras of finite growth, English translation: Math. USSRIzvestija 2 (1968), 1271–1311.Google Scholar
- Kac, V. G. [1978 B] Highest weight representations of infinite dimensional Lie algebras, in Proceeding of ICM, 299–304, Helsinki, 1978.Google Scholar
- Kac, V. G. [1982 B] Some problems on infinite-dimensional Lie algebras and their representations, Lecture Notes in Math. 933 (1982), 141–162.Google Scholar
- Feigin, B. L., Fuchs, D. B.  Skew-symmetric invariant differential operators on a line and Verma modules over the Virasoro algebra, Funkt. analys i ego prilozh. 16 (1982), No. 2, 47–63 (in Russian).Google Scholar
- Feigin, B. L., Fuchs, D. B. [1983 A] Casimir operators in modules over the Virasoro algebra, Doklady AN SSSR 269, (1983) 1057–1060 (in Russian).Google Scholar
- Kaplansky, I.  The Virasoro algebra II, Chicago University, preprint.Google Scholar
- Reiman, A. G., Semenov-Tjan-Shanskii, M. A.  Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Invent. Math. 63 (1981), 423–432.Google Scholar
- Kac, V. G., Peterson, D. H. [1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprint.Google Scholar
- Atiyah, M. F., Pressley, A. N.  Convexity and loop groups, in Arithmetic and Geometry (ed. M. Artin and J. Tate ), 33–64, Birkhäuser, Boston, 1983.Google Scholar
- Goodman, R., Wallach, N.  Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, preprint.Google Scholar
- Moody, R. V.  A simplicity theorem for Chevalley groups defined by generalized Cartan matrices, preprint.Google Scholar