×

zbMATH — the first resource for mathematics

Lie superalgebras. (English) Zbl 0366.17012
Lie superalgebras (author’s notion) are defined as \(\mathbb Z_2\)-graded generalizations of Lie algebras on a vector space \(A_0 +A_1\) such that graded skew symmetry and a graded version of Jacobi-identity
\[ [a, [b,c]] = [[a,b],c] + (-1)^{ik} [b,[a,c]] \]
for \(a\) in \(A_i\), \(b\) in \(A_k\) holds. Sometimes they are called “graded Lie algebras” which is misleading since they are not Lie algebras with a compatible graduation. Mathematicians studied them first some twenty years ago. Recently they gained interest by physicists, especially in the classification of particles with different statistics. This work now is a nearly complete algebraic theory of finite-dimensional superalgebras, giving general constructions as well as classifications of the complex semisimple ones.
In detail: It is shown that any superalgebra \(G\) has a unique maximal solvable ideal (radical) \(R\) such that \(G/R\) is semisimple, i.e. contains no solvable ideals. However, \(G\) in general is not a semidirect sum of \(G\) and \(G/R\). Lie’s theorem that a finite-dimensional irreducible representation of a solvable algebra is one-dimensional no longer is true here. A classification of these representations is given in section 5, and in addition a necessary and sufficient condition for such a representation to be one-dimensional. Furthermore the decomposition of semisimple algebras in a direct sum of simple ones is no longer possible here, but a description of semisimple algebras in terms of simple ones is derived.
Chapters 2–4 contain fi the main part of the work, the principal difficulty of the classification of the complex semisimple algebras lying in the fact that the Killing form may be degenerate. This result is given in two kinds of simple algebras, the classical ones: besides ordinary simple Lie algebras four series of the form \(A(m,n)\), \(B(m,n)\), \(C(n)\), \(D(m,n)\), \(m- n\ne 1\), and two exceptional algebras \(F(4)\), \(G(3)\), and the second kind with vanishing Killing form, two series \(A(n,n)\), \(D(n+1,n)\), two strange series \(P(n)\), \(Q(n)\), and a one-parameter family of 17-dimensional exceptional algebras \(D(1,2,\alpha)\). Also a classification of simple complex 2-graded superalgebras is given, where besides the above ones there are four series \(W(n)\), \(S(n)\), \(\tilde S(n)\) and \(H(n)\) constructed in terms of Grassmann algebras and Hamiltonian vector fields. Recently B. Kostant has given a rigorous globalization of superalgebras in terms of supermanifolds.

MSC:
17B05 Structure theory for Lie algebras and superalgebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65 Infinite-dimensional Lie (super)algebras
17B70 Graded Lie (super)algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andreev, E.M; Vinberg, E.B; Elashvili, A.S, Orbits of largest dimension of semi-simple linear Lie groups, Functional anal. appl., 1, 257-261, (1967)
[2] Berezin, F.A, The method of second quantization, (1966), Academic Press New York · Zbl 0151.44001
[3] Berezin, F.A, Automorphisms of the Grassmann algebra, Math. notes, 1, 180-184, (1967) · Zbl 0211.05002
[4] Berezin, F.A; Kats, G.I, Lie groups with commuting and anticommuting parameters, Math. USSR. sb., 11, 311-326, (1970) · Zbl 0248.22022
[5] Berezin, F.A; Leites, D.A, Supervarieties, Sov. math. dokl., 16, 1218-1222, (1975)
[6] Weisfeiler, B.Yu, Infinite-dimensional filtered Lie algebras and their connection with graded Lie algebras, Functional anal. appl., 2, 88-89, (1968) · Zbl 0245.17006
[7] Weisfeiler, B.Yu; Kac, V.G, Irreducible representations of p-Lie algebras, Functional anal. appl., 5, 111-117, (1971) · Zbl 0237.17003
[8] Weisfeiler, B.Yu; Kac, V.G, Exponentials in Lie algebras of characteristic p, Math. USSR izv., 5, 777-803, (1971) · Zbl 0237.17003
[9] Vinberg, E.B; Onishik, A.L, Seminar on algebraic groups and Lie groups, (1969), [in Russian]
[10] Jacobson, N, Lie algebras, (1962), Wiley-Interscience New York · JFM 61.1044.02
[11] Kac, V.G, Simple irreducible graded Lie algebras of finite growth, Math. USSR izv., 2, 1271-1311, (1968) · Zbl 0222.17007
[12] Kac, V.G, On the classification of simple Lie algebras over a field of non-zero characteristic, Math. USSR izv., 4, 391-413, (1970) · Zbl 0254.17007
[13] Kac, V.G, Some algebras related to the quantum theory of fields, (), 140-141, [in Russian] · Zbl 0497.17007
[14] Kac, V.G, Infinite-dimensional Lie algebras and Dedekind’s η-function, Functional anal. appl., 8, 68-70, (1974) · Zbl 0299.17005
[15] Kac, V.G, Description of filtered Lie algebras associated with graded Lie algebras of Cartan type, Math. USSR izv., 8, 801-835, (1974) · Zbl 0317.17002
[16] Kac, V.G, Classification of simple Lie superalgebras, Functional anal. appl., 9, 263-265, (1975) · Zbl 0331.17001
[17] Leites, D.A, Cohomology of Lie superalgebras, Functional anal. appl., 9, 340-341, (1975) · Zbl 0352.17007
[18] Rudakov, A.N, The automorphism groups of infinite-dimensional simple Lie algebras, Math. USSR izv., 3, 707-722, (1969) · Zbl 0222.17014
[19] Stavraki, G.L, Some non-local model of field selfinteractions and the algebra of field operators, ()
[20] Blattner, R.J, Induced and produced representations of Lie algebras, Trans. amer. math. soc., 144, 457-474, (1969) · Zbl 0295.17002
[21] Block, R.E, Determination of the differentiably simple rings with a minimal ideal, Ann. math., 90, No. 2, 433-459, (1969) · Zbl 0216.07303
[22] Corwin, L; Ne’eman, Y; Sternberg, S, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. mod. phys., 47, 573-604, (1975) · Zbl 0557.17004
[23] Kobayashi, S; Nagano, T, On filtered Lie algebras and their geometric structure, III, J. math. mech., 14, 679-706, (1965) · Zbl 0163.28103
[24] Milnor, J; Moore, J, On the structure of Hopf algebras, Ann. math., 81, 211-264, (1965) · Zbl 0163.28202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.