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Bahturin, Yuri; Mikhalev, Alexander A.; Zaicev, Mikhail

Infinite-dimensional Lie superalgebras. (English) Zbl 0970.17011

Hazewinkel, M. (ed.), Handbook of algebra. Volume 2. Amsterdam: North-Holland. 579-614 (2000).
This survey is based upon the monograph of the authors and V. M. Petrogradsky [Infinite dimensional Lie superalgebras, de Gruyter Exp. Math. Vol. 7, de Gruyter, Berlin (1992; Zbl 0762.17001)]. It also contains references to some recent developments.
1. This review begins with background on colour Lie superalgebras. Let \(G\) be an abelian group and \(\varepsilon:G\times G\to K^*\) a bilinear alternating map to the of the field \(K\). Now a \(L=\oplus_{g\in G}L_g\) is a colour Lie superalgebra if for any \(a\in L_g\), \(b\in L_h\) one has \[ [a,b]+\varepsilon(g,h)[b,a]=0,\quad [[a,b],c]=[a,[b,c]]-\varepsilon(g,h)[b,[a,c]]. \] Ordinary Lie and Lie superalgebras are included as special cases by taking \(G\) of order 1 and 2. Next a construction due to Scheunert is given, it shows that by modifying the operation of a colour Lie superalgebra one can reduce some questions about these algebras to those about ordinary Lie superalgebras. The following basic notions are discussed: enveloping algebra \(U(L)\), Poincaré-Birkhoff-Witt theorem, filtrations on enveloping algebras, restricted Lie superalgebras, solvable, nilpotent, simple superalgebras.
2. Next the review deals with free (colour) Lie superalgebras, their bases and dimension formulas, free products with amalgamated subalgebra, the composition lemma, subalgebras of free colour Lie (\(p\))-superalgebras.
3. Another topic is concerned with properties of the universal enveloping algebra such as the , Hopf structure, regularity and self-injectivity of the restricted enveloping algebra \(u(L)\). The paper also describes necessary and sufficient conditions for existence of a nontrivial identity in ordinary and restricted enveloping algebras.
4. Another paragraph deals with varieties of colour Lie superalgebras. and growth functions for metabelian varieties are given. Varieties of Lie superalgebras over a field of characteristic zero with a polynomial growth of codimensions are described. Grassmann envelopes are defined, and applications of this notion to Lie superalgebras are considered.
5. The final paragraph discusses different finiteness conditions. Lie superalgebras with all irreducible representations of bounded dimension are described. One of the typical finiteness conditions for a variety is that of being Noetherian. Other conditions are: matrix representability, residual finiteness, Hopf condition. Some of these conditions are studied as local, meaning that they are imposed on finitely generated subalgebras only. There are many different results on these finiteness conditions, some of them are equivalently described in terms of identical relations.
For the entire collection see [Zbl 0949.00006].
Reviewer: Victor Petrogradsky (Ulyanovsk)

Cited in 19 Documents

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B75 Color Lie (super)algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17B35 Universal enveloping (super)algebras
17B01 Identities, free Lie (super)algebras

Keywords:

Lie superalgebra; variety; grading; identity; enveloping algebra; free Lie superalgebra; colour Lie superalgebra; restricted algebra; Hilbert series; growth function; codimension growth; survey

Citations:

Zbl 0762.17001
Cite Review PDF

References:

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