# Schur multipliers' for Lie algebras

Schur multipliers for group extensions and for Lie groups also Where are they written for Lie algebras?

• The central extension question is studied in a cohomological spirit in a special (but interesting) case by Wilberd van der Kallen: Infinitesimally central extensions of Chevalley groups, Lect. Notes in Math. 356 (1973). Here the Lie algebras of Chevalley groups come into play in an essential way. Mar 2, 2012 at 23:38
• Basics on the Schur multiplier of Lie algebras are written in Abels' book: MR0903449: Abels, Herbert Finite presentability of S-arithmetic groups. Compact presentability of solvable groups. Lecture Notes in Mathematics, 1261. Springer-Verlag, Berlin, 1987. vi+178 pp. ISBN: 3-540-17975-5 (springer.com/us/book/9783540179757)
– YCor
Sep 22, 2016 at 9:09

Take a look at the Ph.D. Thesis of P.G. Batten:

Multipliers and covers of Lie algebras, North Carolina State University, 1993, dir. by E. Stitzinger; MathSciNet Link.

• That's it, though she calls them factor functions', which google can't find (easily). Care to identify yourself? How came you to be aware of her thesis? Tom Lada was on her committee - I should have asked him. Mar 3, 2012 at 14:17
• I was aware of the work of Batten and Stitzinger on this subject while reviewing a paper of another author. Note that I am not an anonymous user: in my account a link is quoted. Results from group theory having a counterpart in the theory of Lie algebras are among my research interests. Mar 3, 2012 at 17:17

Another answer to your interest in analogues between groups and Lie algebras is in

Ellis, Graham J. Nonabelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras. J. Pure Appl. Algebra 46 (1987), no. 2-3, 111–115.

which gives $H_2(L)$ for a Lie algebra as the kernel of a morphism $L \wedge L \to L$ where $L \wedge L$ is a nonabelian exterior product. This is the Lie algebra analogue of a result for groups proved in

Miller, Clair, `The second homology of a group', Proc. American Math. Soc. 3 (1952) 588-595.

This is part of the development of nonabelian tensor products of groups (and Lie algebras). More references to this are in the bibliography

http://groupoids.org.uk/nonabtens.html

@Jim Stasheff Sept 21, 2016 In a belated answer to Jim's question, there is a "nonabelian tensor product" $G \otimes H$ of groups which act on each other "compatibly", of which an example is when $G,H$ are normal subgroups of a group $Q$: in that case there is a commutator map $[\;,\;]: H \times H \to Q$ with properties for $[gg',h], [g,hh']$ which make it what is called a biderivation. The universal object for biderivations is written $G \otimes H$, and the commutator map then determines a morphism $\kappa: G \otimes H \to Q$ with image $[G,H]$. The kernel of $G \otimes G \to G$ is actually isomorphic to $\pi_3(SK(G,1))$. There are analogues for Lie Algebras.

• What is the meaning of nonabelian in that reference? I skimmed the paper and am missing any mention of the function in relation to extensions. Mar 7, 2012 at 12:58

Throwing a bunch of more references, for what it's worth (haven't looked thoroughly at them). But frankly, for me the "Schur multiplier", at least in the Lie-algebraic context, was always a synonym for the "second (co)homology with trivial coefficients", just defined via the Hopf formula, though probably I am missing some additional data coming by analogy from group theory.

• J. Bichon, G. Carnovale, Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras, J. Pure Appl. Algebra 204 (2006), no.3, 627-665
• L.R. Bosko, On Schur multipliers of Lie algebras and groups of maximal class, Intern. J. Algebra Comput. 20 (2010), N6, 807-821; DOI:10.1142/S0218196710005881
• L.R. Bosko, E.L. Stitzinger, Schur multipliers of nilpotent Lie algebras, arXiv:1103.1812
• H. Mohammadzadeh, B. Edalatzadeh, Some properties on Schur multiplier and cover of a pair of Lie algebras, arXiv:1105.0077
• P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras, Centr. Eur. J. Math. 9 (2011), no.1, 57-64 MR:2011m:17031
• P. Niroomand, F. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011), N4, 1293-1297; arXiv:1001.0176; DOI:10.1080/00927871003652660
• P. Niroomand, F. Russo, A restriction on the Schur multiplier of nilpotent Lie algebras, Electron. J. Lin. Algebra 22 (2011), 1-9 http://www.math.technion.ac.il/iic/ela/ela-articles/22.html#1
• F. Saeedi, A. Salemkar, B. Edalatzadeh, The commutator subalgebra and Schur multiplier of a pair of nilpotent Lie algebras, J. Lie Theory 21 (2011), No.2, 491-498 http://www.heldermann.de/JLT/JLT21/JLT212/jlt21021.htm
• A. Salemkar, V. Alamian, H. Mohammadzadeh, Some properties of the Schur multiplier and covers of Lie algebras, Comm. Algebra 36 (2008), no.2, 697-707; DOI:10.1080/00927870701724193