Schur `multipliers' for Lie algebras Schur multipliers for group extensions and for Lie groups also
Where are they written for Lie algebras?
 A: 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.
A: 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. 
A: 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

