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

Take a look at the Ph.D. Thesis of P.G. Batten: http://www4.ncsu.edu/~stitz/Multipliers%20and%20Covers%20of%20Lie%20Algebras.pdf 


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. 23, 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) 588595. This is part of the development of nonabelian tensor products of groups (and Lie algebras). More references to this are in 


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 Liealgebraic 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.


