Given a central extension for a given Lie algebra, is there any simple way to check that it is/isn't isomorphic to the trivial extension ("simple" meaning, not as tedious [and daunting, for an algebra with many generators] as writing all of the trivial extension basis elements as linear combinations of the given extension, and plugging it into the commutators to solve the resulting system of equations)?

Perhaps, if it is easier, an illustrative example could be used? For example, the central extension of the Galilean algebra has a basis $\{E, P_j, C_j, L_{jk},M\}$, satisfying (I think??) the (nonzero) commutator relations:

$$ [L_{jk},L_{pq}]=i(\delta_{jp}L_{kq}-\delta_{jq}L_{kp}-\delta_{kp}L_{jq}+\delta_{kq}L_{jp}), $$ $$ [L_{jk},P_l]=i(\delta_{jl}P_k-\delta_{kl}P_j), $$ $$ [L_{jk},C_l]=i(\delta_{jl}C_k-\delta_{kl}C_j), $$ $$ [C_j,E]=iP_j, $$ $$ [C_j,P_k]=i\delta_{jk}M $$

(where $1\leq j,k,l,p,q\leq n$). Is there a simple way (say, $n=2$, so that it is not too trivial, but also not too complicated), to check whether or not this is isomorphic to the trivial extension (i.e. take $M=0$ in the above, so that $M$ commutes with everything else)?

[Of course, any references addressing this issue would also be appreciated.]

I'd imagine that there is some sort of answer in terms of Lie group/algebra cohomology (hence the group cohomology tag), but I'm afraid I'm not knowledgable enough of such things to say for sure (feel free to remove the tag, if I'm wrong about that).

EDIT: For those interested, I came across a nice discussion of the example I asked about here (see page 7).


1 Answer 1


Let $\mathfrak{a}$ be an abelian Lie algebra, and $\mathfrak{g}, \mathfrak{h}$ two Lie algebras. For a short sequence of Lie algebras $$ 0\rightarrow \mathfrak{a} \rightarrow \mathfrak{h}\rightarrow \mathfrak{g}\rightarrow 0 $$ $\mathfrak{h}$ is called a central extension of $\mathfrak{g}$ by $\mathfrak{a}$, if $[\mathfrak{a},\mathfrak{h}]=0$, i.e., $\mathfrak{a}\subseteq Z(\mathfrak{h})$. The equivalence classes of central extensions is in one-to-one correspondence with the second cohomology group $H^2(\mathfrak{g},\mathfrak{a})$. This is a vector space, which is not difficult to understand - a useful introduction is, for example, this paper. The equivalence class of trivial (split) central extensions corresponds to the zero element in $H^2(\mathfrak{g},\mathfrak{a})$.

In your example, you have to compute the second cohomology group of the Galilean algebra.


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