Special and general types
Formal Lie groupoids
The notion of Lie algebra extension is a special case of the general notion of Extensions in ∞-Lie algebra cohomology.
A short exact sequences of Lie algebras is a diagram
where are Lie algebras, are homomorphisms of Lie algebras and the underlying diagram of vector spaces is exact, i.e. , and .
We also say that this diagram (and sometimes, loosely speaking, itself) is a Lie algebra extension of by the “kernel” .
Lie algebra extensions may be obtained from Lie group group extensions via the tangent Lie algebra functor.
Classification by nonabelian Lie algebra cocycles
We discuss how Lie algebra extensions are classified by cocycles in nonabelian Lie algebra cohomology.
Each element defines a derivative on by . The rule defines a homomorphism of Lie algebras . Indeed,
for all , for all . The restriction takes (by definition) values in the Lie subalgebra of inner derivatives of . If and are in the same coset, that is , then there is with and such that for all we have and therefore
Thus we obtain a well-defined map .
Choose a -linear section of the projection and denote by the composition where . One considers the problem of reconstructing the group from the knowledge of and . In order to derive the necessary relations we will identify with (as a set).
Indeed, write each element as , by setting . Elements and in that decomposition are unique. Thus we obtain a bijection , . The commutation rule has to be figured out. If , and , then
Now so it can be represented uniquely in the form where can be obtained by evaluating the antisymmetric -bilinear form defined by on . Then formula (1) becomes
Thus all the information about the commutators is encoded in functions and , without knowledge of .
However, not every pair will give some commutation rule on satisfying Jacobi identity, and also some different pairs may lead to the isomorphic extensions.
In order to satisfy the Jacobi identity, this pair needs to form a nonabelian 2-cocycle in the sense of nonabelian Lie algebra cohomology.
Discussion in the generality of super Lie algebras includes