Special and general types
∞-Lie theory (higher geometry)
Formal Lie groupoids
An extension of a Lie algebra is another Lie algebra that is equipped with a surjective Lie algebra homomorphism to
For non-tivial extensions, this homomorphism has a kernel , consisting of those elements of that map to the zero element in . That kernel is a sub-Lie algebra of and hence one says that is an extension of by .
This means equivalently that there is a short exact sequence of Lie algebras of the form
When happens to be abelian, hence when its Lie bracket is trivial, then one speaks of an abelian extension, and when furthermore the Lie bracket of vanishes as soon as already one of its arguments is in , then one has a central extension ( is in the center of ).
Central extensions by the ground field (say ) are equivalently induced by a 2-cocyle in the Lie algebra cohomology of with coefficients in the ground field, say , i.e. by linear maps
satisfying some conditions. The corresponding extension of is then, at the level of underlying vector space, the direct sum , equipped with the Lie bracket given by the formula
for all and . The condition on to be a 2-cocycle is precisely the condition that this formula satisfies the Jacobi identity.
If one regards all Lie algebras here as being special cases of Lie 2-algebras, then the 2-cocycle may itself be thought of as a homomorphism, namely from to the line Lie 2-algebra . With this, then given by the above formula is simply the homotopy fiber of , and the whole story comes down to saying that there is a homotopy fiber sequence of L-∞ algebras of the form
This perspective on Lie algebra extensions makes it evident how the concept generalizes to a concept of L-∞ algebra extensions.
Of course extensions need not be central or even abelian. An important class of non-abelian extensions are semidirect product Lie algebras. These are given by an Lie action of on , hence a homomorphism to the derivations on and with this the bracket on is given by the formula
A short exact sequence 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 cohomology
We discuss how in general 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.
The Heisenberg Lie algebra is an extension of , regarded as an abelian Lie algebra, by with the corresponding 2-cocycle being the canonical commutation relation , , .
More generally, the Kostant Souriau extension exhibits a Poisson bracket on a symplectic manifold as an extension of the Lie algebra of Hamiltonian vector fields.
For more discussion putting these two examples in perspective see also at quantization – Motivation from classical mechanics and Lie theory.
Discussion in the generality of super Lie algebras includes