H2 classifies extensions, and builds new algebras
An extension of g by an abelian ideal A determines an action of g on A, and with that action fixed the extensions are classified by H2(g, A). The construction mirrors the group case exactly: choose a linear section, measure its failure to preserve the bracket, obtain a 2-cocycle. The central extensions produced this way are not curiosities — the Heisenberg algebra and the affine Kac–Moody algebras are built precisely this way.
Learning objectives
- Construct the 2-cocycle from a linear section.
- State the classification by H2.
- Identify central extensions and their cocycle condition.
- Give the standard examples.
Section 01The classification
- Given 0 → A → e → g → 0 with A abelian, choose a linear section s: g → e. A linear splitting always exists over a field.
- Define the action of g on A by x·a = [s(x), a], independent of s since A is abelian.
- Define f(x, y) = [s(x), s(y)] − s([x, y]), which lies in A.
- The Jacobi identity in e forces the 2-cocycle condition on f.
- Changing s by a linear map c: g → A changes f by the coboundary of c.
- The class [f] ∈ H2(g, A) classifies the extension up to equivalence.
Explicitly, f must satisfy the alternating identity x·f(y,z) − y·f(x,z) + z·f(x,y) − f([x,y],z) + f([x,z],y) − f([y,z],x) = 0. This is the Jacobi identity for the twisted bracket, exactly as the group 2-cocycle condition is associativity.
Section 02Central extensions
With trivial action, A is central and the cocycle condition reduces to an alternating bilinear form f satisfying
Heisenberg algebra
The central extension of an abelian Lie algebra with a symplectic form as cocycle. It is the algebra of position and momentum operators in quantum mechanics.
Affine Kac–Moody algebras
The one-dimensional central extension of a loop algebra g ⊗ K[t, t−1], with cocycle given by a residue pairing. The centre is what makes the representation theory rich.
Virasoro algebra
The central extension of the Witt algebra of vector fields on the circle. The central charge appearing in conformal field theory is exactly the H² parameter.
Section 03Vanishing
For a semisimple Lie algebra over a field of characteristic zero, the second Whitehead lemma gives H2(g, A) = 0 for every finite-dimensional module. So every such extension splits.
Semisimplicity, characteristic zero and finite-dimensional coefficients are all needed. The affine and Virasoro algebras are central extensions of infinite-dimensional Lie algebras — precisely the setting where the Whitehead lemmas do not apply, and where the non-vanishing H² is the whole point.
ReferenceFrequently asked questions
Why is a linear section always available?
Because over a field every short exact sequence of vector spaces splits. So unlike the group case, where the section is merely a set map, here the section is linear and only the bracket fails to be preserved. This makes the cocycle bilinear.
What is the central charge?
The parameter labelling which multiple of the generating cocycle is used, when H² with trivial one-dimensional coefficients is one-dimensional. In conformal field theory it is a physical invariant of the theory.
Are all extensions of a Lie algebra classified by H²?
Only those with abelian kernel and a fixed action. For non-abelian kernels an outer action must be specified and an H³ obstruction appears, exactly parallel to the group case.
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