3 Central extensions of Lie algebras

3.1 Central extensions of Lie algebras

Definition 9 Lie algebra extension

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An extension \(\mathfrak {h}\) of a Lie algebra \(\mathfrak {g}\) by a Lie algebra \(\mathfrak {a}\) is a Lie algebra together with a pair of two Lie algebra homomorphisms \(\iota \colon \mathfrak {a}\longrightarrow \mathfrak {h}\) and \(\pi \colon \mathfrak {h}\longrightarrow \mathfrak {g}\) which form a short exact sequence

\begin{align*} 0 \longrightarrow \mathfrak {a}\overset {\iota }{\longrightarrow } \mathfrak {h}\overset {\pi }{\longrightarrow } \mathfrak {g}\longrightarrow 0 , \end{align*}

i.e., such that \(\iota \) is injective, \(\pi \) is surjective, and \(\mathrm{Im}(\iota ) = \mathrm{Ker}(\pi )\).

Definition 10 Lie algebra central extension

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A central extension \(\mathfrak {h}\) of a Lie algebra \(\mathfrak {g}\) by an abelian Lie algebra \(\mathfrak {a}\) is a Lie algebra extension

\begin{align*} 0 \longrightarrow \mathfrak {a}\overset {\iota }{\longrightarrow } \mathfrak {h}\overset {\pi }{\longrightarrow } \mathfrak {g}\longrightarrow 0 \end{align*}

such that \(\mathrm{Im}(\iota )\) is contained in the centre of \(\mathfrak {h}\), i.e., \([\iota (A), W] = 0\) for all \(A \in \mathfrak {a}\), \(W \in \mathfrak {h}\).

3.2 Central extensions determined by 2-cocycles

Definition 11 Central extension determined by a cocycle

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Given a Lie algebra 2-cocycle \(\gamma \in C^2(\mathfrak {g},\mathfrak {a})\), on the vector space

\begin{align*} \mathfrak {h}_\gamma = \mathfrak {g}\oplus \mathfrak {a}\end{align*}

define a bracket by

\begin{align*} [(X, A), (Y, B)]_{\gamma } := \big([X, Y]_{\mathfrak {g}}, \gamma (X, Y) \big) . \end{align*}

Then \(\mathfrak {h}_\gamma \) becomes a Lie algebra with the Lie bracket \([\cdot ,\cdot ]_{\gamma }\).

Lemma 12 Central extension determined by cohomologous cocycles

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Let \(\gamma _1, \gamma _2 \in C^2(\mathfrak {g},\mathfrak {a})\) be two Lie algebra 2-cocycles and \(\mathfrak {h}_{\gamma _1}, \mathfrak {h}_{\gamma _2}\) the central extensions corresponding to these according to Definition 11. If the two 2-cocycles differ by a coboundary, \(\gamma _2 - \gamma _1 = \partial \beta \) with some \(\beta \in C^1(\mathfrak {g},\mathfrak {a})\), then the mapping \(\mathfrak {h}_{\gamma _1} \to \mathfrak {h}_{\gamma _2}\) given by

\begin{align*} (X,A) \mapsto \big( X, A + \beta (X) \big) \end{align*}

is an isomophism of Lie algebras \(\mathfrak {h}_{\gamma _1} \cong \mathfrak {h}_{\gamma _2}\).

Proof ▶

The mapping \(\phi _\beta \colon \mathfrak {h}_1 \to \mathfrak {h}_2\) given by

\begin{align*} \phi _\beta \big((X,A)\big) := \big( X, A + \beta (X) \big) \end{align*}

is clearly linear. It is also bijective, since the similarly defined mapping \(\phi _{-\beta } \colon \mathfrak {h}_2 \to \mathfrak {h}_1\), \(\phi _{-\beta }\big((X,A)\big) := \big( X, A - \beta (X) \big)\), is a two-sided inverse to \(\phi _\beta \). So it remains to verify that this bijective linear map \(\phi _\beta \colon \mathfrak {h}_1 \to \mathfrak {h}_2\) is in fact a homomorphism Lie algebras.

Let \((X, A), (Y, B) \in \mathfrak {g}\oplus \mathfrak {a}= \mathfrak {h}_{\gamma _1}\). The bracket in \(\mathfrak {h}_{\gamma _1}\) of these is, by definition,

\begin{align*} [(X, A), (Y, B)]_{\gamma _1} := \big([X, Y]_{\mathfrak {g}}, \gamma _1(X, Y) \big) . \end{align*}

Applying the mapping \(\phi _\beta \) to this, we get

\begin{align*} \phi _\beta \Big([(X, A), (Y, B)]_{\gamma _1} \Big) = \big([X, Y]_{\mathfrak {g}}, \gamma _1(X, Y) + \beta ([X, Y]_{\mathfrak {g}}) \big) . \end{align*}

On the other hand the Lie bracket in \(\mathfrak {h}_2\) of the images is

\begin{align*} & \big[ \phi _\beta \big(((X, A)\big), \phi _\beta \big((Y, B)\big) \big]_{\gamma _2} \\ = \; & \big[ \big( X, A + \beta (X) \big), \big( Y, B + \beta (Y) \big) \big]_{\gamma _2} \\ = \; & \Big( [X,Y]_{\mathfrak {g}} , \gamma _2(X,Y) \Big) \\ = \; & \Big( [X,Y]_{\mathfrak {g}} , \gamma _1(X,Y) + \beta ([X, Y]_{\mathfrak {g}}) \Big) . \end{align*}

From the equality of these two expressions we see that \(\phi _\beta \) indeed is also a Lie algebra homomorphism.

Lemma 13 Central extension determined by a cocycle is a central extension

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Given a Lie algebra 2-cocycle \(\gamma \in C^2(\mathfrak {g},\mathfrak {a})\), consider the Lie algebra \(\mathfrak {h}_\gamma = \mathfrak {g}\oplus \mathfrak {a}\) as in Definition 11. With the inclusion \(\iota \colon \mathfrak {a}\to \mathfrak {g}\oplus \mathfrak {a}\) in the second direct summand and the projection \(\pi \colon \mathfrak {g}\oplus \mathfrak {a}\to \mathfrak {g}\) to the first direct summand, the Lie algebra \(\mathfrak {h}_\gamma = \mathfrak {g}\oplus \mathfrak {a}\) becomes a central extension of \(\mathfrak {g}\) by \(\mathfrak {a}\), i.e., we have the short exact sequence of Lie algebras

\begin{align*} 0 \longrightarrow \mathfrak {a}\overset {\iota }{\longrightarrow } \mathfrak {h}_\gamma \overset {\pi }{\longrightarrow } \mathfrak {g}\longrightarrow 0 . \end{align*}

Proof ▶

Clearly

\begin{align*} 0 \longrightarrow \mathfrak {a}\overset {\iota }{\longrightarrow } \mathfrak {h}_\gamma \overset {\pi }{\longrightarrow } \mathfrak {g}\longrightarrow 0 \end{align*}

is an exact sequence of vector spaces, and it is straightforward to check with Definition 11 that \(\iota \) and \(\pi \) are Lie algebra homomorphisms.

Theorem 14

Every cohomology class in \(H^2(\mathfrak {g}, \mathfrak {a})\) determines a well-defined isomorphism class of central extensions of the Lie algebra \(\mathfrak {g}\) by \(\mathfrak {a}\) by the rule that the class \([\gamma ] \in H^2(\mathfrak {g}, \mathfrak {a})\) of a cocycle \(\gamma \in C^2(\mathfrak {g}, \mathfrak {a})\) corresponds to the isomorphism class of \(\mathfrak {h}_\gamma \) (Definition 11).

Proof ▶

This follows from Lemmas 13 and 12.