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

define a bracket by

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

Let \(\mathbb {K}\) be a field of characteristic zero. The Heisenberg algebra \(\mathfrak {hei}\) is the Lie algebra over \(\mathbb {K}\) obtained as the central extension of the abelian Lie algebra \(\mathfrak {g}\) with basis \((j_k)_{k \in \mathbb {Z}}\), corresponding to the Heisenberg cocycle \({\gamma }_{\mathfrak {hei}} \in C^2(\mathfrak {g},\mathbb {K})\).

The Heisenberg algebra \(\mathfrak {hei}\) has a basis consisting of \((J_k)_{k \in \mathbb {Z}}\) and \(K\), with Lie brackets determined by the following

for \(k,l \in \mathbb {Z}\).

Let \(\mathfrak {g}\) be the vector space with basis \((j_k)_{k \in \mathbb {Z}}\) over \(\mathbb {K}\), considered as an abelian Lie algebra. The bilinear map \({\gamma }_{\mathfrak {hei}} \colon \mathfrak {g}\times \mathfrak {g}\to \mathbb {K}\) given on basis elements by

is a Lie algebra 2-cocycle, \({\gamma }_{\mathfrak {hei}} \in C^2(\mathfrak {g},\mathbb {K})\). We call \({\gamma }_{\mathfrak {hei}}\) the Heisenberg cocycle.

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

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

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

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

A 1-cocycle of the Lie algebra \(\mathfrak {g}\) with coefficients in the vector space \(\mathfrak {a}\) is a linear map

The set of all such 1-cocycles is denoted by \(C^1(\mathfrak {g}, \mathfrak {a})\).

Given a 1-cocycle \(\beta \in C^1(\mathfrak {g}, \mathfrak {a})\), we define the coboundary \(\partial \beta \) of \(\beta \) to be the bilinear map

given by

We then have \(\partial \beta \in C^2(\mathfrak {g},\mathfrak {a})\). The mapping \(\partial \colon C^1(\mathfrak {g},\mathfrak {a}) \to C^2(\mathfrak {g},\mathfrak {a})\) is linear. Its range is denoted \(B^2(\mathfrak {g},\mathfrak {a}) \subset C^2(\mathfrak {g},\mathfrak {a})\) and called the set of 2-coboundaries of the Lie algebra \(\mathfrak {g}\) with coefficients in \(\mathfrak {a}\).

A 2-cocycle of the Lie algebra \(\mathfrak {g}\) with coefficients in the vector space \(\mathfrak {a}\) is a bilinear map

such that for all \(X \in \mathfrak {g}\) we have the antisymmetry condition

and for all \(X,Y,Z \in \mathfrak {g}\) we have the Leibnitz rule

The set of all such 2-cocycles is denoted by \(C^2(\mathfrak {g}, \mathfrak {a})\).

The vector space

is called the Lie algebra cohomology in degree 2 of \(\mathfrak {g}\) with coefficients in \(\mathfrak {a}\).

For \(k,l \in \mathbb {Z}\), we denote the normal ordered product of the operators \(\mathsf{J}_k\) and \(\mathsf{J}_l\) by

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\), a linear operator

can be defined by the formula

(the sum only has finitely many terms by Lemma 33).

We call the operators \((\mathsf{L}_n)_{n \in \mathbb {Z}}\) the Sugawara operators.

Let \(\mathbb {K}\) be a field of characteristic zero. The Virasoro algebra \(\mathfrak {vir}\) is the Lie algebra over \(\mathbb {K}\) obtained as the central extension of the Witt algebra \(\mathfrak {witt}\) corresponding to the Virasoro cocycle \({\gamma }_{\mathfrak {vir}} \in C^2(\mathfrak {witt},\mathbb {K})\).

The Virasoro algebra \(\mathfrak {vir}\) has a basis consisting of \((L_n)_{n \in \mathbb {Z}}\) and \(C\), with Lie brackets determined by the following

for \(n,m \in \mathbb {Z}\).

The bilinear map \({\gamma }_{\mathfrak {vir}} \colon \mathfrak {witt}\times \mathfrak {witt}\to \mathbb {K}\) given on basis elements of \(\mathfrak {witt}\) by

is called the Virasoro cocycle.

Let \(\mathbb {K}\) be a field (or a commutative ring). The Witt algebra over \(\mathbb {K}\) is the \(\mathbb {K}\)-vector space \(\mathfrak {witt}\) (or free \(\mathbb {K}\)-module) with basis \((\ell _n)_{n \in \mathbb {Z}}\) and a \(\mathbb {K}\)-bilinear bracket \(\mathfrak {witt}\times \mathfrak {witt}\to \mathfrak {witt}\) given on basis elements by

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm). Then for any \(k,l \in \mathbb {Z}\) we have

For any \(n \in \mathbb {N}\), we have

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

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

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then for any \(n, m \in \mathbb {Z}\), we have

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\) and \(k \in \mathbb {Z}\), we have

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\) and \(k, m \in \mathbb {Z}\), we have

where \(\mathbb {I}_{{\mathrm{condition}}}\) is defined as \(1\) if the condition is true and \(0\) otherwise.

The cohomology class \([{\gamma }_{\mathfrak {hei}}] \in H^2(\mathfrak {g},\mathbb {K})\) of the Heisenberg cocycle is nonzero.

The set \(C^1(\mathfrak {g}, \mathfrak {a})\) of 1-cocycles of \(\mathfrak {g}\) with coefficients in \(\mathfrak {a}\) forms a vector space over \(\mathbb {K}\).

For any \(\gamma \in C^2(\mathfrak {g},\mathfrak {a})\) and any \(X,Y \in \mathfrak {g}\), we have the skew-symmetry property

The set \(C^2(\mathfrak {g}, \mathfrak {a})\) of 2-cocycles of \(\mathfrak {g}\) with coefficients in \(\mathfrak {a}\) forms a vector space over \(\mathbb {K}\).

If \(\mathfrak {g}\) is abelian, i.e., \([\mathfrak {g},\mathfrak {g}] = 0\), then the canonical projection

is a linear isomorphism.

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(n \in \mathbb {Z}\) and any \(v \in V\), there are only finitely many \(k \in \mathbb {Z}\) such that \({\mathbb {:} \mathsf{J}_{n-k} \, \mathsf{J}_k \mathbb {:}} \, v \ne 0\).

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc). Then for any \(v \in V\) there exists an \(N\) such that whenever \(\max \{ k,l\} \ge N\) we have \({\mathbb {:} \mathsf{J}_{k} \, \mathsf{J}_{l} \mathbb {:}} \, v = 0\).

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

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the local truncation condition (HeiTrunc), and suppose that \(\mathsf{A} \colon V \to V\) is a linear operator. Then for any \(n \in \mathbb {Z}\), the action of the commutator \([\mathsf{L}_n, \mathsf{A}]\) on any \(v \in V\) is given by the series

where only finitely many of the terms are nonzero.

The Virasoro cocycle is a 2-cocycle, \({\gamma }_{\mathfrak {vir}} \in C^2(\mathfrak {witt},\mathbb {K})\).

The cohomology class \([{\gamma }_{\mathfrak {vir}}] \in H^2(\mathfrak {witt},\mathbb {K})\) of the Virasoro cocycle is nonzero.

If \(\mathbb {K}\) is a field of characteristic zero, then \(\mathfrak {witt}\) is a Lie algebra over \(\mathbb {K}\).

(In the case when \(\mathbb {K}\) is a commutative ring, the \(\mathfrak {witt}\) is also a Lie algebra assuming the \(\mathbb {K}\) has characteristic zero and that for \(c \in \mathbb {K}\) and \(X \in \mathfrak {witt}\) we have \(c \cdot X = 0\) only if either \(c = 0\) or \(X = 0\).)

For any Witt algebra 2-cocycle \(\gamma \in C^2(\mathfrak {witt},\mathbb {K})\) with coefficients in \(\mathbb {K}\), we have

for all \(n,m,k \in \mathbb {Z}\).

For any 2-cocycle \(\gamma \in C^2(\mathfrak {witt},\mathbb {K})\), there exists a coboundary \(\partial \beta \) with \(\beta \in C^1(\mathfrak {witt},\mathbb {K})\) such that

for some scalar \(r \in \mathbb {K}\).

Let \(\gamma \in C^2(\mathfrak {witt},\mathbb {K})\) be a Witt algebra 2-cocycle such that \(\gamma (\ell _0 , \ell _n) = 0\) for all \(n \in \mathbb {Z}\). Then for any \(n,m \in \mathbb {Z}\) with \(n+m \ne 0\), we have

The Lie algebra 2-cohomology \(H^2(\mathfrak {witt},\mathbb {K})\) of the Witt algebra \(\mathfrak {witt}\) with coefficients in \(\mathbb {K}\) is one-dimensional and spanned by the class of the Virasoro cocycle \({\gamma }_{\mathfrak {vir}}\),

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).

Suppose that \((\mathsf{J}_{k})_{k \in \mathbb {Z}}\) satisfy the commutation relations (HeiComm) and the local truncation condition (HeiTrunc). Then there exists a representation of the Virasoro algebra \(\mathfrak {vir}\) with central charge \(c = 1\) on \(V\) (i.e., the central element \(C \in \mathfrak {vir}\) acts as \(c \, \mathrm{id}_V\) with \(c = 1\)) where the basis elements \(L_n\) of \(\mathfrak {vir}\) act by the Sugawara operators \((\mathsf{L}_n)_{n \in \mathbb {Z}}\).