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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 \(\alpha \in \mathbb {K}\). The charged Fock space of charge \(\alpha \) is the Verma module \(\mathscr {V}^{\eta }\) associated to the linear functional \(\eta \colon \mathfrak {hei}^0 \to \mathbb {K}\) with

We denote the charged Fock space by \(\mathscr {F}^{\alpha }\). The highest weight vector \(\mathbb {v}^{\eta }\) is (also) called the vacuum vector of the charged Fock space, and we denote it by \(\mathbb {v}^{\alpha }\).

The charged Fock space of charge \(\alpha \) becomes a representation of the Virasoro algebra \(\mathfrak {vir}\) with central charge \(c = 1\) via the Sugawara construction (Theorem 47).

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 triangular decomposition

of \(\mathfrak {hei}\) is defined so that \(\mathfrak {hei}^0\) is spanned by \(J_0, K \in \mathfrak {hei}\), \(\mathfrak {hei}^+\) is spanned by \(J_k\) for \(k {\gt} 0\), and \(\mathfrak {hei}^-\) is spanned by \(J_k\) for \(k {\lt} 0\).

(Without further comment, for the Heisenberg algebra we always use this triangular decomposition.)

The vector \(\mathbb {v}^{\eta } := 1 + J^{\eta } \in \mathscr {V}^{\eta }\) (the equivalence class of the unit element of \(A\)) is called the highest weight vector of \(\mathscr {V}^{\eta }\). It is cyclic, i.e., it generates the whole Verma module \(\mathbb {v}^{\eta }\) as an \(A\)-module, and it satisfies

for all \(i \in I\), where \(a_i \in A\) and \(r_i \in \mathbb {K}\) are the algebra elements and scalars in the collection \(\eta = (a_i, r_i)_{i \in I}\).

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

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

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

given by

We then have \(\partial \beta \in \mathrm{Z}^2(\mathfrak {g},\mathfrak {a})\). The mapping \(\partial \colon \mathrm{C}^1(\mathfrak {g},\mathfrak {a}) \to \mathrm{Z}^2(\mathfrak {g},\mathfrak {a})\) is linear. Its range is denoted \(\mathrm{B}^2(\mathfrak {g},\mathfrak {a}) \subset \mathrm{Z}^2(\mathfrak {g},\mathfrak {a})\) 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 \(\mathrm{Z}^2(\mathfrak {g}, \mathfrak {a})\).

The vector space

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

Let \(\mathfrak {g}\) be a Lie algebra over \(\mathbb {K}\), with a triangular decomposition \(\mathfrak {g}= \mathfrak {g}^0 \oplus \mathfrak {g}^+ \oplus \mathfrak {g}^-\). Let \(\eta \colon \mathfrak {g}^0 \to \mathbb {K}\) be a linear functional on the Cartan part. Then the Verma module associated to the "highest weight" \(\eta \) is the generalized Verma module for the universal enveloping algebra \(\mathscr {U}(\mathfrak {g})\), associated with the data consisting of the pairs \((H, \eta (H)) \in \mathscr {U}(\mathfrak {g}) \times \mathbb {K}\) for \(H \in \mathfrak {g}^0\) and pairs \((E,0) \in \mathscr {U}(\mathfrak {g}) \times \mathbb {K}\) for \(E \in \mathfrak {g}^+\). The Verma module is (by a mild abuse of notation) still denoted by \(\mathscr {V}^{\eta }\).

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 40).

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

Let \(\mathfrak {g}\) be a Lie algebra over \(\mathbb {K}\). A triangular decomposition of \(\mathfrak {g}\) is a decomposition

of \(\mathfrak {g}\) into a vector space direct sum of three vector subspaces: Cartan part (or Cartan subalgebra) \(\mathfrak {g}^0\subseteq \mathfrak {g}\), the upper part \(\mathfrak {g}^+ \subseteq \mathfrak {g}\), and the lower part \(\mathfrak {g}^- \subseteq \mathfrak {g}\).

(Note that in this definition we do not yet require \(\mathfrak {g}^0, \mathfrak {g}^+, \mathfrak {g}^- \subseteq \mathfrak {g}\) to be Lie subalgebras, with the Cartan subalgebra being abelian and the upper and lower parts being nilpotent. In intended use cases, we typically have these properties, however.)

Let \(A\) be an algebra over a commutative ring \(\mathbb {K}\). Let \(\eta = (a_i, r_i)_{i \in I} \in (A \times \mathbb {K})^I\) be a family of pairs of algebra elements \(a_i \in A\) and scalars \(r_i \in \mathbb {K}\). The generalized Verma module associated to the data \(\eta \) is the quotient

where \(J^{\eta } \subset A\) is the left ideal (an \(A\)-submodule) of \(A\) generated by the elements \(a_i - r_i \, 1 \in A\). Note that \(\mathscr {V}^{\eta }\) is a (left) \(A\)-module.

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.

A triangular decomposition

of \(\mathfrak {vir}\) is defined so that \(\mathfrak {vir}^0\) is spanned by \(L_0, C \in \mathfrak {vir}\), \(\mathfrak {vir}^+\) is spanned by \(L_n\) for \(n {\gt} 0\), and \(\mathfrak {vir}^-\) is spanned by \(L_n\) for \(n {\lt} 0\).

(Without further comment, for the Virasoro algebra we always use this triangular decomposition.)

Let \(c, h \in \mathbb {K}\). The Virasoro Verma module with central charge \(c\) and conformal weight \(h\) is the Verma module \(\mathscr {V}^{\eta }\) associated to the linear functional \(\eta \colon \mathfrak {vir}^0 \to \mathbb {K}\) with

We denote the Virasoro Verma module by \(\mathscr {V}^{c,h}\).

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.

Let \(a \in \mathscr {U}(\mathfrak {hei})\) be an arbitrary element of the universal enveloping algebra of the Heisenberg algebra \(\mathfrak {hei}\). Then there exists a \(k_0=k_0(a) \in \mathbb {Z}\) such that for all \(k \ge k_0\) we have

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

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

The set \(\mathrm{Z}^2(\mathfrak {g}, \mathfrak {a})\) 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.

Let \(\alpha \in \mathbb {K}\) and let \(v \in \mathscr {F}^{\alpha }\) be a vector in the charged Fock space with charge \(\alpha \). Then there exists a \(k_0 \in \mathbb {Z}\) such that for all \(k \ge k_0\) we have

(In other words, the charged Fock space satisfies the local truncation condition (HeiTrunc) needed for the Sugawara construction.)

The vacuum vector \(\mathbb {v}^{\alpha }\) of the charged Fock space of charge \(\alpha \in \mathbb {K}\) satisfies

In particular (by the universal property of Verma modules) there exists a Virasoro-module map

such that \(\mathbb {v}^{c=1,h=\alpha ^2/2} \mapsto \mathbb {v}^{\alpha }\).

Suppose that \(M\) is an \(A\)-module and \(v \in M\) is a vector such that

for all \(i \in I\), where \(a_i \in A\) and \(r_i \in \mathbb {K}\) are the algebra elements and scalars in the collection \(\eta = (a_i, r_i)_{i \in I}\). Then there exists a (unique) \(A\)-module homomorphism \(\phi \colon \mathscr {V}^{\eta } \to M\) such that \(\phi (\mathbb {v}^{\eta }) = v\). The range of the map \(\phi \) is the submodule generated by \(v\) in \(M\).

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

The cohomology class \([{\gamma }_{\mathfrak {vir}}] \in \mathrm{H}^2(\mathfrak {witt},\mathbb {K})\) 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 \(\gamma \in \mathrm{Z}^2(\mathfrak {witt},\mathbb {K})\) with coefficients in \(\mathbb {K}\), we have

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

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

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

Let \(\gamma \in \mathrm{Z}^2(\mathfrak {witt},\mathbb {K})\) 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 \(\mathrm{H}^2(\mathfrak {witt},\mathbb {K})\) 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 \((\mathsf{L}_n)_{n \in \mathbb {Z}}\).