4 Witt algebra and its 2-cohomology

4.1 Definition of the Witt algebra

Definition 15 Witt algebra

✓

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

\begin{align*} [\ell _n , \ell _m] = (n-m) \, \ell _{n+m} . \end{align*}

With some assumptions on \(\mathbb {K}\), the Witt algebra \(\mathfrak {witt}\) with the above bracket is an \(\infty \)-dimensional Lie algebra over \(\mathbb {K}\).

Lemma 16 Witt algebra is a Lie algebra

✓

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 \(X = 0\).)

Proof ▶

By construction, the bracket in Definition 15 is bilinear. It is antisymmetric on the basis vectors \(\ell _n\), \(n \in \mathbb {Z}\), so by bilinearity the bracket is antisymmetric. It remains to check that the bracket satisfies Leibnitz rule (or the Jacobi identity), i.e., that for any \(X, Y, X \in \mathfrak {witt}\) we have

\begin{align*} \big[X, [Y, Z] \big] = \big[[X, Y] , Z \big] + \big[Y , [X, Z] \big] . \end{align*}

This formula is trilinear in \(X,Y,Z\), so it suffices to verify it on basis vectors \(X = \ell _n\), \(Y = \ell _m\), \(Z = \ell _k\). Calculating, with Definition 15, we have

\begin{align*} \big[\ell _n, [\ell _m, \ell _k] \big] = \big[\ell _n, (m-k) \ell _{m+k} \big] = (m-k) \big(n-(m+k)\big) \, \ell _{n+m+k} \end{align*}

and

\begin{align*} & \big[[\ell _n, \ell _m] , \ell _k \big] + \big[\ell _m , [\ell _n, \ell _k] \big] \\ = \; & \big[(n-m) \ell _{n+m} , \ell _k \big] + \big[\ell _m , (n-k) \ell _{n+k}] \big] \\ = \; & (n-m) \big( n+m-k \big) \, \ell _{n+m+k} + (n-k) \big( m-(n+k) \big) \, \ell _{m+n+k} . \end{align*}

Noting that

\begin{align*} (n-m) \big( n+m-k \big) + (n-k) \big( m-(n+k) \big) = (m-k) \big(n-(m+k)\big) , \end{align*}

the Leibniz rule follows.

4.2 Virasoro cocycle

In this section we assume that \(\mathbb {K}\) is a field of characteristic zero and \(\mathfrak {witt}\) is the Witt algebra over \(\mathbb {K}\) as in Definition 15.

Definition 17 Virasoro cocycle

✓

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

\begin{align*} {\gamma }_{\mathfrak {vir}}(\ell _n,\ell _m) = \frac{n^3 - n}{12} \, \delta _{n+m,0} \end{align*}

is called the Virasoro cocycle.

Lemma 18 The Virasoro cocycle is a 2-cocycle

✓

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

Proof ▶

By the construction if Definition 17, \({\gamma }_{\mathfrak {vir}} \colon \mathfrak {witt}\times \mathfrak {witt}\to \mathbb {K}\) is bilinear. It’s antisymmetry on basis elements of the Witt algebra is easily checked, so \({\gamma }_{\mathfrak {vir}}\) is antisymmetric. It remains to prove the Leibniz rule for \({\gamma }_{\mathfrak {vir}}\), i.e., that for \(X,Y,X \in \mathfrak {witt}\), we have

\begin{align} \label{eq:LieTwoCocycle.leibniz} {\gamma }_{\mathfrak {vir}}(X,[Y,Z]) = \; & {\gamma }_{\mathfrak {vir}}([X,Y],Z) + {\gamma }_{\mathfrak {vir}}(Y,[X,Z]) . \end{align}

This formula is trilinear in \(X,Y,Z\), so it suffices to verify it for basis vectors \(X=\ell _n\), \(Y=\ell _m\), \(Z=\ell _k\). We calculate

\begin{align} {\gamma }_{\mathfrak {vir}}(\ell _n , [\ell _m , \ell _k]) = \; & {\gamma }_{\mathfrak {vir}}\big( \ell _n , (m-k) \ell _{m+k} \big) \\ = \; & (m-k) \, \frac{n^3 - n}{12} \delta _{n+m+k,0} . \end{align}

and

\begin{align} & {\gamma }_{\mathfrak {vir}}([\ell _n , \ell _m] , \ell _k) + {\gamma }_{\mathfrak {vir}}(\ell _m , [\ell _n , \ell _k]) \\ = \; & {\gamma }_{\mathfrak {vir}}((n-m) \ell _{n+m} , \ell _k) + {\gamma }_{\mathfrak {vir}}(\ell _m , (n-k) \ell _{n+k}) \\ = \; & (n-m) \, \frac{(n+m)^3 - (n+m)}{12} \delta _{n+m+k,0} + (n-k) \, \frac{m^3-m}{12} \delta _{n+m+k,0} . \end{align}

Both of the above results are nonzero only if \(k=-(n+m)\), in which case \(m-k = 2m+n\) and \(n-k = 2n+m\), so it suffices to note that

\begin{align*} (2m+n) \, (n^3 - n) = (n-m) \, \big( (n+m)^3 - (n+m) \big) + (2 n + m) (m^3 - m) \end{align*}

to verify the Leibniz rule for \({\gamma }_{\mathfrak {vir}}\).

Lemma 19 The Virasoro cocyle is nontrivial

✓

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

Proof ▶

Assume, by way of contradiction, that \({\gamma }_{\mathfrak {vir}} \in B^2(\mathfrak {witt},\mathbb {K})\), i.e., that \({\gamma }_{\mathfrak {vir}} = \partial \beta \) for some \(\beta \in C^1(\mathfrak {witt},\mathbb {K})\). Then, in particular, for every \(n \in \mathbb {Z}\) we would have

\begin{align*} {\gamma }_{\mathfrak {vir}}(\ell _n,\ell _{-n}) = \beta \big( [\ell _n, \ell _{-n}] \big) = 2 n \, \beta ( \ell _0 ) . \end{align*}

By Definition 17, this would imply

\begin{align*} \frac{n^3-n}{12} = 2 n \, \beta ( \ell _0 ) \end{align*}

for all \(n \in \mathbb {Z}\). Considering for example \(n=3\) and \(n=6\), we then get

\begin{align*} 2 = 6 \, \beta (\ell _0) \qquad \text{ and } \qquad \frac{35}{2} = 12 \, \beta (\ell _0) , \end{align*}

which obviously yield a contradiction.

4.3 Witt algebra 2-cohomology

Lemma 20 Witt algebra 2-cocycle condition for basis

✓

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

\begin{align*} (m-k) \, \gamma \big( \ell _n , \ell _{m+k} \big) + (k-n) \, \gamma \big( \ell _{m} , \ell _{n+k} \big) + (n-m) \, \gamma \big( \ell _k , \ell _{n+m} \big) \; = \; 0 \end{align*}

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

Proof ▶

Direct calculation, using Definitions 15 and 2.

Lemma 21 Witt algebra 2-cocycle support assuming normalization

✓

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

\begin{align*} \gamma (\ell _n , \ell _m) = 0. \end{align*}

Proof ▶

Apply Lemma 20 with \(k=0\). The last term vanishes, and by skew-symmetry of \(\gamma \), the first two terms simplify to yield

\begin{align*} (m+n) \, \gamma (\ell _n , \ell _m) = 0 , \end{align*}

which, assuming \(n+m \ne 0\), yields the asserted equation \(\gamma (\ell _n , \ell _m) = 0\).

Lemma 22 Normalization of Witt algebra 2-cocycles

✓

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

\begin{align*} \gamma + \partial \beta \; = \; r \cdot {\gamma }_{\mathfrak {vir}} \end{align*}

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

Proof ▶

Let \(\gamma \in C^2(\mathfrak {witt},\mathbb {K})\) be a Witt algebra 2-cocycle. Define a Witt algebra 1-cocycle \(\beta \in C^1(\mathfrak {witt},\mathbb {K})\) by linear extension of

\begin{align*} \beta (\ell _n) = \begin{cases} -\frac{1}{2} \gamma (\ell _1, \ell _{-1}) & \text{ if } n = 0 \\ \frac{1}{n} \gamma (\ell _0, \ell _n) & \text{ if } n \ne 0 . \end{cases}\end{align*}

For any \(n \ne 0\), we calculate

\begin{align*} \big( \gamma + \partial \beta \big) (\ell _0 , \ell _n) = \; & \gamma (\ell _0 , \ell _n) + \beta ([\ell _0,\ell _n]) \\ = \; & \gamma (\ell _0 , \ell _n) - n \, \beta (\ell _n) \\ = \; & \gamma (\ell _0 , \ell _n) - n \, \frac{1}{n}\gamma (\ell _0, \ell _n) = 0 . \end{align*}

This property and Lemma 21 imply that

\begin{align*} \big( \gamma + \partial \beta \big) (\ell _0 , \ell _n) = 0 \end{align*}

whenever \(n+m \ne 0\).

We will show the asserted equation with

\begin{align*} r = 2 \, \big( \gamma + \partial \beta \big) ( \ell _2, \ell _{-2}) . \end{align*}

By comparison with the Virasoro cocycle \({\gamma }_{\mathfrak {vir}}\) of Definition 17, and using skew-symmetry, it remains to show that for any \(n \in \mathbb {N}\) we have

\begin{align*} \big( \gamma + \partial \beta \big) (\ell _n , \ell _{-n}) = r \; \frac{n^3 - n}{12} . \end{align*}

The case \(n = 0\) is a direct consequence of antisymmetry. The case \(n = 1\) follows using the definition of \(\beta \) and the calculation

\begin{align*} \big( \gamma + \partial \beta \big) (\ell _1 , \ell _{-1}) = \; & \gamma (\ell _1 , \ell _{-1}) + \beta ([\ell _1,\ell _{-1}]) \\ = \; & \gamma (\ell _1 , \ell _{-1}) + 2 \, \beta (\ell _0) \\ = \; & \gamma (\ell _1 , \ell _{-1}) - 2 \, \frac{1}{2} \gamma (\ell _1, \ell _{-1}) = 0 . \end{align*}

The case \(n = 2\) follows directly by the choice of \(r\). We prove the equality in the cases \(n \ge 3\) by induction on \(n\). Assume the equation for smaller values of \(n\). Apply Lemma 20 to \(\gamma + \partial \beta \) with \(m = 1-n\) and \(k = -1\) to get

\begin{align*} 0 = \; & (2-n) \, \big( \gamma + \partial \beta \big) (\ell _n , \ell _{-n}) + (-1-n) \, \big( \gamma + \partial \beta \big) (\ell _{1-n} , \ell _{n-1}) + (2 n - 1) \, \big( \gamma + \partial \beta \big) (\ell _1 , \ell _{-1}) \\ = \; & (2-n) \, \big( \gamma + \partial \beta \big) (\ell _n , \ell _{-n}) - (-1-n) \, r \frac{(n-1)^3 - (n-1)}{12} \\ = \; & (2-n) \, \big( \gamma + \partial \beta \big) (\ell _n , \ell _{-n}) + \frac{r}{12} (n+1) n (n-1) (n-2) . \end{align*}

where in the seciond step we used the induction hypothesis. Since \(2-n \ne 0\), this can be solved for

\begin{align*} \big( \gamma + \partial \beta \big) (\ell _n , \ell _{-n}) = \; & - \frac{r}{12} \frac{(n+1) n (n-1) (n-2)}{2-n} = r \frac{n^3 - n}{12} , \end{align*}

completing the induction step.

Lemma 23 Witt algebra 2-cohomology is spanned by the Virasoro cocycle

✓

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 \({\gamma }_{\mathfrak {vir}}\),

\begin{align*} H^2(\mathfrak {witt},\mathbb {K}) \; = \; \mathbb {K}\cdot [{\gamma }_{\mathfrak {vir}}] . \end{align*}

Proof ▶

This follows directly from Lemmas 22 and 19.