# A Brief Introduction to Cohomology of Lie Algebra

Let \(G\) be a compact simply connected Lie group and \(\mathfrak{g}\) its Lie algebra. In this posts, we will discuss the motivation of Lie algebra cohomology of \(\mathfrak{g}\) and its connection with the de Rham cohomology of \(G\).

We fix a field \(\mathbb{k}\). All Lie algebras and vector spaces will be considered over \(\kk\).

## 1 Representations of Lie Algebras

Let \(\mathfrak{g}\) be a Lie algebra, \(M\) be a vector space \(M\), and \(\Lgl(M)\) be the space of endomorphisms of \(M\). The vector space \(\Lgl(M)\) admits a Lie algebra structure given by \([A, B]=A\circ B-B\circ A\).

**Definition 1 (Representations of a Lie Algebra) **The vector space \(M\) is said to be a **representation** of \(\mathfrak{g}\) if there is a Lie algebra homomorphism \(\rho: \Lg\to \Lgl(M)\), i.e. \(\rho\) is a linear map such that \(\rho([x, y])=[\rho(x),\rho(y)]\).

**Definition 2 (Modules over a Lie Algebra) **A vector space \(M\) is said to admit a \(\Lg\)-module structure if there is a \(\kk\)-bilinear map \(\pi: \Lg\otimes_{\kk}M\to M\) such that \([x,y]\cdot m=x\cdot(y\cdot m)-y\cdot (x\cdot m)\), where \(x\cdot m:=\pi(x, m)\).

One can check directly that a representation \(\rho:\Lg\to \Lgl(M)\) defines a \(\Lg\)-module structure on \(M\): \(g\cdot m=\rho(g)(m)\). Conversely, a \(\Lg\)-module structure on \(M\) defines a representation.

**Example 1 (Adjoint representation) **Let \(\Lg\) be a Lie algebra. Consider the map \(\ad: \Lg\to \Lgl(\Lg)\) defined by \(x\mapsto \ad_x:=[x, -]\). By Jacobian identity, the map \(\ad\) defines representation \(\Lg\) of \(\Lg\) called an **adjoint representation**. Indeed, \[
\begin{aligned}
\ad([x, y])(z)=&[[x, y], z]\\
=&[x, [y, z]]-[y, [x,z]]\\
=&\ad_x(\ad_y(z))-\ad_y(\ad_x(z))\\
=&[\ad_x,\ad_y](z).
\end{aligned}
\]

Note that an adjoint is a \(\kk\)-derivation of \(\Lg\), that is \(\ad_x\) satisfies the Leibniz rule for the Lie bracket

\[ \ad_x[g,h]=[\ad_x(g),h]+[g,\ad_x(h)]. \]

**Example 2 (Coadjoint representation) **Given any Lie algebra \(\Lg\), denote by \(\Lg^*\) the dual vector space, the space for \(\kk\)-linear functions on \(\Lg\). We define \(<\xi, x>=\xi(x)\), where \(\xi\in\Lg^*\) and \(x\in \Lg\). Consider the map \(\ad^*: \Lg \to \Lgl(\Lg^*)\) defined by \[
x \mapsto \ad_x^*: \Lg^* \to \Lg^*,
\] where \(\ad_x^*\) is given by \(\ad_x^*(\xi)(y)=-<\xi, [x, y]>=-\xi([x, y])\) for any \(\xi\in \Lg^*\) and \(y\in\Lg\).

The map \(\ad^*\) defines representation of \(\Lg\) in \(\Lg^*\) called an coadjoint representation of \(\Lg\). For any \(x, y, z\) in \(\Lg\) and \(\xi\) in \(\Lg^*\), we see that \[ \begin{aligned} \ad_{[x, y]}^*(\xi)(z)=&-<\xi, [[x, y], z]>\\ =&-<\xi, [x, [y, z]]>+<\xi, [y, [x,z]]>\\ =&\ad_x^*(\xi)([y,z])-\ad_y^*(\xi)([x,z])\\ =&-\ad_y^*(\ad_x^*(\xi))(z)+\ad_x^*(\ad_y^*(\xi))(z)\\ =&[\ad_x^*,\ad_y^*](\xi)(z). \end{aligned} \]

Note that the negative sign in the definition of \(\ad_x^*\) is necessary so that \(\ad\) is a Lie algebra homomorphism.

**Example 3 (Tensor product of representations) **Given two representations \(\varphi: \Lg\to\Lgl(M)\) and \(\psi: \Lg\to \Lgl(N)\), there is a natural representation on the tensor product \(M\otimes N\) given by \[
(\varphi\otimes\psi)(g)=\varphi(g)\otimes I+I\otimes\psi(g),
\] for any \(g\) in \(\Lg\). We can check that \(\varphi\otimes\psi\) is a Lie algebra morphism by direct calculations, i.e. \(\varphi\otimes\psi([g, h])=[(\varphi\otimes\psi)(g), (\varphi\otimes\psi)(h)]\) for any \(g\) and \(h\) in \(\Lg\). Indeed, we have

\[ \begin{aligned} \varphi\otimes\psi([g, h])=&\varphi([g, h])\otimes I+I\otimes\psi([g, h])\\ =&([\varphi(g), \varphi(h)])\otimes I+I\otimes([\psi(g), \psi(h)])\\ =&(\varphi(g)\varphi(h)-\varphi(h)\varphi(g))\otimes I\\ &+I\otimes(\psi(g)\psi(h)-\psi(h)\psi(g)), \end{aligned} \] and \[ \begin{aligned} &[(\varphi\otimes\psi)(g), (\varphi\otimes\psi)(h)]\\ =&(\varphi\otimes\psi)(g) (\varphi\otimes\psi)(h)-(\varphi\otimes\psi)(h) (\varphi\otimes\psi)(g)\\ =&(\varphi(g)\otimes I+I\otimes\psi(g))(\varphi(h)\otimes I+I\otimes\psi(h))\\ &-(\varphi(h)\otimes I+I\otimes\psi(h))(\varphi(g)\otimes I+I\otimes\psi(g))\\ =&(\varphi(g)\varphi(h)\otimes I+\varphi(h)\otimes\psi(g)+\varphi(g)\otimes\psi(h)+I\otimes\psi(g)\psi(h))\\ &-(\varphi(h)\varphi(g)\otimes I+\varphi(g)\otimes\psi(h)+\varphi(h)\otimes\psi(g)+I\otimes\psi(h)\psi(g))\\ =&(\varphi(g)\varphi(h)-\varphi(h)\varphi(g))\otimes I\\ &+I\otimes(\psi(g)\psi(h)-\psi(h)\psi(g)). \end{aligned} \]

## 2 Cohomology of Lie Algebra

One motivation of Cohomology of Lie Algebra is the de Rham cohomology.

Let \(G\) be a connected Lie group and \(\Lg\) the associated Lie algebra, i.e. the tangent space \(T_eG\) equipped with the Lie bracket \([X, Y](f):=X(Y(f))-Y(X(f))\). Note that the tangent space \(T_eG\) can be identified with the space of left-invariant vector fields on \(G\). Taking the dual, we may identify \(\Lg^*\) with space of left-invariant differential forms on \(G\). Then there is a complex \[ 0\to\wedge^0\Lg^*=\RR \to \wedge^1\Lg^*=\Lg^*\to \wedge^2\Lg^*\to \cdots \] which is isomorphic to the subcomplex of the de Rham complex \[ 0\to \RR \to \Omega_L^1(G)^G\to \Omega_L^2(G)^G\to \cdots, \] where \(\Omega_L^k(G)^G\) are spaces of left-invariant differential \(k\)-forms on \(G\).

Let \(H^k(\Lg)\) and \(H^k_L(G)\) be the cohomology groups of the above complex respectively.

By Theorem 15.1 of (Chevalley and Eilenberg 1948), we know that that \[ H^k_L(G)\cong H^k(\Lg). \] If in addition that \(G\) is also compact, then by Theorem 15.2 of (Chevalley and Eilenberg 1948), we know that \[ H^k_{dR}(G)\cong H^k_L(G)\cong H^k(\Lg), \] where \(H^k_{dR}(G)\) is the de Rham cohomology group.

Another motivation is from the study of extensions of Lie algebras. We refer the reader to (Knapp 1988) for detailed explorations.

Viewing \(\wedge^k\Lg^*\) as the space \(\Hom_{\RR}(\Lg, \RR)\) of multilinear alternating forms from \(\Lg\to \RR\), where \(\RR\) is the \(\Lg\)-module associated to the trivial representation of \(\Lg\) in \(\RR\), we can generalized the complex \((\wedge^*\Lg^*,d)\) to general \(\Lg\)-modules.

Let \(M\) be a \(\Lg\)-module. For \(k\geq 1\), the space \(C^k(\Lg, M)\) of \(k\)-cochains on \(\Lg\) with values in \(M\) is defined to be the space \(\Hom_{\kk}(\wedge^k\Lg, M)\) of multilinear alternating maps from \(\Lg\to M\). The space \(C^0(\Lg, M)\) of \(0\)-cochains is defined to be \(M\).

There is a coboundary operator \(d: C^k(\Lg, M)\to C^{k+1}(\Lg, M)\) defined by \[ \begin{aligned} d \omega\left(X_1 \wedge \cdots \wedge X_{k}\right) &=\sum_{j=0}^{k}(-1)^{j+1} X_{j}\cdot\left(\omega\left(X_1 \wedge \cdots \wedge \hat{X}_{j} \wedge \cdots \wedge X_{k}\right)\right) \\ &+\sum_{r<s} (-1)^{r+s}\omega\left(\left[X_{r}, X_{s}\right] \wedge X_1 \wedge \cdots \wedge \hat{X}_{r} \wedge \cdots \wedge \hat{X}_{s} \wedge \cdots \wedge X_{k}\right). \end{aligned} \]

The definition of \(d\) is determined by the conditions \(\d f=\sum (X_if)\omega_i\), \(d^2=0\), and the Leibniz rule \(\d(\xi\wedge\eta)=\d\xi\wedge\eta+(-1)^{\deg \xi}\xi\wedge\d\eta\), where \(f\in \Hom_{\kk}(\Lg, M)\), \(\{X_i\}\) is a basis of \(\Lg\) and \(\omega_i\in\Lg^*\) is the dual of \(X_i\). Note that the sign \((-1)^{\deg\xi}\) is necessary so that \(\d^2=0\).

To see that \(d\) is determined by those conditions, assume that \(\Lg\) is of dimension 2 with a basis \(X\) and \(Y\). Denote by \(\omega\) and \(\mu\) the dual basis. Suppose that \([X, Y]=aX+bY\). From the conditions \(\d f=\sum (X_if)\omega_i\) and \(d^2=0\), for any \(f\in \Hom_{\kk}(\Lg, M)\), we get \[ \begin{aligned} 0=&\d^2f\\ =&\d((Xf)\omega+(Yf)\mu)\\ =&\d(Xf)\wedge\omega+\d(Yf)\wedge\mu+(Xf)\d\omega+(Yf)\d\mu\\ =&Y(Xf)\mu\wedge\omega+X(Y(f))\omega\wedge\mu+(Xf)\d\omega+(Yf)\d\mu\\ =&[X, Y](f)\omega\wedge\mu+(Xf)\d\omega+(Yf)\d\mu\\ =&X(f)(a\omega\wedge\mu+\d\omega)+Y(f)(b\omega\wedge\mu+\d\mu). \end{aligned} \] Since \(f\) is arbitrary, we see that \[\d\omega(X\wedge Y)=-a\omega\wedge\mu(X\wedge Y)=-\omega([X, Y]),\] and \[\d\mu(X\wedge Y)=-b\omega\wedge\mu(X\wedge Y)=-\mu([X, Y).\]

The Leibniz rule is necessary for defining higher degree coboundary maps.

The cochain complex \((C^∗(\Lg, M), d)\) is called the **Chevalley-Eilenberg complex**.

**Definition 3 **The space of **\(k\)-cocycles** is defined to be \[
Z^{k}(\mathfrak{g}, M):=\ker\d=\left\{\omega \in C^{k}(\mathfrak{g}, M) \mid \d\omega=0\right\}.
\]

The space of **\(k\)-coboundaries** is defined to be

\[
B^{k}(\mathfrak{g}, M):=\im \d=\left\{\d\omega \mid \omega \in C^{k-1}(\mathfrak{g}, M)\right\}.
\]

The **\(k\)-th cohomology** space of \(\mathfrak{g}\) with values in \(M\) is defined as the quotient vector space \[
H^{k}(\mathfrak{g}, M):=Z^{k}(\mathfrak{g}, M)/B^{k}(\mathfrak{g}, M)
\]

**Definition 4 (The Universal Enveloping Algebra) **Let \(\Lg\) be a lie algebra. The quotient algebra \(U(\Lg)\) defined as \[
U(\Lg)=T(\Lg)/([x, y]-x\otimes y+y\otimes)
\] is called the universal enveloping algebra of \(\Lg\).

A \(\kk\)-module \(M\) is a \(\Lg\)-module if and only if \(M\) is a \(U(\Lg)\)-module. This result provides another approach to compute Lie algebra cohomology in terms of free resolution of \(U(\Lg)\)-modules. For details, we refer the reader to (Knapp 1988).

### 2.1 Cohomology of \(\Lg\) in degree 0

For any \(\Lg\)-module \(M\), the \(0\)-th cohomology space of \(\Lg\) with values in \(M\) is \[ H^0(\Lg, M)=Z^0(\Lg, M)=\{m\in M\mid X\cdot m=0 \text{ for all } X \text{ in }\Lg\}. \]

### 2.2 Cohomology of \(\Lg\) in degree 1

In degree 1, the space of cochains is \(\Hom_{\kk}(\Lg, M)\). Let \(\omega\) be a 1-cochain. Then \[ \d\omega(X, Y)=X\cdot \omega (Y) - Y\cdot \omega(X)-\omega([X, Y]). \]

The space of cocycles is \[ Z^1(\Lg, M)=\{\omega \mid \omega([X, Y])=X\cdot \omega (Y) - Y\cdot \omega(X)\text{ for all } X, Y \in \Lg\}. \]

The space of coboundaries is \[ B^1(\Lg, M)=\{\omega \mid \omega(X)=X\cdot m \text{ for some } m\in M\}. \]

**Example 4 **Consider \(\Lg\) as a \(\Lg\)-module via the adjoint representation. Then \[Z^1(\Lg, M)=\{D: \Lg\to \Lg\mid D([X, Y])=[X, D(Y)]+[D(X), Y]\}\] is the space of derivations of \(\Lg\) and \[B^1(\Lg, M)=\{D:\Lg\to \Lg\mid D(X)=[X, Y_D]\}=\{\ad_Y\mid Y\in \Lg\}\] is the space of **inner derivations** of \(\Lg\). The cohomology space \(H^1(\Lg, \Lg)\) is known as the space of **outer derivations**.

Note that those spaces of derivations admit Lie algebra structure.

**Example 5 **Consider \(\Lg^*\) as a \(\Lg\)-module via the coadjoint representation. Then \[
Z^1(\Lg, M)=\{\rho: \Lg\to \Lg^*\mid \rho([X, Y])=\ad_X^*(\rho(Y))-\ad_Y^*(\rho(X))\}
\] and \[
\begin{aligned}
B^1(\Lg, M)=&\{\rho:\Lg\to \Lg^*\mid \rho(X)=\ad_X^*(\xi) \text{ for some }\xi\in\Lg^*\}\\
=&\{\ad^*(\xi):\Lg\to \Lg^*\mid \xi\in\Lg^*\}.
\end{aligned}
\]

**Example 6 **Consider \(\Lsl(2, \mathbb{C})=<e,f,h>\) with the relations \([e, f]=h\), \([h, e]=2e\) and \([h, f]=-2f\). Let \(D:\Lg\to \Lg\) be a linear map such that \[
D\begin{pmatrix}
e\\
f\\
h
\end{pmatrix}
=\begin{pmatrix}
x_e& x_f &x_h\\
y_e&y_f&y_h\\
z_e&z_f&z_h
\end{pmatrix}\begin{pmatrix}
e\\
f\\
h
\end{pmatrix}.
\] A direct calculation shows that \(D\) is a Lie algebra derivation of \(\Lg\) is and only if \(D=\ad_X\), where \(X=y_h e-x_h f+\frac12x_e\), and \(x_f=y_e=z_h=0\).

Consequently, \(H^1(\Lsl(2,\mathbb{C}, \Lsl(2,\mathbb{C}))=0\), where the \(\Lsl(2,\mathbb{C})\)-module structure on itself is defined by the adjoint representation.

## References

*Transactions of the American Mathematical Society*63: 85–124. https://doi.org/d58jp5.

*Lie Groups, Lie Algebras, and Cohomology*. Princeton, N.J: Princeton University Press.