Why are Lie algebras "rigid" objects?

Question

I read the following motivation for quantum groups on wikipedia:

The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed".

Why is that? Could one point me towards the concerned theorems?

Answer 1

Here's an algebraic approach to deformations:

Definition. Let $({\mathfrak g},[-,-])$ be a finite-dimensional Lie algebra over a field ${\mathbb k}$. An infinitesimal deformation (of order $2$) of ${\mathfrak g}$ is a ${\mathbb k}[t]/(t^2)$-Lie algebra structure on ${\mathfrak g}\otimes_{\mathbb k} {\mathbb k}[t]/(t^2)$ restricting to ${\mathfrak g}$ upon annihilating $t$.

In other words, we seek for Lie brackets $\widetilde{[-,-]}$ on ${\mathfrak g}\otimes_{\mathbb k} {\mathbb k}[t]/(t^2)$ which have the form $$(\ddagger)\qquad\widetilde{[X,Y]_\psi}\ \ =\ \ [X,Y] + t\psi(X,Y)\quad\text{ for some }\quad \psi: {\mathfrak g}\otimes_{\mathbb k}{\mathfrak g}\to{\mathfrak g};$$ however, not any such $\psi$ will yield a Lie algebra structure, since the validity of antisymmetry and the Jacobi identity will put restrictions on it - see below.

Note that there is always the constant deformation given by the ${\mathbb k}[t]/(t^2)$-linear extension of $$\widetilde{[X,Y]}_\text{triv}\quad:=\quad [X,Y]\quad\text{ for }\quad X,Y\in{\mathfrak g}.$$

Definition. A morphism of infinitesimal deformations $$({\mathfrak g}\otimes_{\mathbb k}{\mathbb k}[t]/(t^2),\widetilde{[-,-]})\to ({\mathfrak g}\otimes_{\mathbb k}{\mathbb k}[t]/(t^2),\widehat{[-,-]})$$ is a homomorphism of ${\mathbb k}[t]/(t^2)$-Lie algebras restricting to the identity on ${\mathfrak g}$ upon annihilating $t$.

In particular, call an infinitesimal deformation trivial if it is isomorphic to the constant deformation.

Theorem: Any infinitesimal deformation of a semisimple Lie algebra is trivial.

This follows from Lie algebra cohomology once one has made the above notions more explicit.

1. A bilinear map $\psi: {\mathfrak g}\otimes_{\mathbb k}{\mathfrak g}\to{\mathfrak g}$ gives rise to a Lie algebra structure on ${\mathfrak g}\otimes_{\mathbb k}{\mathbb k}[t]/(t^2)$ via $(\ddagger)$ if and only if it is antisymmetric and a $2$-cocycle in the Cartan-Eilenberg complex computing the Lie algebra cohomology $\text{H}^{\ast}({\mathfrak g};({\mathfrak g},\text{ad}))$ of the adjoint representation.

2. An isomorphism of deformations between the constant deformation and $({\mathfrak g}\otimes_{\mathbb k}{\mathbb k}[t]/(t^2),\widetilde{[-,-]}_\psi)$ has the form $X\mapsto X + t\varphi(X)$ for some $\varphi: {\mathfrak g}\to{\mathfrak g}$, and this morphism being compatible with the bracket turns out to be equivalent to $\psi=\text{d}\varphi$ in the Chevalley-Eilenberg complex.

Hence, if $\text{H}^2({\mathfrak g};({\mathfrak g},\text{ad}))=0$, any infinitesimal deformation is trivial. For semisimple Lie algebras, this is the case by the following

Theorem (Whitehead). If ${\mathfrak g}$ is semisimple, then $\text{H}^2({\mathfrak g};V)=0$ for all f.d. representations $V$ of ${\mathfrak g}$.