The Lie algebra of a linear algebraic group

Let k be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group) over k. By definition, Lie(G) is the Lie algebra of all derivations of k[G] that commute with the left action of G. As in the Lie group case, it can be identified with the tangent space to G at the identity element.

Enveloping algebra

There is the following general construction for a Hopf algebra. Let A be a Hopf algebra. The finite dual of A is the space of linear functionals on A with kernels containing left ideals of finite codimensions. Concretely, it can be viewed as the space of matrix coefficients.

Definition

Let X = Spec A be an affine scheme over a field k and let I_x be the kernel of the restriction map ${\displaystyle A\to k(x)}$, the residue field of x. By definition, a distribution f supported at x'' is a k-linear functional on A such that ${\displaystyle f(I_{x}^{n})=0}$ for some n. (Note: the definition is still valid if k is an arbitrary ring.)

Now, if G is an algebraic group over k, we let Dist(G) be the set of all distributions on G supported at the identity element (often just called distributions on G). If f, g are in it, we define the product of f and g, demoted by f * g, to be the linear functional

${\displaystyle k[G]{\overset {\Delta }{\to }}k[G]\otimes k[G]{\overset {f\otimes g}{\to }}k\otimes k=k}$

where Δ is the comultiplication that is the homomorphism induced by the multiplication ${\displaystyle G\times G\to G}$. The multiplication turns out to be associative (use ${\displaystyle 1\otimes \Delta \circ \Delta =\Delta \otimes 1\circ \Delta }$) and thus Dist(G) is an associative algebra, as the set is closed under the muplication by the formula:

(*) ${\displaystyle \Delta (I_{1}^{n})\subset \sum _{r=0}^{n}I_{1}^{r}\otimes I_{1}^{n-r}.}$

It is also unital with the unity that is the linear functional ${\displaystyle k[G]\to k,\phi \mapsto \phi (1)}$, the Dirac's delta measure.

The Lie algebra Lie(G) sits inside Dist(G). Indeed, by definition, Lie(G) is the tangent space to G at the identity element 1; i.e., the dual space of ${\displaystyle I_{1}/I_{1}^{2}}$. Thus, a tangent vector amounts to a linear functional on I₁ that has no constant term and kills the square of I₁ and the formula (*) implies ${\displaystyle [f,g]=f*g-g*f}$ is still a tangent vector.

Let ${\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}$ be the Lie algebra of G. Then, by the universal property, the inclusion ${\displaystyle {\mathfrak {g}}\hookrightarrow \operatorname {Dist} (G)}$ induces the algebra homomorphism:

${\displaystyle U({\mathfrak {g}})\to \operatorname {Dist} (G).}$

When the base field k has characteristic zero, this homomorphism is an isomorphism.^[1]

Examples

Correspondence

• For any closed subgroups H, 'K of G, if k is perfect and H is irreducible, then

${\displaystyle H\subset K\Leftrightarrow \operatorname {Dist} (H)\subset \operatorname {Dist} (K).}$

• If V is a G-module (that is a representation of G), then it admits a natural structure of Dist(G)-module, which in turns gives the module structure over ${\displaystyle {\mathfrak {g}}}$.
• Any action G on an affine algebraic variety X induces the representation of G on the coordinate ring k[G]. In particular, the conjugation action of G induces the action of G on k[G]. One can show Iⁿ
  ₁ is stable under G and thus G acts on (k[G]/Iⁿ
  ₁)^* and whence on its union Dist(G). The resulting action is called the adjoint action of G.