Classical complex affine algebraic actions and rational modules

A complex affine algebraic group is a nonempty affine algebraic set $G$ equipped with a group law whose multiplication $G\times G\to G$ and inversion $G\to G$ are morphisms. Write $e$ for its identity. Neither $G$ nor any affine algebraic set below is required to be irreducible. Its coordinate algebra $H=\mathbb{C}[G]$ has maps $\mathrm{\Delta }h(g,h^{\mathrm{\prime }})=h(g{h}^{\mathrm{\prime }})$, $\epsilon (h)=h(e)$, and $S(h)(g)=h(g^{-1})$, using Products of affine algebraic sets have tensor-product coordinate rings to identify product rings.

The group identities give $(\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }=(\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }$ and $(\epsilon \otimes \mathrm{id})\mathrm{\Delta }=\mathrm{id}=(\mathrm{id}\otimes \epsilon )\mathrm{\Delta }$: evaluate on $(g,h,k)$ and $g$, respectively. Likewise multiplying the two factors of $(S\otimes \mathrm{id})\mathrm{\Delta }$ or $(\mathrm{id}\otimes S)\mathrm{\Delta }$ evaluates to $h(e)$, so both are $\epsilon (h)1$. These are identities of coordinate functions by the product-ring lemma and polynomial-function identification, and $S^2=\mathrm{id}$ follows from inversion squared.

An algebraic (or rational) left action on an affine algebraic set $X$ is a morphism $a:G\times X\to X$ satisfying $a(e,x)=x$ and $a(g,a(h,x))=a(gh,x)$. Here morphisms and coordinate rings are those of A morphism from an open subset of a classical affine variety to an affine variety and The coordinate ring of a classical affine algebraic set. An equivariant morphism $u:X\to Y$ satisfies $u(gx)=gu(x)$.

A rational $G$-module is a complex vector space with a linear left action $r$ of $G$ such that every vector belongs to a finite-dimensional stable subspace $W$ on which $r:G\to GL(W)$ is a morphism. The zero subspace is permitted. For a finite-dimensional $W$, regular matrix coefficients and the group law express this condition; $GL(W)$ has coordinate ring $\mathbb{C}[t_{ij},\det (t_{ij}{)}^{-1}]$.

The left action on functions is always $(r(g)f)(x)=f(g^{-1}x)$. Its right-comodule convention is $c:V\to V\otimes H$ with $(c\otimes \mathrm{id})c=(\mathrm{id}\otimes \mathrm{\Delta })c$ and $(\mathrm{id}\otimes \epsilon )c=\mathrm{id}$. Evaluation of the second factor at $g$ gives $r(g)$. The direct action pullback instead lands in $H\otimes \mathbb{C}[X]$ and evaluates to $f(gx)$; these two conventions must be distinguished.