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Coconnected Hopf algebras: the coordinate ring of U_n and passage to quotients

Statement

Let $k$ be a field and $n \geq 1$ . Then:

(a) $O (U_{n}) = k [X_{i j} ∣ 1 \leq i < j \leq n]$ is a coconnected Hopf algebra: assigning weight $j - i$ to each generator $X_{i j}$ and letting $C_{r}$ be the span of the monomials of weight at most $r$ gives a filtration satisfying $Δ (C_{r}) \subseteq \sum_{i + j = r} C_{i} \otimes_{k} C_{j}$ ;

(b) if $A \to B$ is a surjective morphism of commutative Hopf algebras over $k$ and $A$ is coconnected, then $B$ is coconnected;

(c) assuming the Axiom of Choice (The Axiom of Choice), if $H \subseteq U_{n}$ is a closed subgroup scheme, then $O (H)$ is a quotient of $O (U_{n})$ and hence coconnected.

Facts & Assumptions

Given: A field $k$ , an integer $n \geq 1$ , and the Hopf algebra $O (U_{n}) = k [X_{i j} : i < j]$ of The upper unitriangular group scheme U_n and its coordinate ring with the displayed comultiplication.

[F1]

A commutative Hopf algebra $A$ is coconnected when it has an increasing filtration $(C_{r})$ with $C_{0} = k \cdot 1$ , $⋃_{r} C_{r} = A$ and $Δ (C_{r}) \subseteq \sum_{i + j = r} C_{i} \otimes_{k} C_{j}$ ; the filtration need not be finite in each degree. (Coconnected commutative Hopf algebras)

[F2]

$O (U_{n})$ is a polynomial algebra on the entries strictly above the diagonal with $Δ (X_{i j}) = X_{i j} \otimes 1 + 1 \otimes X_{i j} + \sum_{i < l < j} X_{i l} \otimes X_{l j}$ , $ε (X_{i j}) = 0$ , and counit/antipode compatible with these formulas. (The upper unitriangular group scheme U_n and its coordinate ring)

[F3]

Assuming AC for the geometric closed-subscheme/quotient-ring conversion, a closed subgroup scheme $H \subseteq U_{n}$ has coordinate ring $O (H) = O (U_{n}) / I$ for the Hopf ideal $I$ of functions vanishing on $H$ , and the quotient of a commutative Hopf algebra by a Hopf ideal carries the quotient Hopf algebra structure. (Closed subgroup schemes of an affine group scheme correspond to Hopf ideals, Hopf ideals, kernels and quotients of commutative Hopf algebras, The general linear group scheme and its coordinate ring)

Proof

Given: A field $k$ , an integer $n \geq 1$ , and the Hopf algebra $O (U_{n})$ .

1.1F1F2algebra

Declare the weight of the monomial $\prod X_{i j}^{a_{i j}}$ to be $\sum_{i < j} a_{i j} (j - i)$ , and let $C_{r}$ be the $k$ -span of the monomials of weight at most $r$ . Then $C_{0} = k \cdot 1$ , the $C_{r}$ increase, and $⋃_{r} C_{r} = O (U_{n})$ because every polynomial is a finite sum of monomials of bounded weight. On the generators, [F2] gives $Δ (X_{i j}) = X_{i j} \otimes 1 + 1 \otimes X_{i j} + \sum_{i < l < j} X_{i l} \otimes X_{l j}$ , and the three kinds of terms have total weight $j - i$ , $j - i$ , and $(l - i) + (j - l) = j - i$ ; hence $Δ (X_{i j}) \in \sum_{a + b = j - i} C_{a} \otimes C_{b}$ . Since $Δ$ is a $k$ -algebra homomorphism from the tensor product and the weights add under multiplication, $Δ (\prod X_{i j}^{a_{i j}}) \in \sum_{a + b = r} C_{a} \otimes C_{b}$ for a monomial of weight $r$ , and the condition $Δ (C_{r}) \subseteq \sum_{a + b = r} C_{a} \otimes C_{b}$ follows for all $r$ by linearity. This proves (a).

1.2F1algebra

Let $π : A \to B$ be a surjective morphism of commutative Hopf algebras and let $(C_{r})$ be a coconnected filtration on $A$ . Put $D_{r} = π (C_{r})$ . Then $D_{0} = k \cdot 1_{B}$ because $π$ preserves units and $C_{0} = k \cdot 1_{A}$ ; the $D_{r}$ increase and exhaust $B$ because $π$ is surjective; and, since $π \otimes π$ is surjective onto $B \otimes B$ with $(π \otimes π) (C_{i} \otimes C_{j}) = D_{i} \otimes D_{j}$ , the comultiplication of $B$ satisfies $Δ_{B} (D_{r}) = (π \otimes π) Δ_{A} (C_{r}) \subseteq \sum_{i + j = r} D_{i} \otimes D_{j}$ . Hence $B$ is coconnected, which proves (b).

2.1F3step 1.1step 1.2∎

Assume AC and let $H \subseteq U_{n}$ be a closed subgroup scheme. By [F3] the coordinate ring of $H$ is the quotient $O (U_{n}) / I$ by the Hopf ideal $I$ of functions vanishing on $H$ , and $π : O (U_{n}) \to O (H)$ is a surjective morphism of commutative Hopf algebras. By [step 1.1] $O (U_{n})$ is coconnected, so [step 1.2] applied to $π$ shows that $O (H)$ is coconnected. This proves (c) and completes the proof.

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
J. S. Milne, Algebraic Groups (v2.00, 20 December 2015 author-hosted preliminary edition) (standard reference, not scraped)