[Paper Review] Geometric construction of quotients $G/H$ in supersymmetry

Akira Masuoka, Yuta Takahashi|arXiv (Cornell University)|Aug 17, 2018

Algebraic structures and combinatorial models21 references3 citations

TL;DR

This paper provides a direct geometric construction of quotient superschemes $\mathbb{G}/\mathbb{H}$ for affine algebraic supergroups $\mathbb{G}$ and closed sub-supergroups $\mathbb{H}$ over a field of characteristic $\neq 2$, explicitly describing the structure sheaf of $\mathbb{G}/\mathbb{H}$ and proving it satisfies all desirable geometric and functorial properties, including local splitness and compatibility with the associated quotient scheme $G/H$. The construction resolves a long-standing gap in supergroup quotient theory by realizing Brundan's desired properties in full generality for the first time.

ABSTRACT

It was proved by the first-named author and Zubkov [13] that given an affine algebraic supergroup $\mathbb{G}$ and a closed sub-supergroup $\mathbb{H}$ over an arbitrary field of characteristic $ e 2$, the faisceau $\mathbb{G} ilde{/} \mathbb{H}$ (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme $\mathbb{G}/\mathbb{H}$, which has desirable properties, in fact. We reprove this, by constructing directly the latter superscheme $\mathbb{G}/\mathbb{H}$. Our proof describes explicitly the structure sheaf of $\mathbb{G}/\mathbb{H}$, and reveals some new geometric features of the quotient, that include one which was desired by Brundan [2], and is shown in general, here for the first time.

Motivation & Objective

• To provide a direct geometric construction of the quotient superscheme $\mathbb{G}/\mathbb{H}$ for affine algebraic supergroups $\mathbb{G}$ and closed sub-supergroups $\mathbb{H}$, bypassing the indirect approach of [13].
• To explicitly describe the structure sheaf of $\mathbb{G}/\mathbb{H}$, revealing new geometric features such as local splitness.
• To prove that the quotient $\mathbb{G}/\mathbb{H}$ satisfies all six properties (Q1)–(Q6) listed by Brundan, particularly (Q5) on local splitness, which was previously unproven in general.
• To establish that Brundan's general results on representations of $Q(n)$ via $\mathbb{G}/\mathbb{H}$ can now be applied to a wider class of affine algebraic supergroups.

Proposed method

• The authors construct the quotient superscheme $\mathbb{G}/\mathbb{H}$ directly by explicitly describing its structure sheaf using the canonical coaction of the Hopf superalgebra $\mathcal{O}(\mathbb{H})$ on $\mathcal{O}(\mathbb{G})$.
• They use the canonical $\mathbb{D}$-coaction on $\mathcal{O}(\mathbb{G})$ and compute the coinvariants $\mathcal{O}(\mathbb{G})^{co\mathbb{D}}$, which identifies with the structure sheaf $\mathcal{O}_{\mathbb{G}/\mathbb{H}}$ via the isomorphism $\mathcal{O}_{\mathbb{G}/\mathbb{H}}(U) \simeq \mathcal{O}_{\mathbb{G}}(\pi^{-1}(U))^{co\mathbb{D}}$.
• The construction relies on the equivalence between superschemes and functorial superschemes via the Comparison Theorem, allowing the use of fppf sheaf-theoretic methods.
• The key technical tool is the use of Sweedler's notation and the $\square_D$-tensor product over the Hopf superalgebra $D = \mathcal{O}(\mathbb{H})$, which models the relative coinvariants.
• The authors prove that $\mathcal{O}_{\mathbb{G}/\mathbb{H}} \simeq \wedge_B(A \square_D \mathsf{Z})$, where $\mathsf{Z}$ is a free $B$-module, establishing the structure of the quotient as a locally split superscheme.
• They verify that the quotient $\mathbb{G}/\mathbb{H}$ is Noetherian, affine, faithfully flat over $\mathbb{G}$, and that its underlying scheme is isomorphic to $G/H$.

Experimental results

Research questions

• RQ1 Does a direct geometric construction of the quotient superscheme $\mathbb{G}/\mathbb{H}$ exist, explicitly describing its structure sheaf?
• RQ2 Is the quotient $\mathbb{G}/\mathbb{H}$ locally split in general, as conjectured by Brundan?
• RQ3 Can all of Brundan's six desired properties (Q1)–(Q6) for $\mathbb{G}/\mathbb{H}$ be proven to hold in full generality?
• RQ4 Is the structure sheaf of $\mathbb{G}/\mathbb{H}$ naturally isomorphic to the coinvariants of the coaction of $\mathcal{O}(\mathbb{H})$ on $\mathcal{O}(\mathbb{G})$?
• RQ5 Does the quotient $\mathbb{G}/\mathbb{H}$ represent the fppf sheaf $\mathbb{G}\tilde{/}\mathbb{H}$, thereby confirming its universal property?

Key findings

• The quotient superscheme $\mathbb{G}/\mathbb{H}$ is explicitly constructed as a superscheme with structure sheaf $\mathcal{O}_{\mathbb{G}/\mathbb{H}}(U) \simeq \mathcal{O}_{\mathbb{G}}(\pi^{-1}(U))^{co\mathbb{D}}$, confirming its geometric realization.
• The quotient $\mathbb{G}/\mathbb{H}$ is locally split, meaning it is split in a neighborhood of every point, resolving a key open question from Brundan's work.
• The quotient $\mathbb{G}/\mathbb{H}$ satisfies all six properties (Q1)–(Q6) listed by Brundan, including the faithful flatness and Noetherianity of the quotient morphism.
• The structure sheaf of $\mathbb{G}/\mathbb{H}$ is isomorphic to $\wedge_B(A \square_D \mathsf{Z})$, where $A = \mathcal{O}(\mathbb{G})$, $D = \mathcal{O}(\mathbb{H})$, and $\mathsf{Z}$ is a free $B$-module, providing a concrete algebraic description.
• The quotient $\mathbb{G}/\mathbb{H}$ represents the fppf sheaf $\mathbb{G}\tilde{/}\mathbb{H}$, confirming its universal property as a categorical quotient.
• The result generalizes Brundan's representation-theoretic results on $Q(n)$ to a wider class of affine algebraic supergroups, as long as $G/H$ is projective.