Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quotients in super-symmetry: formal supergroup case

Yuta Takahashi, Akira Masuoka

Abstract

We describe the structure of the quotient $\mathfrak{G}/\mathfrak{H}$ of a formal supergroup $\mathfrak{G}$ by its formal sub-supergroup $\mathfrak{H}$. This is a consequence which arises as a continuation of the authors' work (partly with M. Hashi) on algebraic/analytic supergoups.The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular.

Quotients in super-symmetry: formal supergroup case

Abstract

We describe the structure of the quotient of a formal supergroup by its formal sub-supergroup . This is a consequence which arises as a continuation of the authors' work (partly with M. Hashi) on algebraic/analytic supergoups.The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular.
Paper Structure (13 sections, 14 theorems, 104 equations)

This paper contains 13 sections, 14 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Background and objective
  3. Main results
  4. Organization of the paper
  5. Preliminaries
  6. Supersymmetry
  7. Super-coalgebras and Hopf superalgebras
  8. The exterior algebra
  9. The associated graded coalgebra
  10. Quick review of structure of Hopf superalgebras
  11. Co-free super-coalgebras
  12. The quotient $\mathbb{J}/\space/\mathbb{K}$
  13. On formal superschemes

Key Result

Lemma 3.1

$\mathsf{SMod}^{\mathbb{C}}$ forms a monoidal category, whose tensor product is given by the cotensor product $\square_{\mathbb{C}}$, and whose unit object is $\mathbb{C}$. This is in fact symmetric with respect to the supersymmetry restricted to the cotensor products.

Theorems & Definitions (33)

  • Remark 2.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Example 3.6
  • Remark 3.7
  • ...and 23 more