Algebraic supergroups of Cartan type

TL;DR

This paper constructs and classifies connected affine algebraic supergroups associated with Cartan type Lie superalgebras, providing explicit descriptions and a classification result for these supergroups.

Contribution

It introduces a construction of supergroups G_V from Cartan type Lie superalgebras and proves their classification among connected affine algebraic supergroups with such tangent Lie superalgebras.

Findings

Explicit construction of supergroups G_V for Cartan type Lie superalgebras

Classification of all connected affine algebraic supergroups with Cartan type tangent Lie superalgebras

Detailed description of the supergroup associated with g := W(n)

Abstract

I present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup constructed in this way associated with g := W(n) and its standard representation is described somewhat more in detail. In addition, *** an "Erratum" is added here *** after the main text to fix a mistake which was kindly pointed out to the author by prof. Masuoka after the paper was published: this "Erratum" is accepted for publication in "Forum Mathematicum", it appears here in its final form (but prior to proofreading). In it, I also explain more in detail the *Existence Theorem* for algebraic supergroups of Cartan type which comes out of the main result in the original paper.