nLab
orthosymplectic super Lie algebra

Contents

Context

Super-Geometry

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The orthosymplectic supergroup Osp(N|2p)Osp(N|2p) is the sub-supergroup of the general linear supergroup GL(N|2p)GL(N|2p) on those elements which preserve the canonical graded-symmetric bilinear form on N|2p\mathbb{R}^{N|2p}, i.e. the form which is the canonical symmetric bilinear form on the even elements N\mathbb{R}^{N}, is the canonical symplectic form on the odd elements in 0|2p\mathbb{R}^{0|2p} and is zero on mixed pairs of elements.

The corresponding super Lie algebras are called the orthosymplectic Lie algebras 𝔬𝔰𝔭(N|2p)\mathfrak{osp}(N|2p). Over a field of characteristic zero these constitute the infinite BB- and DD-series in the classification of simple super Lie algebras. They are closely related to superconformal symmetry (e.g. DAuria-Ferrara-Lledo-Varadarajan 00), see at supersymmetry – Classification – Superconformal symmetry.

Examples

A\phantom{A}ddA\phantom{A}A\phantom{A}super anti de Sitter spacetimeA\phantom{A}
A\phantom{A}4A\phantom{A}OSp(8|4)Spin(3,1)×SO(7)\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;
A\phantom{A}5A\phantom{A}SU(2,2|5)Spin(4,1)×SO(5)\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;
A\phantom{A}7A\phantom{A}OSp(6,2|4)Spin(6,1)×SO(4)\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;

References

General

In superstring theory

Representation theory

The representation theory (singleton representations) of the orthosymplectic group is discussed in

  • Hermann Nicolai, Ergin Sezgin, Singleton representations of Osp(N,4)Osp(N,4), Physics Letters B, Volume 143, Issues 4–6, 16 August 1984, Pages 389-395

Last revised on August 1, 2018 at 18:11:44. See the history of this page for a list of all contributions to it.