nLab
metric Lie representation

Contents

Context

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

Given a Lie algebra (𝔤,[,])(\mathfrak{g}, [-,-]) and a Lie algebra representation 𝔤VρV\mathfrak{g} \otimes V \overset{\rho}{\to} V, and given non-degenerate inner products, hence “metrics”, both on 𝔤\mathfrak{g} and on VV, one may ask that all structure is compatible with these metrics. For the Lie algebra this means to have a metric Lie algebra and for the representation this means to have an orthogonal representation. Hence together this is an orthogonal representation of a metric Lie algebra, or metric Lie representation, for short.

Definition

The following table shows the data in a metric Lie representation equivalently

  1. in category theory-notation;

  2. in Penrose notation (string diagrams);

  3. in index notation:

graphics from Sati-Schreiber 19c

Properties

Structures induced from metric Lie representations

The following mathematical structures are induced from the data of metric Lie representations:

See there for more.

References

On the Faulkner construction (which became known as the M2-brane 3-algebra constructed from a metric Lie representation):

Last revised on December 28, 2019 at 18:49:16. See the history of this page for a list of all contributions to it.