nLab
metric Lie algebra

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

Defintion

A metric Lie algebra or quadratic Lie algebra over some ground field 𝔽\mathbb{F} is

  1. Lie algebra 𝔤\mathfrak{g}

  2. a bilinear form 𝔤𝔤g𝔽\mathfrak{g} \otimes \mathfrak{g} \overset{g}{\longrightarrow} \mathbb{F}

such that gg

  1. is symmetric and invariant under the adjoint action, hence is an invariant polynomial on 𝔤\mathfrak{g};

  2. is non-degenerate as a bilinear form, in that its tensor product-adjunct 𝔤𝔤 *\mathfrak{g} \to \mathfrak{g}^\ast is a linear isomorphism.

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

Examples

Applications

Lie algebra weight systems on chord diagrams

“Most” weight systems on chord diagrams come from metric Lie representations over metric Lie algebras: these are the Lie algebra weight systems.

References

General

On the Faulkner construction:

See also

Relation to M2-brane 3-Lie algebras

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to Lie algebra representations of metric Lie algebras is due to

The full identification of M2-brane 3-algebras with dualizable Lie algebra representations over metric Lie algebras is due to

reviewed in

  • José Figueroa-O'Farrill, slide 145 onwards in: Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf, pdf)

further explored in

and putting to use the Faulkner construction that was previously introduced in (Faulkner 73)

See also:

Last revised on February 8, 2020 at 12:32:25. See the history of this page for a list of all contributions to it.