∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
A metric Lie algebra or quadratic Lie algebra over some ground field $\mathbb{F}$ is
Lie algebra $\mathfrak{g}$
a bilinear form $\mathfrak{g} \otimes \mathfrak{g} \overset{g}{\longrightarrow} \mathbb{F}$
such that $g$
is symmetric and invariant under the adjoint action, hence is an invariant polynomial on $\mathfrak{g}$;
is non-degenerate as a bilinear form, in that its tensor product-adjunct $\mathfrak{g} \to \mathfrak{g}^\ast$ is a linear isomorphism.
(But $g$ may be indefinite.)
The following table shows the data in a metric Lie representation equivalently
in category theory-notation;
in index notation:
graphics from Sati-Schreiber 19c
“Most” weight systems on chord diagrams come from metric Lie representations over metric Lie algebras: these are the Lie algebra weight systems.
On the Faulkner construction:
See also
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
further explored in
and putting to use the Faulkner construction that was previously introduced in (Faulkner 73)
See also:
Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)
Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)
Christian Saemann, appendix A of Lectures on Higher Structures in M-Theory (arXiv:1609.09815)
Last revised on December 2, 2023 at 09:30:58. See the history of this page for a list of all contributions to it.