Fix the Killing form and generators for a Lie subalgebra

[tscrim]

Currently the Killing form does not work for a Lie subalgebra since the adjoint_matrix() of an element is not returning coordinates in the basis for the subalgebra.

It was also noted that the adjoint_matrix() is returning the transpose of what it should be in order to define a representation. We fix this on this ticket since we also want to fix it for the subalgebras as well.

We also fix another issue that the generators of a Lie subalgebra are not elements of the subalgebra, but instead belong to the ambient algebra.

CC: @fchapoton @darijgr @heluani @ehaka @soehms

Component: algebra

Keywords: lie algebra, subalgebra

Author: Travis Scrimshaw

Branch/Commit: c7fb79d

Reviewer: Darij Grinberg

Issue created by migration from https://trac.sagemath.org/ticket/34006

[tscrim]

Branch: public/lie_algebras/fix_subalgebra_vector-34006

[tscrim]

comment:1

Here is a fix that is fairly simple. Unfortunately I don't know of a more general solution then the whack-a-mole way of fixing such issues as we come across them.

See also #33836 for a similar issue.

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New commits:

141bc3e

Using the coordinate_vector when working with subalgebra vectors.

[tscrim]

Commit: 141bc3e

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c7fb79d

Fixing the adjoint matrix being the transposed version.

[sagetrac-git]

Changed commit from 141bc3e to c7fb79d

[tscrim]

Reviewer: Darij Grinberg

[darijgr]

comment:4

The changes look good to me. Thanks for doing this, Travis!

[tscrim]

comment:5

Thanks. Positive review once the bot comes back (morally) green?

[tscrim]

comment:6

Morally green patchbot.

[vbraun]

Changed branch from public/lie_algebras/fix_subalgebra_vector-34006 to c7fb79d