Implement finite-dimensional subalgebras with basis

[tscrim]

We currently have no generic class for subalgebras. In this ticket, we should implement basic features for magmatic subalgebras of a magmatic algebra with a basis:

• finding a basis,
• centralizers of a subset of elements, and
• some improvements for submodules.

This will also promote some methods such as center to the category of magmatic algebras.

Depends on #19448

CC: @nthiery @darijgr @avirmaux

Component: algebra

Keywords: subalgebras

Author: Travis Scrimshaw

Branch/Commit: public/algebras/subalgebras-19359 @ 0f159aa

Issue created by migration from https://trac.sagemath.org/ticket/19359

[tscrim]

comment:1

This will draw from the corresponding code for Lie algebras currently in #14901.

[tscrim]

Commit: 0f159aa

[tscrim]

comment:3

Some things that might need a discussion:

• I made the lift map from a submodule a coercion.
• I made a distinction between a subalgebra and a unital subalgebra, where the latter enforces that unit match in the subalgebra.
• Move code up to magamatic algebras, but it should be okay since it only involves products of 2 elements.
• How to better incorporate axiomatic information to the category of the subalgebras. For example, subalgebras of algebras of finite groups are always unital, but this is not reflected currently in the category setup because it seemed like too much of a headache right now.

Thoughts?

For a future ticket, we should implement idealizer (equivalent to a normalizer for Lie algebras).

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New commits:

0f159aa

Adding basic support for subalgebras

[tscrim]

Branch: public/algebras/subalgebras-19359

[tscrim]

Dependencies: #19448

[tscrim]

comment:4

I pulled some changes into #19448 an will need to rebase this over #19448.

[jhpalmieri]

comment:5

Is there a good way to extend this to work for some infinite-dimensional subalgebras? From the mathematical point of view, k[x] should be constructable as a subalgebra of k[x,y]. (I don't know how well the Sage implementations of these fit into the category framework.) From my point of view, at some point I should rewrite the Steenrod algebra stuff to fit into the category framework, and it has plenty of interesting infinite-dimensional subalgebras with explicit descriptions of their bases. At least some of these subalgebras are available in Sage. The Steenrod algebra also happens to be graded and finite-dimensional in each graded piece, for what that's worth.

[tscrim]

comment:6

The better way would be to just write a separate class (probably inheriting from CombinatorialFreeModule), implement a lift (and retract?) method which is the natural inclusion into the Steenrod algebra, have it be in the corresponding Subobjects() category, and make sure it knows the coercion.

Although for graded algebras with finite-dimensional components, this should be doable in a generic fashion (and would be useful for the combinatorial Hopf algebras). Something to work on after #17367.