Implement representations for Virasoro algebra

[tscrim]

We implement the chargeless representations and Verma modules for the Virasoro algebra. Along the way, we also add witt to the catalog and change the bracket convention for the Witt and Virasoro algebras.

A note on the change to the bracket convention: Our current bracket convention is non-standard with the rest of the literature and causes formulas to change in subtle ways. So the easiest thing to do is change it, and I think this will be better in the long run.

CC: @sagetrac-sage-combinat @darijgr

Component: algebra

Keywords: lie algebras

Author: Travis Scrimshaw

Branch/Commit: 999dcfb

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/23037

[tscrim]

Branch: public/lie_algebras/reprs_virasoro-23037

[tscrim]

Commit: 8069b18

[tscrim]

New commits:

65a57d4	Improving doc and check for (p)Witt algebras.
e614c79	Fixed `__mul__` for Lie algebra elements to handle actions.
c7a49c4	Implement chargeless represenation.
8069b18	Changing bracket convention for Witt and Virasoro. Implementing Verma modules.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

aaeb1c9

Updating documentation.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 8069b18 to aaeb1c9

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from aaeb1c9 to 4c17c8c

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

4c17c8c

Merge branch 'public/lie_algebras/reprs_virasoro-23037' of git://trac.sagemath.org/sage into public/lie_algebras/reprs_virasoro-23037

[fchapoton]

comment:4

There is one it.next that should be next(it) for py3 sake, no ?

and doc does not build, probably because of

+    this for `\{d_i \mid -10 \leq i < 10\}::

EDIT: typo "+ REFERNCES::"

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 4c17c8c to 2fdd3e4

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

2fdd3e4

Use next() and typo fixes.

[tscrim]

comment:6

Yes, thanks. Fixed.

[fchapoton]

comment:7

another issue here:

+    REFERENCES::

has too many colons

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

4b9ef94

No double colon on references.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 2fdd3e4 to 4b9ef94

[tscrim]

comment:9

Good catch. Fixed.

[tscrim]

Description changed:

--- 
+++ 
@@ -1,3 +1,3 @@
-We implement the chargeless representations and Verma modules for the Virasoro algebra. Along the way, we also add `witt` to the catalog, fix an issue with actions for Lie algebra elements, and change the bracket convention for the Witt and Virasoro algebras.
+We implement the chargeless representations and Verma modules for the Virasoro algebra. Along the way, we also add `witt` to the catalog and change the bracket convention for the Witt and Virasoro algebras.

 A note on the change to the bracket convention: Our current bracket convention is non-standard with the rest of the literature and causes formulas to change in subtle ways. So the easiest thing to do is change it, and I think this will be better in the long run.

[fchapoton]

comment:11

A strange sentence here in Verma modules:

+    The Virasoro algebra admits a triangle decomposition of

Maybe you mean of the shape ?

Otherwise, looks good.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 4b9ef94 to 8dfb361

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

8dfb361

Tweak documentation.

[tscrim]

comment:13

Replying to @fchapoton:

A strange sentence here in Verma modules:

+    The Virasoro algebra admits a triangle decomposition of

Maybe you mean of the shape ?

No, it is a triangular decomposition in the same way of simple Lie algebras or Lie groups by the root system (positive, negative, and 0 parts). I made some tweaks to the documentation; they probably don't help much, but maybe they do.

Otherwise, looks good.

Thank you very much for looking at this.

[fchapoton]

comment:14

I understand the triangular decomposition, sure.

You introduced a typo : "trianglar"

I would simply remove the "of"

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

999dcfb

Fixes in virasoro.VermaModule.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 8dfb361 to 999dcfb

[tscrim]

comment:16

Ah, sorry, I misunderstood your comment:11. Here is the fix from comment:14.

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:17

Thanks. Then good to go.

[tscrim]

comment:18

Thank you again very much for the review!

[vbraun]

Changed branch from public/lie_algebras/reprs_virasoro-23037 to 999dcfb