• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

191 submissions , 151 unreviewed
4,796 questions , 1,987 unanswered
5,288 answers , 22,472 comments
1,470 users with positive rep
773 active unimported users
More ...

  GKO (or coset) construction - all possible highest weights $h$

+ 4 like - 0 dislike

I am reading the famous paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive.

From a compact simple Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\mathfrak{h}$, they obtain a representation $Vir(\mathfrak{g},\mathfrak{h})$ of the Virasoro algebra. The unitary highest weight irreducible representations of a Virasoro algebra are labelled by $(c,h)$, with $c$ the central charge and $h$ the highest weight. In the paper, they show that $c$ can take any value in the series enter image description here

In a second moment, they show that $h$ can take any value in the series enter image description here

They prove this last result using character theory. But what I do not understand is the idea behind this last proof. They start it with the following paragraph: enter image description here

In particular, I do not understand how to make sense of the highlighted sentence: what do they mean with "decompose with respect to" in this context; how such decomposition helps us at all; and how exactly does (2.20) come to be.

This post imported from StackExchange MathOverflow at 2019-08-21 22:28 (UTC), posted by SE-user Soap
asked May 5, 2019 in Mathematics by Soap (40 points) [ no revision ]
retagged Aug 22, 2019

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights