nLab
positive energy representation
Context
Representation theory

Ingredients
Definitions
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Geometric representation theory
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Theorems
Contents
Definition
Let $G$ be a compact Lie group and write $L G$ for its loop group . See there for details and notation.

Write

$t_\theta \colon L G \to L G$

for the automorphism which rotates loops by an angle $\theta$ .

The corresponding semidirect product group we write $S^1 \rtimes L G$

Definition
Let $V$ be a topological vector space . A linear representation

$S^1 \to Aut(V)$

of the circle group is called positive if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a linear operator with positive spectrum .

A linear representation

$\rho : L G \to Aut(V)$

is said to have positive energy or to be a positive energy representation if it extends to a representation of the semidirect product group $S^1 \rtimes L G$ such that the restriction to $S^1$ is positive.

Properties
Relation to quantum group representations
See this MO discussion .

References
The standard textbook on loop groups is

Andrew Pressley, Graeme Segal , Loop groups Oxford University Press (1988)
A review talk is

Discussion in the context of string theory (the Witten genus ) is in

Kefeng Liu , section 2.2 of On modular invariance and rigidity theorems , J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID , pdf )
Last revised on August 12, 2014 at 00:24:23.
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