nLab
positive energy representation
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

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 $L G \rtimes S^1$

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 $L G \rtimes S^1$ 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 October 17, 2019 at 08:41:44.
See the history of this page for a list of all contributions to it.