positive energy representation




Let GG be a compact Lie group and write LGL G for its loop group. See there for details and notation.


t θ:LGLG t_\theta \colon L G \to L G

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

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


Let VV be a topological vector space. A linear representation

S 1Aut(V) S^1 \to Aut(V)

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

A linear representation

ρ:LGAut(V) \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 LGS 1L G \rtimes S^1 such that the restriction to S 1S^1 is positive.


Relation to quantum group representations

See this MO discussion.



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)

Relation to ad-equivariant K-theory

On twisted ad-equivariant K-theory of compact Lie groups and the identification with the Verlinde ring of positive energy representations of their loop group:

Last revised on November 1, 2020 at 22:58:30. See the history of this page for a list of all contributions to it.