arXiv subjects

Valerio Toledano-Laredo

Publications and source records attributed to Valerio Toledano-Laredo.

22 records · Page 2Linked to original sources

arXiv2002

Flat Connections and Quantum Groups

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture when g=sl(n) and when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection D and Cherednik's generalisation of the KZ connection to finite reflection groups.

math.QA↗

arXiv2001

A Kohno-Drinfeld theorem for quantum Weyl groups

Let g be a complex, simple Lie algebra and t a Cartan subalgebra of g. A new unitary, flat connection on t with values in any finite-dimensional g-module V and simple poles along the root hyperplanes was recently introduced by J. Millson and myself. This connection depends upon a complex parameter h and I conjectured that its monodromy is equivalent to the quantum Weyl group representation of the braid group of type g defined by Lusztig, Kirillov-Reshetikhin and Soibelman via the quantum group U_{h}g. In this paper, I prove this conjecture for g=sl_{n}.

math.QA↗

arXiv2001

Integrating Unitary Representations of Infinite-Dimensional Lie Groups

We show that in the presence of suitable commutator estimates, a projective unitary representation of the Lie algebra of a connected and simply connected Lie group G exponentiates to G. Our proof does not assume G to be finite--dimensional or of Banach-Lie type and therefore encompasses the diffeomorphism groups of compact manifolds. We obtain as corollaries short proofs of Goodman and Wallach's results on the integration of positive energy representations of loop groups and Diff(S^{1}) and of Nelson's criterion for the exponentiation of unitary representations of finite-dimensional Lie algebras.

math.RT↗

arXiv2001

Positive Energy Representations of the Loop Groups of Non Simply Connected Lie Groups

We classify and construct all irreducible positive energy representations of the loop group of a compact, connected and simple Lie group and show that they admit an intertwining action of Diff(S^{1}).

math.QA↗