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Register a new userAffine Lie algebras and a conditioned space-time Brownian motion in an affine Weyl chamber
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Manon Defosseux
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We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $mathfrak{g}$ considering tensor product of irreducible highest weight modules of $mathfrak{g}$ and specializations of the characters involving the Weyl vector $rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $mathfrak{g}$.
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We construct a sequence of Markov processes on the set of dominant weights of the Affine Lie algebra $hat{mathfrak{sl}_2}(C)$ which involves tensor product of irreducible highest weight modules of $hat{mathfrak{sl}_2}(C)$ and show that it converges towards a Doobs space-time harmonic transformation of a space-time Brownian motion.
This is a summary (in French) of my work about brownian motion and Kac-Moody algebras during the last seven years, presented towards the Habilitation degree.
Pitmans theorem states that if {Bt, t $ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$le$t Bs, t $ge$ 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever positive. In this paper one gives a similar representation for the Brownian motion in an interval. Due to the double barrier, it is much more involved and only asymptotic. This uses the fact that the interval is an alcove of the Affine Lie algebra A 1 1 .
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
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