Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$, PhD thesis California Institute of Technology 2008

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dc.creatorLevaillant, Claire
dc.date2009-01-25
dc.date.accessioned2026-07-07T12:34:27Z
dc.date.available2026-07-07T12:34:27Z
dc.descriptionGiven two nonzero complex parameters $l$ and $m$, we construct by the mean of knot theory a matrix representation of size $\chl$ of the BMW algebra of type $A_{n-1}$ with parameters $l$ and $m$ over the field $\Q(l,r)$, where $m=\unsurr-r$. As a representation of the braid group on $n$ strands, it is equivalent to the Lawrence-Krammer representation that was introduced by Lawrence and Krammer to show the linearity of the braid groups. We prove that the Lawrence-Krammer representation is generically irreducible, but that for some values of the parameters $l$ and $r$, it becomes reducible. In particular, we show that for these values of the parameters $l$ and $r$, the BMW algebra is not semisimple. When the representation is reducible, the action on a proper invariant subspace of the Lawrence-Krammer space must be a Hecke algebra action. It allows us to describe the invariant subspaces when the representation is reducible.
dc.description172 pages, 10 figures, 6 tables, 2 programs
dc.identifierhttps://arxiv.org/abs/0901.3908
dc.identifierhttp://arxiv.org/abs/0901.3908
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/217440
dc.subjectRepresentation Theory
dc.subjectQuantum Algebra
dc.subject16G30
dc.titleIrreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$, PhD thesis California Institute of Technology 2008
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