113 research outputs found
Degenerations of spherical subalgebras and spherical roots
We obtain several structure results for a class of spherical subgroups of
connected reductive complex algebraic groups that extends the class of strongly
solvable spherical subgroups. Based on these results, we construct certain
one-parameter degenerations of the Lie algebras corresponding to such
subgroups. As an application, we exhibit explicit algorithms for computing the
set of spherical roots of such a spherical subgroup.Comment: v2: 45 pages, revised extended version with new Section 6 containing
an optimization of the initial algorith
Boundary algebras and Kac modules for logarithmic minimal models
Virasoro Kac modules were initially introduced indirectly as representations
whose characters arise in the continuum scaling limits of certain transfer
matrices in logarithmic minimal models, described using Temperley-Lieb
algebras. The lattice transfer operators include seams on the boundary that use
Wenzl-Jones projectors. If the projectors are singular, the original
prescription is to select a subspace of the Temperley-Lieb modules on which the
action of the transfer operators is non-singular. However, this prescription
does not, in general, yield representations of the Temperley-Lieb algebras and
the Virasoro Kac modules have remained largely unidentified. Here, we introduce
the appropriate algebraic framework for the lattice analysis as a quotient of
the one-boundary Temperley-Lieb algebra. The corresponding standard modules are
introduced and examined using invariant bilinear forms and their Gram
determinants. The structures of the Virasoro Kac modules are inferred from
these results and are found to be given by finitely generated submodules of
Feigin-Fuchs modules. Additional evidence for this identification is obtained
by comparing the formalism of lattice fusion with the fusion rules of the
Virasoro Kac modules. These are obtained, at the character level, in complete
generality by applying a Verlinde-like formula and, at the module level, in
many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.
Rational points of universal curves
In this paper we prove a version of Grothendieck's section conjecture for the
restriction of the universal complete curve over M_{g,n}, g > 4, to the
function field k(M_{g,n}) where k is, for example, a number field. In this
version, the fundamental group of the closed fiber is replaced by its ell-adic
unipotent completion when n > 1.Comment: 60 pages. This is a minor revision of the previous version. To appear
in JAM
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