884,691 research outputs found
Integral Bases for the Universal Enveloping Algebras of Map Algebras
Given a finite-dimensional, complex simple Lie algebra we exhibit an integral
form for the universal enveloping algebra of its map algebra, and an explicit
integral basis for this integral form. We also produce explicit commutation
formulas in the universal enveloping algebras of the map algebras of sl_2 that
allow us to write certain elements in Poincare-Birkhoff-Witt order.Comment: 20 page
The loop expansion of the Kontsevich integral, the null move and S-equivalence
This is a substantially revised version. The Kontsevich integral of a knot is
a graph-valued invariant which (when graded by the Vassiliev degree of graphs)
is characterized by a universal property; namely it is a universal Vassiliev
invariant of knots. We introduce a second grading of the Kontsevich integral,
the Euler degree, and a geometric null-move on the set of knots. We explain the
relation of the null-move to S-equivalence, and the relation to the Euler
grading of the Kontsevich integral. The null move leads in a natural way to the
introduction of trivalent graphs with beads, and to a conjecture on a rational
version of the Kontsevich integral, formulated by the second author and proven
in joint work of the first author and A. Kricker.Comment: AMS-LaTeX, 20 pages with 31 figure
A Characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2
An integral quadratic polynomial (with positive definite quadratic part) is
called almost universal if it represents all but finitely many positive
integers. In this paper, we provide a characterization of almost universal
ternary quadratic polynomials with conductor 2
Counting integral points on universal torsors
Manin's conjecture for the asymptotic behavior of the number of rational
points of bounded height on del Pezzo surfaces can be approached through
universal torsors. We prove several auxiliary results for the estimation of the
number of integral points in certain regions on universal torsors. As an
application, we prove Manin's conjecture for a singular quartic del Pezzo
surface.Comment: 48 page
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