11 research outputs found
New Approach to Arakelov Geometry
This work is dedicated to a new completely algebraic approach to Arakelov
geometry, which doesn't require the variety under consideration to be
generically smooth or projective. In order to construct such an approach we
develop a theory of generalized rings and schemes, which include classical
rings and schemes together with "exotic" objects such as F_1 ("field with one
element"), Z_\infty ("real integers"), T (tropical numbers) etc., thus
providing a systematic way of studying such objects.
This theory of generalized rings and schemes is developed up to construction
of algebraic K-theory, intersection theory and Chern classes. Then existence of
Arakelov models of algebraic varieties over Q is shown, and our general results
are applied to such models.Comment: 568 pages, with hyperlink
A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra
Given a -dimensional Lie algebra over a field ,
together with its vector space basis , we give a formula,
depending only on the structure constants, representing the infinitesimal
generators, in , where is a formal
variable, as a formal power series in with coefficients in the Weyl algebra
. Actually, the theorem is proved for Lie algebras over arbitrary rings
.
We provide three different proofs, each of which is expected to be useful for
generalizations. The first proof is obtained by direct calculations with
tensors. This involves a number of interesting combinatorial formulas in
structure constants. The final step in calculation is a new formula involving
Bernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions of
certain spaces of tensors are also calculated. The second method of proof is
geometric and reduces to a calculation of formal right-invariant vector fields
in specific coordinates, in a (new) variant of formal group scheme theory. The
third proof uses coderivations and Hopf algebras.Comment: v2: expositional improvements (significant in sections 5,6); v3:
minor expositional improvements (including in notation, and in introduction);
v4: final version, to appear in Journal of Algebra (4 minor differences from
v3 due wrong uploaded file in v3