Re: Nonrenormalization vs Renormalization 17: Valuations, Adeles, Ideles, Dedekind Zeta Functions, Quantum Statistical Mechanics, Euler Characteristics, Unit Spheres

From Osher Doctorow mdoctorow@xxxxxxxxxxx

The role of groups acting transitively on the unit sphere (see below)
in Alesker's paper is largely to establish various isomorphisms
involving Val^G(V) and Val^infinity(V), where Val^G(V) is the subspace
of Val(V) of G-invariant convex valuations, G is a compact subgroup of
the orthogonal group, Val(V) is the space of continuous translation
invariant convex valuations, Val^infinity(V) is the subset of smooth
convex valuations in the sense that the map GL(V) to Val(V) defined by
g --> g(phi) is infinitely differentiable for each convex valuation phi
in Val(V).

Alesker points out that there's an explicit classification of compact
connected Lie groups acting transitively on the sphere due to A. Borel
and Montgomery-Samelson, including 6 infinite series (SO(n), U(n),
SU(n), Sp(n) * Sp(1), Sp(n) * U(1); 3 exceptions G2, Spin(7), Spin(9);
etc.

Also there's a formal analogy of Poincare duality and the "hard
Lefschetz theorem" in the algebra of valuations with the cohomology
algebra of compact Kahler manifolds, the Poincare duality being a
fundamental property of general compact oriented manifolds while the
hard Lefschetz theorem is a fundamental property of general compact
Kahler manifolds (p. 12 of Alesker).

Osher Doctorow

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