Science > Physics > Nonrenormalization vs Renormalization 17: Valuations, Adeles, Ideles, Dedekind Zeta Functions, Quantum Statistical Mechanics, Euler Characteristics, Unit Spheres
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
19 Apr 2006 11:58:12 PM |
| Object: |
Nonrenormalization vs Renormalization 17: Valuations, Adeles, Ideles, Dedekind Zeta Functions, Quantum Statistical Mechanics, Euler Characteristics, Unit Spheres |
From Osher Doctorow
In Section 14.4 "Adeles and ideles intuition" of this thread, Apr 16,
2002 1:47PM and 1:16PM, I defined valuations and pointed out with
reference to Paula B. Cohen's paper (1998) their relationship to adeles
and ideles and quantum statistical mechanics.
Now take a look at Semyon Alesker's (U. Tel Aviv, Israel) "Theory of
valuations on manifolds: a survey," math.MG/0603372 v1 15 Mar 2006,
which relates valuations to Euler characteristics, unit spheres, the
group GL(V) of invertible linear transformations on a finite
dimensional real vector space V, smooth densities on smooth manifolds
X, convexity, etc. The paper is referred to by Alesker as a "survey",
but it includes survey of many of his own results which were proven
elsewhere as well as those by various other researchers.
It turns out that there are several very close relationships between
valuations and Euler characteristics. For example, any smooth density
on smooth manifold X belongs to V^infinity(X), and the Euler
characteristic belongs to V^infinity(X), where the latter is defined as
a set of smooth valuations which turns out to be a linear space and is
the main subject of Alesker's paper. V^infinity(X) carries a very
important multiplicative structure from its cross-product (actually,
canonical product) with itself to itself which is continuous,
commutative, associative, such that the EUler characteristic is the
unit in the algebra V^infinity(X), etc.
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Nonrenormalization vs Renormalization 17: Valuations, Adeles, Ideles, Dedekind Zeta Functions, Quantum Statistical Mechanics, Euler Characteristics, Unit Spheres |
20 Apr 2006 12:31:56 AM |
|
|
From Osher Doctorow
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
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Nonrenormalization vs Renormalization 17: Valuations, Adeles, Ideles, Dedekind Zeta Functions, Quantum Statistical Mechanics, Euler Characteristics, Unit Spheres |
20 Apr 2006 12:44:16 AM |
|
|
From Osher Doctorow
A group action from G x X to X is transitive if for each pair x1, x2 of
elements of set X there's an element g of group G such that gx1 = x2.
The action of G permutes the elements of X. The group orbit for a
given x1 is the set of gx1 for all g.
See Wolfram's "Transitive group action," "Group action," "Group orbit,"
"Transitive," "Isotropy group," "Transitive group" for concise and
relatively painless summaries of things related to transitive group
actions.
Osher Doctorow
.
|
|
|
|
|
|
Related Articles |
|
|
