4,399 research outputs found
A generalization of Strassen's Positivstellensatz
Strassen's Positivstellensatz is a powerful but little known theorem on
preordered commutative semirings satisfying a boundedness condition similar to
Archimedeanicity. It characterizes the relaxed preorder induced by all monotone
homomorphisms to in terms of a condition involving large powers.
Here, we generalize and strengthen Strassen's result. As a generalization, we
replace the boundedness condition by a polynomial growth condition; as a
strengthening, we prove two further equivalent characterizations of the
homomorphism-induced preorder in our generalized setting.Comment: 24 pages. v6: condition (d) in Theorem 2.12 has been correcte
On the Spectrum of Holonomy Algebras
Connections on a trivial bundle MxG can be identified with their holonomy
maps, i.e. with homomorphisms of a groupoid of paths in M into the gauge group
G.
For a connected compact G, various algebras depending on the set of the
smooth connections through their holonomy maps have been introduced in the
literature, called cylindrical and holonomy algebras. We discuss the relations
between these algebras and the consistence of their spectra.Comment: 23 pages, no figures, to appear in J.Geom.Phy
Homomorphisms on infinite direct product algebras, especially Lie algebras
We study surjective homomorphisms f:\prod_I A_i\to B of
not-necessarily-associative algebras over a commutative ring k, for I a
generally infinite set; especially when k is a field and B is
countable-dimensional over k.
Our results have the following consequences when k is an infinite field, the
algebras are Lie algebras, and B is finite-dimensional:
If all the Lie algebras A_i are solvable, then so is B.
If all the Lie algebras A_i are nilpotent, then so is B.
If k is not of characteristic 2 or 3, and all the Lie algebras A_i are
finite-dimensional and are direct products of simple algebras, then (i) so is
B, (ii) f splits, and (iii) under a weak cardinality bound on I, f is
continuous in the pro-discrete topology. A key fact used in getting (i)-(iii)
is that over any such field, every finite-dimensional simple Lie algebra L can
be written L=[x_1,L]+[x_2,L] for some x_1, x_2\in L, which we prove from a
recent result of J.M.Bois.
The general technique of the paper involves studying conditions under which a
homomorphism on \prod_I A_i must factor through the direct product of finitely
many ultraproducts of the A_i.
Several examples are given, and open questions noted.Comment: 33 pages. The lemma in section 12.1 of the previous version was
incorrect, and has been removed. (Nothing else depended on it.) Other changes
are improvements in wording, et
On strongly just infinite profinite branch groups
For profinite branch groups, we first demonstrate the equivalence of the
Bergman property, uncountable cofinality, Cayley boundedness, the countable
index property, and the condition that every non-trivial normal subgroup is
open; compact groups enjoying the last condition are called strongly just
infinite. For strongly just infinite profinite branch groups with mild
additional assumptions, we verify the invariant automatic continuity property
and the locally compact automatic continuity property. Examples are then
presented, including the profinite completion of the first Grigorchuk group. As
an application, we show that many Burger-Mozes universal simple groups enjoy
several automatic continuity properties.Comment: Typos and a minor error correcte
Rigidity of Poisson Lie group actions
n this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum maps (when the acting Lie group G is endowed with an arbitrary Poisson structure) combining a rigidity result for canonical Hamiltonian actions (\cite{MMZ}) and a linearization theorem(\cite{GW}). These results have applications to quantization of symmetries since these infinitesimal momentum maps appear as the classical limit of quantum momentum maps (\cite{BEN}).Peer ReviewedPreprin
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