243 research outputs found
Torsors and ternary Moufang loops arising in projective geometry
We give an interpretation of the construction of torsors from preceding work
(Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of
classical projective geometry. For the Desarguesian case, this leads to a
reformulation of certain results from lot.cit., whereas for the Moufang case
the result is new. But even in the Desarguesian case it sheds new light on the
relation between the lattice structure and the algebraic structures of a
projective space.Comment: 15 p., 5 figure
The splitting process in free probability theory
Free cumulants were introduced by Speicher as a proper analog of classical
cumulants in Voiculescu's theory of free probability. The relation between free
moments and free cumulants is usually described in terms of Moebius calculus
over the lattice of non-crossing partitions. In this work we explore another
approach to free cumulants and to their combinatorial study using a
combinatorial Hopf algebra structure on the linear span of non-crossing
partitions. The generating series of free moments is seen as a character on
this Hopf algebra. It is characterized by solving a linear fixed point equation
that relates it to the generating series of free cumulants. These phenomena are
explained through a process similar to (though different from) the
arborification process familiar in the theory of dynamical systems, and
originating in Cayley's work
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
Remarks on the GNS Representation and the Geometry of Quantum States
It is shown how to introduce a geometric description of the algebraic
approach to the non-relativistic quantum mechanics. It turns out that the GNS
representation provides not only symplectic but also Hermitian realization of a
`quantum Poisson algebra'. We discuss alternative Hamiltonian structures
emerging out of different GNS representations which provide a natural setting
for quantum bi-Hamiltonian systems.Comment: 20 page
The Whitehead group of the Novikov ring
The Bass-Heller-Swan-Farrell-Hsiang-Siebenmann decomposition of the Whitehead
group of a twisted Laurent polynomial extension
of a ring is generalized to a decomposition of the
Whitehead group of a twisted Novikov ring of power series
. The decomposition involves a summand
which is an abelian quotient of the multiplicative group
of Witt vectors . An example
is constructed to show that in general the natural surjection is not an isomorphism.Comment: Latex file using Diagrams.tex, 36 pages. To appear in "K-theory
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