3,163 research outputs found
Hopf algebras under finiteness conditions
This is a brief survey of some recent developments in the study of infinite
dimensional Hopf algebras which are either noetherian or have finite
Gelfand-Kirillov dimension. A number of open questions are listed.Comment: Comments welcom
Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras
-algebraic Weyl quantization is extended by allowing also degenerate
pre-symplectic forms for the Weyl relations with infinitely many degrees of
freedom, and by starting out from enlarged classical Poisson algebras. A
powerful tool is found in the construction of Poisson algebras and
non-commutative twisted Banach--algebras on the stage of measures on the not
locally compact test function space. Already within this frame strict
deformation quantization is obtained, but in terms of Banach--algebras
instead of -algebras. Fourier transformation and representation theory of
the measure Banach--algebras are combined with the theory of continuous
projective group representations to arrive at the genuine -algebraic
strict deformation quantization in the sense of Rieffel and Landsman. Weyl
quantization is recognized to depend in the first step functorially on the (in
general) infinite dimensional, pre-symplectic test function space; but in the
second step one has to select a family of representations, indexed by the
deformation parameter . The latter ambiguity is in the present
investigation connected with the choice of a folium of states, a structure,
which does not necessarily require a Hilbert space representation.Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Three dimensional quantum algebras: a Cartan-like point of view
A perturbative quantization procedure for Lie bialgebras is introduced and
used to classify all three dimensional complex quantum algebras compatible with
a given coproduct. The role of elements of the quantum universal enveloping
algebra that, analogously to generators in Lie algebras, have a distinguished
type of coproduct is discussed, and the relevance of a symmetrical basis in the
universal enveloping algebra stressed. New quantizations of three dimensional
solvable algebras, relevant for possible physical applications for their
simplicity, are obtained and all already known related results recovered. Our
results give a quantization of all existing three dimensional Lie algebras and
reproduce, in the classical limit, the most relevant sector of the complete
classification for real three dimensional Lie bialgebra structures given by X.
Gomez in J. Math. Phys. Vol. 41. (2000) 4939.Comment: LaTeX, 15 page
On the Lie enveloping algebra of a post-Lie algebra
We consider pairs of Lie algebras and , defined over a common
vector space, where the Lie brackets of and are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
Unimodular graded Poisson Hopf algebras
Let be a Poisson Hopf algebra over an algebraically closed field of
characteristic zero. If is finitely generated and connected graded as an
algebra and its Poisson bracket is homogeneous of degree , then
is unimodular; that is, the modular derivation of is zero. This is a
Poisson analogue of a recent result concerning Hopf algebras which are
connected graded as algebras.Comment: 14 pages; preliminary version, comments welcom
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