3,766 research outputs found
Pre-alternative algebras and pre-alternative bialgebras
We introduce a notion of pre-alternative algebra which may be seen as an
alternative algebra whose product can be decomposed into two pieces which are
compatible in a certain way. It is also the "alternative" analogue of a
dendriform dialgebra or a pre-Lie algebra. The left and right multiplication
operators of a pre-alternative algebra give a bimodule structure of the
associated alternative algebra. There exists a (coboundary) bialgebra theory
for pre-alternative algebras, namely, pre-alternative bialgebras, which
exhibits all the familiar properties of the famous Lie bialgebra theory. In
particular, a pre-alternative bialgebra is equivalent to a phase space of an
alternative algebra and our study leads to what we called -equations in a
pre-alternative algebra, which are analogues of the classical Yang-Baxter
equation.Comment: 34 page
Quasi-Hamiltonian groupoids and multiplicative Manin pairs
We reformulate notions from the theory of quasi-Poisson g-manifolds in terms
of graded Poisson geometry and graded Poisson-Lie groups and prove that
quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then
interpret this result within the theory of Dirac morphisms and multiplicative
Manin pairs, to connect our work with more traditional approaches, and also to
put it into a wider context suggesting possible generalizations.Comment: 39 page
Mirror Symmetry and Generalized Complex Manifolds
In this paper we develop a relative version of T-duality in generalized
complex geometry which we propose as a manifestation of mirror symmetry. Let M
be an n-dimensional smooth real manifold, V a rank n real vector bundle on M,
and nabla a flat connection on V. We define the notion of a nabla-semi-flat
generalized complex structure on the total space of V. We show that there is an
explicit bijective correspondence between nabla-semi-flat generalized complex
structures on the total space of V and nabla(dual)-semi-flat generalized
complex structures on the total space of the dual of V. Similarly we define
semi-flat generalized complex structures on real n-torus bundles with section
over an n-dimensional base and establish a similar bijective correspondence
between semi-flat generalized complex structures on pair of dual torus bundles.
Along the way, we give methods of constructing generalized complex structures
on the total spaces of vector bundles and torus bundles with sections. We also
show that semi-flat generalized complex structures give rise to a pair of
transverse Dirac structures on the base manifold. We give interpretations of
these results in terms of relationships between the cohomology of torus bundles
and their duals. We also study the ways in which our results generalize some
well established aspects of mirror symmetry as well as some recent proposals
relating generalized complex geometry to string theory.Comment: Small additions, references adde
Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that
compositions of smooth canonical relations are not always smooth, building a
category suitable for functorial quantization. To apply their construction to
more examples, we introduce a notion of highly selective category, in which
only certain morphisms and certain pairs of these morphisms are "good". We then
apply this notion to the category of linear canonical
relations and the result of our version of the WW
construction, identifying the morphisms in the latter with pairs
consisting of a linear canonical relation and a nonnegative integer. We put a
topology on this category of indexed linear canonical relations for which
composition is continuous, unlike the composition in itself.
Subsequent papers will consider this category from the viewpoint of derived
geometry and will concern quantum counterparts
Higgs bundles over elliptic curves
In this paper we study -Higgs bundles over an elliptic curve when the
structure group is a classical complex reductive Lie group. Modifying the
notion of family, we define a new moduli problem for the classification of
semistable -Higgs bundles of a given topological type over an elliptic curve
and we give an explicit description of the associated moduli space as a finite
quotient of a product of copies of the cotangent bundle of the elliptic curve.
We construct a bijective morphism from this new moduli space to the usual
moduli space of semistable -Higgs bundles, proving that the former is the
normalization of the latter. We also obtain an explicit description of the
Hitchin fibration for our (new) moduli space of -Higgs bundles and we study
the generic and non-generic fibres
Banach Lie-Poisson spaces and reduction
The category of Banach Lie-Poisson spaces is introduced and studied. It is
shown that the category of W*-algebras can be considered as one of its
subcategories. Examples and applications of Banach Lie-Poisson spaces to
quantization and integration of Hamiltonian systems are given. The relationship
between classical and quantum reduction is discussed.Comment: 58 pages, to apear in Comm.Math.Phy
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