158 research outputs found
On the universality of the Carter and McLenaghan formula
It is shown that the formula of the isometry generators of the spinor
representation given by Carter and McLenaghan is universal in the sense that
this holds for any representation either in local frames or even in natural
ones. The point-dependent spin matrices in natural frames are introduced for
any tensor representation deriving the covariant form of the isometry
generators in these frames.Comment: 7 pages, no figure
Generalized Lie bialgebroids and Jacobi structures
The notion of a generalized Lie bialgebroid (a generalization of the notion
of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has
associated a canonical generalized Lie bialgebroid. As a kind of converse, we
prove that a Jacobi structure can be defined on the base space of a generalized
Lie bialgebroid. We also show that it is possible to construct a Lie
bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce
a duality theorem. Finally, some special classes of generalized Lie
bialgebroids are considered: triangular generalized Lie bialgebroids and
generalized Lie bialgebras.Comment: 32 page
Jacobi structures revisited
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra
associated with a vector bundle which satisfy a property similar to that of the
Jacobi brackets, are introduced. They turn out to be equivalent to generalized
Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as
odd Jacobi brackets on the supermanifolds associated with the vector bundles.
Jacobi bialgebroids are defined in the same manner. A lifting procedure of
elements of this Grassmann algebra to multivector fields on the total space of
the vector bundle which preserves the corresponding brackets is developed. This
gives the possibility of associating canonically a Lie algebroid with any local
Lie algebra in the sense of Kirillov.Comment: 20 page
A supergeometric approach to Poisson reduction
This work introduces a unified approach to the reduction of Poisson manifolds
using their description by graded symplectic manifolds. This yields a
generalization of the classical Poisson reduction by distributions
(Marsden-Ratiu reduction). Further it allows one to construct actions of strict
Lie 2-groups and to describe the corresponding reductions.Comment: 40 pages. Final version accepted for publicatio
The graded Jacobi algebras and (co)homology
Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in
the context of graded Jacobi brackets on graded commutative algebras. This
unifies varios concepts of graded Lie structures in geometry and physics. A
method of describing such structures by classical Lie algebroids via certain
gauging (in the spirit of E.Witten's gauging of exterior derivative) is
developed. One constructs a corresponding Cartan differential calculus (graded
commutative one) in a natural manner. This, in turn, gives canonical generating
operators for triangular Jacobi algebroids. One gets, in particular, the
Lichnerowicz-Jacobi homology operators associated with classical Jacobi
structures. Courant-Jacobi brackets are obtained in a similar way and use to
define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi
structure. All this offers a new flavour in understanding the
Batalin-Vilkovisky formalism.Comment: 20 pages, a few typos corrected; final version to be published in J.
Phys. A: Math. Ge
A geometric construction of the exceptional Lie algebras F4 and E8
We present a geometric construction of the exceptional Lie algebras F4 and E8
starting from the round 8- and 15-spheres, respectively, inspired by the
construction of the Killing superalgebra of a supersymmetric supergravity
background. (There is no supergravity in the paper.)Comment: 12 page
Dual branes in topological sigma models over Lie groups. BF-theory and non-factorizable Lie bialgebras
We complete the study of the Poisson-Sigma model over Poisson-Lie groups.
Firstly, we solve the models with targets and (the dual group of the
Poisson-Lie group ) corresponding to a triangular -matrix and show that
the model over is always equivalent to BF-theory. Then, given an
arbitrary -matrix, we address the problem of finding D-branes preserving the
duality between the models. We identify a broad class of dual branes which are
subgroups of and , but not necessarily Poisson-Lie subgroups. In
particular, they are not coisotropic submanifolds in the general case and what
is more, we show that by means of duality transformations one can go from
coisotropic to non-coisotropic branes. This fact makes clear that
non-coisotropic branes are natural boundary conditions for the Poisson-Sigma
model.Comment: 24 pages; JHEP style; Final versio
Poisson-Jacobi reduction of homogeneous tensors
The notion of homogeneous tensors is discussed. We show that there is a
one-to-one correspondence between multivector fields on a manifold ,
homogeneous with respect to a vector field on , and first-order
polydifferential operators on a closed submanifold of codimension 1 such
that is transversal to . This correspondence relates the
Schouten-Nijenhuis bracket of multivector fields on to the Schouten-Jacobi
bracket of first-order polydifferential operators on and generalizes the
Poissonization of Jacobi manifolds. Actually, it can be viewed as a
super-Poissonization. This procedure of passing from a homogeneous multivector
field to a first-order polydifferential operator can be also understood as a
sort of reduction; in the standard case -- a half of a Poisson reduction. A
dual version of the above correspondence yields in particular the
correspondence between -homogeneous symplectic structures on and
contact structures on .Comment: 19 pages, minor corrections, final version to appear in J. Phys. A:
Math. Ge
On quasi-Jacobi and Jacobi-quasi bialgebroids
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships
with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct
quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also
that the structures induced on their base manifolds are related via a quasi
Poissonization
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