20,458 research outputs found
Cocommutative Com-PreLie bialgebras
A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie
product satisfying some compatibilities with the product and coproduct. We here
give a classification of connected, cocommutative Com-PreLie bialgebras over a
field of characteristic zero: we obtain a main family of symmetric algebras on
a space V of any dimension, and another family available only if V is
one-dimensional. We also explore the case of Com-PreLie bialgebras over a group
algebra and over a tensor product of a group algebra and of a symmetric
algebra.Comment: arXiv admin note: text overlap with arXiv:1501.0637
On the law of the supremum of L\'evy processes
We show that the law of the overall supremum of
a L\'evy process before the deterministic time is equivalent to the
average occupation measure \mu_t(dx)=\int_0^t\p(X_s\in dx)\,ds, whenever 0 is
regular for both open halflines and . In this case,
\p(\bar{X}_t\in dx) is absolutely continuous for some (and hence for all)
, if and only if the resolvent measure of is absolutely continuous. We
also study the cases where 0 is not regular for one of the halflines
or . Then we give absolute continuity criterions for
the laws of , and ,
where is the time at which the supremum occurs before . The proofs of
these results use an expression of the joint law \p(g_t\in ds,X_t\in
dx,\bar{X}_t\in dy) in terms of the entrance law of the excursion measure of
the reflected process at the supremum and that of the reflected process at the
infimum. As an application, this law is made (partly) explicit in some
particular instances
Deformation of the Hopf algebra of plane posets
We describe and study a four parameters deformation of the two products and
the coproduct of the Hopf algebra of plane posets. We obtain a family of
braided Hopf algebras, generally self-dual. We also prove that in a particular
case (when the second parameter goes to zero and the first and third parameters
are equal), this deformation is isomorphic, as a self-dual braided Hopf
algebra, to a deformation of the Hopf algebra of free quasi-symmetric
functions.Comment: 28 pages. Second versio
Free and cofree Hopf algebras
We first prove that a graded, connected, free and cofree Hopf algebra is
always self-dual; then that two graded, connected, free and cofree Hopf
algebras are isomorphic if, and only if, they have the same Poincar\'e-Hilbert
formal series. If the characteristic of the base field is zero, we prove that
the Lie algebra of the primitive elements of such an object is free, and we
deduce a characterization of the formal series of free and cofree Hopf algebras
by a condition of growth of the coefficients. We finally show that two graded,
connected, free and cofree Hopf algebras are isomorphic as (non graded) Hopf
algebras if, and only if, the Lie algebra of their primitive elements have the
same number of generators.Comment: 22 pages. To be published in "Journal of Pure and Applied Algebra
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