1. CJM 2013 (vol 66 pp. 481)
||On the Hadamard Product of Hopf Monoids|
Combinatorial structures that compose and decompose give rise to Hopf monoids
in Joyal's category of species. The Hadamard product of two Hopf monoids
is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states
that if one factor is connected and the other is free as a monoid,
their Hadamard product is free (and connected).
The second provides an explicit basis for the Hadamard
product when both factors are free.
The first main result is obtained by showing the existence of a one-parameter deformation
of the comonoid structure and appealing to a rigidity result of Loday and Ronco
that applies when the parameter is set to zero.
To obtain the second result, we introduce an operation on species that is intertwined
by the free monoid functor with the Hadamard product.
As an application of the first result, we deduce that the Boolean transform
of the dimension sequence of a connected Hopf monoid is nonnegative.
Keywords:species, Hopf monoid, Hadamard product, generating function, Boolean transform
Categories:16T30, 18D35, 20B30, 18D10, 20F55
2. CJM 2013 (vol 66 pp. 205)
||Generalized Frobenius Algebras and Hopf Algebras|
"Co-Frobenius" coalgebras were introduced as dualizations of
We previously showed
that they admit
left-right symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasi-co-Frobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) co-Frobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)co-Frobenius coalgebra $C$. We
give many examples of co-Frobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
q-homological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, co-Frobenius
Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10
3. CJM 2012 (vol 65 pp. 241)
||Lagrange's Theorem for Hopf Monoids in Species|
Following Radford's proof of Lagrange's theorem for pointed Hopf algebras,
we prove Lagrange's theorem for Hopf monoids in the category of
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence of a Hopf monoid
in the form of certain polynomial inequalities, and of
a set-theoretic Hopf monoid in the form of certain linear inequalities.
The latter express that the binomial transform of the sequence must be nonnegative.
Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarÃ©-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement
Categories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35
4. CJM 2002 (vol 54 pp. 970)
||On Graded Categorical Groups and Equivariant Group Extensions |
In this article we state and prove precise theorems on the homotopy
classification of graded categorical groups and their homomorphisms.
The results use equivariant group cohomology, and they are applied to
show a treatment of the general equivariant group extension problem.
Categories:18D10, 18D30, 20E22, 20F29