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Search: All articles in the CJM digital archive with keyword Hopf algebra

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1. CJM Online first

Grinberg, Darij
Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$ of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an " immaculate tableau" is defined similarly to be a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary; but each row has to weakly increase). Dual immaculate functions have been introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties. In this note, we prove a conjecture of Mike Zabrocki which provides an alternative construction for the dual immaculate functions in terms of certain "vertex operators". The proof uses a dendriform structure on the ring $\operatorname*{QSym}$; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras $\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.

Keywords:combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux

2. CJM 2013 (vol 66 pp. 205)

Iovanov, Miodrag Cristian
Generalized Frobenius Algebras and Hopf Algebras
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. 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)

Aguiar, Marcelo; Lauve, Aaron
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 connected species. 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, Poincaré-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 2004 (vol 56 pp. 871)

Schocker, Manfred
Lie Elements and Knuth Relations
A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of $\Sym_n$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution product
Categories:17B01, 05E10, 20C30, 16W30

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