
	Where results make sense

Topic: Comultiplications

	[No title]
		We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$.
		We also give necessary and sufficient conditions for a comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$.
		In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\aut F$ on the set of comultiplications of $F$.
	math.wesleyan.edu /~mhovey/archive/all96 (7847 words)

	Hopf algebra and metric/duality
		That is: the antipode is now defined in terms of multiplication and comultiplication.
		The left-hand side says: three things are multiplied together, two comultiplications are done, and then two of the outgoing strands are traced with two of the incoming strands.
		/ \ / \ \ / \_/ which you might think of as a comultiplication in which the outgoing indices are then traced together (I think of it as a multiplication with the incoming indices traced together, but I think I do things upside down from the way the seminar has been going).
	www.lns.cornell.edu /spr/2001-03/msg0031933.html (334 words)

	Transactions of the American Mathematical Society
		Abstract: By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group
		In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group
		We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.
	www.ams.org /tran/1998-350-04/S0002-9947-98-01916-3/home.html (399 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		As an application we prove that any two isomorphic superalgebras with different Cartan matrices have isomorphic q-deformations (as associative superalgebras) and their standard comultiplications are connected by such twisting.
		We show that Drinfeld's formulas of a comultiplication for the second realization are a twisting of the standard comultiplication by factors of the universal R-matrix.
		Finally, properties of the Drinfeld's comultiplication are considered.
	www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-th:9404036 (114 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The comultiplication, the counit and the antipode will be denoted by $\phi$, $\e$ and $\k$ respectively.
		Similarly, the symbol $a^{(1)}\otimes \dots a^{(n)}$ denotes the result of a $(n-1)$-fold comultiplication of $a\in\cal A$ (due to the coassociativity property of $\phi$ this is independent of the way in which comultiplications are performed).
		In this case, as shown in Proposition~\ref{pro:B12}, the comultiplication map admits a natural extension $\widehat{\phi}\colon \Gamma^\wedge\rightarrow\Gamma^\wedge\grten \Gamma^\wedge$, which is a graded-differential algebra homomorphism.
	www.math.unam.mx /~micho/papers/bundles1.tex (12851 words)

	Department of Mathematics - Mauricio Gutierrez (Site not responding. Last check: 2007-10-21)
		Normal forms for basis-conjugating automorphisms of a free group
		Comultiplications on free groups and wedges of circles
		Two complexes with fundamental group a semidirect product of cyclics
	www.tufts.edu /as/math/gutierrez.html (67 words)

	Journal of Pure and Applied Algebra. (Site not responding. Last check: 2007-10-21)
		Annetta Aramova, Kristina Crona, Emanuela De Negri, Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions, Journal of Pure and Applied Algebra 150 (3) (2000) pp.
		Martin Arkowitz, Equivalence classes of homotopy-associative comultiplications of finite complexes, Journal of Pure and Applied Algebra 102 (2) (1995) pp.
		Arlettaz, P. Zelewski, Divisible homology classes in the special linear group of a number field, Journal of Pure and Applied Algebra 109 (3) (1996) pp.
	www1.elsevier.com /cdweb/journals/00224049/viewer.htt?viewtype=authors&rangeselected=2 (566 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		We will give the definition of a Hopf $G$-coalgebra and explain (very roughly) how these objects are used to construct such invariants (based on the thesis of Verilizier).
		Then, we will show that a Hopf $G$-coalgebra is a multiplier Hopf algebra $A$, together with a homomorphism of the multiplier Hopf algebra $K(G)$ of complex functions with finite support on $G$ into the center of the multiplier algebra $M(A)$ of $A$, compatible with the comultiplications.
		We will look at the various notions and properties in the theory of Hopf $G$-coalgebras used to construct these invariants and see how they translate into the framework of multiplier Hopf algebras.
	www.ms.u-tokyo.ac.jp /~yasuyuki/vd.htm (147 words)

	[No title]
		The cohomology of coalgebras is defined using a simplicial complex generated by repeated applications of $S$, i.e.
		if $C$ and $D$ arecoalgebras, the groups $H^n(C,D)$ are defined by a complex $(C,S^D)$ whoseboundary map $d$ depends on the comultiplications* on $C$ and $D$ (11.4below).
		The comultiplication $c$ is in $(C,SC)$, and the {\em coalgebraic} deformations of $c$ are given by deformations $G$ in $(C,SC)$ satisfying $\gkd G = G\circ G$ This gives equation 11.1 below, analogous to 9.1, and leads us into deformation theory.
	www.math.mcgill.ca /fox/Intro.tex (4460 words)

	[No title]
		A key observation due to Fr\o nsdal \cite{Fron1} is that the $RLL$ relations for the elliptic algebras of both types, \eqref{vRLL2} and \eqref{fRLL}, arise by the same mechanism as above.
		Namely, there exist two types of twistors which give rise to different comultiplications on the quantum affine algebras $U_q(\goth{g})$, and the resultant quasi-Hopf algebras are nothing but the two types of elliptic quantum groups.
		\lb{R3} \end{eqnarray} Here $\Delta'=\sigma\circ\Delta$ ($\sigma(a\otimes b)=b\otimes a$) is the opposite comultiplication.
	azusa.shinshu-u.ac.jp /~odake/paper/9712029v3.tex (6023 words)

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