#18675: Add 'connected' as a class for graded Hopf algebras with basis.
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Reporter: kdilks | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.8
Component: algebra | Resolution:
Keywords: days65 | Merged in:
Authors: Jean-Baptiste | Reviewers: zabrocki
Priez | Work issues:
Report Upstream: N/A | Commit:
Branch: | 174d2288c77874c17750339dfef5400a8dccce8c
public/ticket/18675 | Stopgaps:
Dependencies: |
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Comment (by darij):
To follow the conventions we currently have, your Hopf algebras should be
called "Hopf superalgebras" or "super-Hopf algebras". Symmetric functions,
as well as most other combinatorial Hopf algebras we currently have,
cannot inherit from that class, due to them satisfying the sign-free
bialgebra axiom. (Unless we double their degrees, but that is a totally
new can of worms and incompatible with existing combinatorial literature.)
John: Please see almost any paper on combinatorial Hopf algebras for an
example of signless graded Hopf algebras being used. See, for example,
Theorem 3.8.3 in http://preprints.ihes.fr/2006/M/M-06-40.pdf , or
Proposition 4.4 in http://home.gwu.edu/~wschmitt/papers/iha.pdf , or
Theorem 5.6.4 in Radford's "Hopf algebras", or Example 2.3 in
http://www.math.cornell.edu/~maguiar/a.pdf . Generally, I rarely see
people double a `ZZ`-grading to make an even object work with the Koszul
rule; instead they usually drop the Koszul sign rule. IMHO it is also not
a good idea to require `Z`-graded superobjects to have their `Z`-grading
refine their `Z/2`-grading.
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Ticket URL: <http://trac.sagemath.org/ticket/18675#comment:12>
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