#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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Changes (by jhpalmieri):
* cc: darij (added)
Comment:
I don't think it can be a subcategory because, unless the algebra is
evenly graded, it can't satisfy both `\tau(f \otimes g) = g \otimes f` and
`\tau (f \otimes g) = (-1)^{deg(f) deg(g)} g \otimes f`. We seem to have
two conflicting notions of what a graded Hopf algebra is. My version comes
from topology and dates back to the 1950s. The version without the sign
looks to be much more recent, mainly arising (as far as I can tell) in
combinatorics. In "[http://web.mit.edu/~darij/www/algebra/HopfComb-
sols.pdf Hopf algebras in combinatorics]" by Grinberg and Reiner, they at
least refer (Definition 1.11, equation (1.7)) to the sign, although they
have decided to omit it, saying that one can artificially double all
degrees to deal with this issue.
I really will insist that this sign be one of the axioms of a graded Hopf
algebra, unless you can provide citations that convince me otherwise. Your
version could be called "unsigned", maybe?
`UnsignedGradedHopfAlgebrasWithBasis`?
`GradedUnsignedHopfAlgebrasWithBasis`?
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Ticket URL: <http://trac.sagemath.org/ticket/18675#comment:10>
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