Double deformation
As if deforming your mathematics in one parameter weren't enough, they're now doing it in two. So p, q deformed integers are [n]p,q = (pn - qn)/(p - q). Time to go through weeks 183-188 of This Week's Finds to see if there's any categorification to be had. This reminds me of an idea I had when reading in week 184 about the Euler characteristic of complex flag manifold in n dimensions being n!. Noting from p. 67 of Cartier's excellent Mathemagics paper that infinity factorial is the square root of 2.pi, can we conclude that the Euler characteristic of the infinite-dimensional complex flag manifold is the square root of 2.pi? Stranger things happen here.
2 Comments:
It would take more evidence to convince me that this "double deformation idea" is really interesting.
It's easy to make up ideas in mathematics, but the really interesting ideas are not usually things you "make up". Instead, they press themselves upon you with the force of necessity, often from several directions at once.
q-deformation is unavoidable, whether you're doing combinatorics, group representation theory, string theory, statistical mechanics, or knot theory. You need to understand q-deformation to understand why the same patterns keep popping up in all these subjects!
I don't see that this "double deformation" is similarly unavoidable. Maybe it will turn out to be, but so far it just feels "made up". Time will tell.
What you say resonates with what I've been saying in some of my recent papers about a distinction between mathematical results to parallel happenstantial and law-like ones in the sciences. A bolder idea is to tie this distinction to the capacity for categorification.
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