Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I moving the following old discussion from dg-algebra to here:
A previous version of this entry gave rise to the following discussion
+–{.query} Zoran, why would you not say that this is ’following the product rule from ordinary calculus’, as I wrote? Not that this can be proved like the product rule can, but it's an easy mnemonic (and a similar one works for direct sums too). —Toby
I find it very confusing for me at least. The Leibniz rule is about the coproduct in a single algebra; here one has several algebras with different differentials, not a single derivative operators, and not acting on a tensor square of a single algebra, so it is a bit far. If $A=B$ then I would be happy, but otherwise it is too general. —Zoran
You mean that if $A = B$, then the Leibniz rule is a special case of this? Then surely it is also a special case of the more general case without $A = B$? Anyway, I think that it's more an example of categorification than generalisation. —Toby
For some special algebras this is true. For example, the dual of symmetric algebra as a Hopf algebra can be identified with the infinite order formal differential operators with constant coefficients (the isomorphism is given by evaluation at zero). Thus the Leibniz rule for derivatives is indeed the dual coproduct to the product on the symmetric algebras. There are braided etc. generalizations to this, and a version for computing the coproduct on a dual of enveloping algebras. In physics the addition of momenta and angular momenta for multiparticle systems is exactly coming from this kind of coproduct. But in all these cases the operators whose product you are taking live in a representation of a single algebra. — Zoran
=–
For example, see my article arxiv/0711.0149 and
Max Karoubi, M. Suarez-Alvarez, Twisted Kähler differential forms, JPAA 181, Issues 2-3 (2003), 279-289, MR2004m:55009, doi
M. Karoubi, Braiding of differential forms and homotopy types, C. R. Acad. Sci. Paris Sér. I 331 (2000), pp. 757–762, MR2001j:55010, doi
1 to 3 of 3