nLab universal differential envelope


kk-linear derivations of a (not necessarily commutative) associative unital kk-algebra AA with values in AA-bimodules are often considered: these are kk-linear maps d:AMd: A\to M satisfying the Leibniz identity: d(ab)=d(a)b+ad(b)d(a b)=d(a) b+ a d(b). One would like to construct a universal construction of that kind.


The triple (A,M,d)(A,M,d) is then called a differential calculus over AA, provided MM is in the kk-linear span of all expressions of the form ad(b 1)d(b 2)d(b n)a d(b_1) d(b_2)\cdots d(b_n) where a,b 1,,b nAa,b_1,\ldots,b_n\in A. Morphisms of differential calculi over AA are straightforward to define. A universal object in that category is the universal differential calculus and ΩA=M\Omega A = M in that case is called the universal differential envelope of AA. It is constructed as the quotient of the tensor algebra TAT A modulo the ideal generated by the (“augmentation”) kernel IAAI\subset A\otimes A of the multiplication map AAAA\otimes A\to A; then da=1aa1d a = 1\otimes a - a\otimes 1.

This is a bit confusing. The subbimodule II should rather be the degree-1 component of ΩA\Omega A (see Bourbaki, Algèbre III, paragraph Problème universel pour les dérivations: cas non commutatif). —Benoit Jubin

Eric: I’m probably confused, but AA is degree-0 and AAA\otimes A is degree-1.

Benoit Jubin: I agree. To me, we should have ΩA=AI\Omega A = A \oplus I \oplus \ldots, whereas the article currently proposes ΩA=TA/I\Omega A = TA / \langle I \rangle.

The bimodule ΩA\Omega A is in fact a kk-algebra, generated by degree-11 elements (in AA); differential d:AΩAd:A\to \Omega A can be naturally extended along the inclusion AΩAA\hookrightarrow \Omega A to a differential d:ΩAΩAd:\Omega A\to \Omega A satisfying the Leibniz rule, by setting d1=0d 1 = 0, d(wy)=d(w)y+(1) pwd(y)d (w y) = d(w) y + (-1)^p w d(y), where w,yw,y are forms and wΩ pAw\in\Omega^p A is homogeneous.



Eric: I have some notes on my personal web, but they need work before transferring them here.

Last revised on April 27, 2020 at 23:02:21. See the history of this page for a list of all contributions to it.