nLab
universal differential envelope

Motivation

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

Definitions

The triple $(A,M,d)$ is then called a differential calculus over $A$, provided $M$ is in the $k$-linear span of all expressions of the form $ad(b_1)d(b_2)\cdots d(b_n)$ where $a,b_1,\dots ,b_n\in A$. Morphisms of differential calculi over $A$ are straightforward to define. A universal object in that category is the universal differential calculus and $\Omega A=M$ in that case is called the universal differential envelope of $A$. It is constructed as the quotient of the tensor algebra $TA$ modulo the ideal generated by the (“augmentation”) kernel $I\subset A\otimes A$ of the multiplication map $A\otimes A\to A$; then $da=1\otimes a-a\otimes 1$.

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

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

Benoit Jubin: I agree. To me, we should have $\Omega A=A\oplus I\oplus \dots$, whereas the article currently proposes $\Omega A=\mathrm{TA}/⟨I⟩$.

The bimodule $\Omega A$ is in fact a $k$-algebra, generated by degree-$1$ elements (in $A$); differential $d:A\to \Omega A$ can be naturally extended along the inclusion $A\hookrightarrow \Omega A$ to a differential $d:\Omega A\to \Omega A$ satisfying the Leibniz rule, by setting $d1=0$, $d(wy)=d(w)y+(-1{)}^{p}wd(y)$, where $w,y$ are forms and $w\in {\Omega }^{p}A$ is homogeneous.

References

• A. Connes, Noncommutative Geometry
• R. Coquereaux and D. Kastler, Remarks on the differential envelopes of associative algebras
• F. Bonechi, R. Giachetti, R. Maciocco, E. Sorace, M.Tarlini, Cohomological Properties of Differential Calculi on Hopf Algebras
• H. C. Baehr, A. Dimakis, F. Müller-Hoissen, Differential Calculi on Commutative Algebras
• F. Müller-Hoissen, Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics

Discussion

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