##
**Remarks on the differential envelopes of associative algebras.**
*(English)*
Zbl 0743.16020

Let \(A\) be a \(\mathbb{Z}/2\)-graded complex algebra. This paper deals with some aspects of what the authors call the differential envelopes associated to \(A\), also known as universal bigraded differential algebras; these are solutions (in suitable categories) of the following universal problem: find a \(\mathbb{Z}/2\)-graded algebra whose homomorphisms of degree 0 to a \(\mathbb{Z}/2\)-graded algebra \(B\) are in one-to-one correspondence with pairs \((\alpha,\Delta)\) where \(\alpha\) is a homomorphism of degree 0 from \(A\) to \(B\), and \(\Delta\) is a linear map of degree 1 from \(A\) to \(B\) which is a graded derivation with respect to \(\alpha\). In this generality, one gets a solution \(\Omega(A)\); if \(A\) is unital and the derivation \(\Delta\) is required to vanish on the unit, one gets a solution \(\Omega A\) which is of course a quotient of \(\Omega(A)\). Since \(\Omega A\) has no unit (even if \(A\) has one), one may adjoin a unit and get \(\tilde\Omega(A)\) (with obvious extensions of the gradings and differential). If \(\tilde A\) is the algebra obtained from \(A\) by adjoining a unit 1, the authors first show that \(\tilde\Omega(A)\) is isomorphic to \(\Omega\tilde A\). Next, for \(A\) unital, the quotient map \(\Omega(A)\to\Omega A\) is split via a map \(\theta\) (associated to the inclusion \(\alpha: A\to\Omega(A)\) and the differential \(\Delta(a)= 1\cdot da\cdot 1\); the range of \(\theta\) is very precisely identified. Finally, if \(\underline A\) denotes the ungraded algebra underlying \(A\), it is shown that \(\Omega(A)\) [resp. \(\Omega A\)] can be recovered from \(\Omega (\underline{A})\) [resp. \(\Omega\underline{A}]\) by a fairly simple modification of the product. These results might be of interest for the development of cyclic cohomology for \(\mathbb{Z}/2\)-graded algebras. (For \(A\) ungraded, \(\Omega(A)\) was crucial in A. Connes’ seminal paper [Publ. Math., Inst. Hautes Etud. Sci. 62, 257-360 (1985; Zbl 0592.46056)]; afterwards for \(A\) \(\mathbb{Z}/2\)-graded, \(\Omega(A)\) was exploited by D. Kastler [Cyclic cohomology within the differential envelope (Hermann, Paris, 1988; Zbl 0662.55001)].).

### MSC:

16W50 | Graded rings and modules (associative rings and algebras) |

16W25 | Derivations, actions of Lie algebras |

17B35 | Universal enveloping (super)algebras |

16D90 | Module categories in associative algebras |

16S30 | Universal enveloping algebras of Lie algebras |

16S32 | Rings of differential operators (associative algebraic aspects) |

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |