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)].).


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
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