##
**The Frölicher-Nijenhuis bracket in non-commutative differential geometry.**
*(English)*
Zbl 0830.58002

The authors carry over to a quite general noncommutative setting some of the basic tools of differential geometry. From the very beginning they use the setting of convenient vector spaces developed by Frölicher and Kriegl. The reasons for this are the following: if the noncommutative theory should contain some version of differential geometry, a manifold \(M\) should be represented by the algebra \(C^\infty (M,\mathbb{R})\) of smooth functions on it. The simplest considerations of groups needs products, and \(C^\infty (M\times N,\mathbb{R})\) is a certain completion of the algebraic tensor product \(C^\infty (M,\mathbb{R}) \times C^\infty (N,\mathbb{R})\). Now the setting of convenient vector spaces offers in its multilinear version a monoidally closed category, i.e. there is an appropriate tensor product which has all the usual properties with respect to bounded multilinear mappings. So multilinear algebra is carried into this kind of functional analysis without loss.

In the first section the authors give a short description of the setting of convenient spaces elaborating those aspects which they will need later. Then they repeat the usual construction of noncommutative differential forms for convenient algebras in the second section. There they consider triples \((A, \Omega^A_*, d)\) where \((\Omega^A_*, d)\) is a graded differential algebra with \(\Omega^A_0 = A\) and \(\Omega^A_n = 0\) for negative \(n\). They call \((\Omega^A_*, d)\) a differential algebra for \(A\). A universal construction of such an algebra \(\Omega^A_*\) for a commutative algebra \(A\) is described byE. Kunz in ‘Kähler differentials’ Vieweg, Wiesbaden (1986; Zbl 0587.13014). They present a noncommutative version of the construction of Kunz, since they need more information.

Next they show that the bimodule \(\Omega_n (A)\) represents the functor of the normalized Hochschild \(n\)-cocycles. In the third section they introduce the noncommutative version of the Frölicher-Nijenhuis bracket by investigating all bounded graded derivations of the algebra of differential forms. This bracket is then used to formulate the concept of integrability and involutiveness for distributions and to indicate a route towards a theorem of Frobenius. This is then used to discuss bundles and connections in the noncommutative setting and to go some steps towards a noncommutative Chern-Weil homomorphism. In the final section the authors give a brief description of the noncommutative version of the Schouten-Nijenhuis bracket and describe Poisson structures.

In the first section the authors give a short description of the setting of convenient spaces elaborating those aspects which they will need later. Then they repeat the usual construction of noncommutative differential forms for convenient algebras in the second section. There they consider triples \((A, \Omega^A_*, d)\) where \((\Omega^A_*, d)\) is a graded differential algebra with \(\Omega^A_0 = A\) and \(\Omega^A_n = 0\) for negative \(n\). They call \((\Omega^A_*, d)\) a differential algebra for \(A\). A universal construction of such an algebra \(\Omega^A_*\) for a commutative algebra \(A\) is described byE. Kunz in ‘Kähler differentials’ Vieweg, Wiesbaden (1986; Zbl 0587.13014). They present a noncommutative version of the construction of Kunz, since they need more information.

Next they show that the bimodule \(\Omega_n (A)\) represents the functor of the normalized Hochschild \(n\)-cocycles. In the third section they introduce the noncommutative version of the Frölicher-Nijenhuis bracket by investigating all bounded graded derivations of the algebra of differential forms. This bracket is then used to formulate the concept of integrability and involutiveness for distributions and to indicate a route towards a theorem of Frobenius. This is then used to discuss bundles and connections in the noncommutative setting and to go some steps towards a noncommutative Chern-Weil homomorphism. In the final section the authors give a brief description of the noncommutative version of the Schouten-Nijenhuis bracket and describe Poisson structures.

Reviewer: V.G.Angelov (Sofia)

### MSC:

46L85 | Noncommutative topology |

46L87 | Noncommutative differential geometry |

17B70 | Graded Lie (super)algebras |

17B66 | Lie algebras of vector fields and related (super) algebras |