arXiv subjects
Publications and source records attributed to Michel Dubois-Violette.
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
We construct N-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ generalizing thereby the usual complex (N=2) of differential forms. These complexes arise naturally in the description of higher spin gauge fields. Although, for $N\geq 3$, the generalized cohomology of these N-complexes is non trivial, we prove a generalization of the Poincaré lemma. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma.
We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent operator Q such that $Q^h=0$ ($h=k+2=2,3,...$ being the height of the current algebra representation) acting on an extended state space. The physical subquotient is identified with the direct sum $\oplusinf^{h-1}_{n=1}\ker(Q^n)/\im(Q^{h-n})$.
We study the generalized homology associated with a nilpotent endomorphism $d$ satisfying $d^N=0$. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalization of the notion of graded differential algebra in this context.
The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite dimensional state space. A generalized BRS operator $A$ such that $A^h=0 (h=k+2=3,4,...$ being the height of the current algebra representation) acts in a (2h-1)-dimensional indefinite metric space $H_I$ of quantum group invariant vectors. The generalized cohomologies $Ker A^n/ Im A^{h-n} (n=1,..., h-1)$ are 1-dimensional. Their direct sum spans the physical subquotient of $H_I$.
We analyse the structure of the $κ=0$ limit of a family of algebras $A_κ$ describing noncommutative versions of space-time, with $κ$ a parameter of noncommutativity. Assuming the Poincaré covariance of the $κ=0$ limit, we show that, besides the algebra of functions on Minkowski space, $A_0$ must contain a nontrivial extra factor $A^I_0$ which is Lorentz covariant and which does not commute with the functions whenever it is not commutative. We give a general description of the possibilities and analyse some representative examples.
We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.
We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some examples. In particular we construct the $q$-analog of the simplicial differential on forms, the $q$-analog of the Hochschild differential and the $q$-analog of the universal differential envelope of an associative unital algebra.
We recall the definition of $q$-differential algebras and discuss some representative examples. In particular we construct the $q$-analog of the Hochschild coboundary. We then construct the universal $q$-differential envelope of a unital associative algebra and study its properties. The paper also contains general results on $d^N=0$.
We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework.
We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.
We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.
A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
In commutative differential geometry the Frölicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Frölicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracket and the Frölicher-Nijenhuis calculus will be developped for several kinds of differential graded algebras based on derivations, which were indroduced by \cit!{6}.
We characterize the derivation $d:A\to Ω^1_{\der}(A)$ by a universal property introducing a new class of bimodules.
We define a new cohomology for associative algebras which we compute for algebras with units.
We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on Donaldson invariants.
Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.