We study Galois extensions M (co-)H ⊂ M for H -(co)module algebras M if H is a FrobeniusHopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf-Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. We find that the Galois extensions N ⊂ M over some Frobenius Hopf algebroid are precisely the balanced depth 2 Frobenius extensions. We prove that the Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized Brzeziński-Militaru theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's theorem and to get a monoidal embedding of the module category M E of the endomorphism Hopf algebroid E = End N M N into N M op N .

Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter-Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter-Drinfel'd category is monoidal and pre-braided just as in the case of bialgebroids, and is embedded into the one-sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of [2] and [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg's theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co-)monadic point of view. IntroductionBicoalgebroids were introduced by Brzeziński and Militaru in [2] as the structure that dualizes bialgebroids (in fact, Takeuchi's × R -bialgebras) in the sense of reversing arrows. This notion is not to be confused with the different kinds of bialgebroid-duals that were later introduced in [8]. It would seem that the study of bicoalgebroids hasn't been taken up vigorously since their inception; in our view, they merit attention for at least two reasons. First, it is well established that a bialgebroid may be thought of as a non-commutative analogue of the algebra of functions on a groupoid. It follows that a bicoalgebroid, in turn, should be regarded as a non-commutative analogue of the groupoid itself. This raises the hope that classical constructions on groupoids may find their non-commutative generalizations more easily in the context of bicoalgebroids. Secondly, just as bialgebroids play a fundamental role in depth-two extensions of algebras, it is expected that bicoalgebroids feature prominently in extensions of coalgebras (from a different approach, in [9] Kadison constructs bialgebroids from depth 2 extensions of coalgebras). To complete the picture, the dual Hopf-Galois theory of [16] for extensions of coalgebras should generalize (from bialgebras) to bicoalgebroids, giving a dual version of bialgebroid-Galois theory. Further work in this latter direction is deferred to a subsequent publication.Central to this paper is the introduction of scalar extension for bicoalgebroids. Incidentally, the construction that was shown in [1] to be a non-commutative version of scalar extension was defined (for Hopf-algebras) in [2] -alongside with bicoalgebroids. Bicoalgebroids; comodules and modulesThroughout, k will be a field and the category M = M k of k-modules will serve as our underlying category. The unadorned ⊗ always means ⊗ k .We use the ubiquitous Sweedler notation for coproducts and coactions. For a coalgebra C, ∆, ε , the coproduct ∆ : C → C ⊗ C on elements is denoted ∆(c) = c (1) ⊗ c (2) , with an implicit finite summation understood, ...

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The two-particle density matrix contains all information, which is "readable" by one- and two-particle operators. The report discusses the building of a two-electron density matrix, which is perfectly N-representable by construction. The 2-matrix of a pure quantum state is obtained without the intermediate determination of the wave function. The procedure is illustrated by computations on atomic and small molecular systems.

The two-particle density matrix contains all information which is "readable" by one-and two-particle operators. This work discusses the building of a two-electron density matrix, which is perfectly N-representable by construction. The 2-matrix of a pure quantum state is obtained without the intermediate determination of the wave function. The procedure is illustrated by computations on atomic and small molecular systems.