Jiang-Hua Lu

HOPF ALGEBROIDS AND QUANTUM GROUPOIDS

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the R-matrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which have Poisson groupoids as their semi-classical limits. The example of quantum sl(2) is worked out in details.

On the set-theoretical Yang-Baxter equation

We propose a general way of constructing set-theoretical solutions of the YangBaxter equation. We study the properties of the construction. We also show that our construction includes the earlier ones given by Weinstein-Xu and Etingof-SchedlerSoloviev.

Moment maps at the quantum level

We introduce the notion of moment maps for quantum groups acting on their module algebras. When the module algebras are quantizations of Poisson manifolds, we prove that the construction at the quantum level is a quantization of that at the semi-classical level. We also prove that the corresponding smashed product algebras are quantizations of the semi-direct product Poisson structures.

Classical Dynamical r-Matrices¶and Homogeneous Poisson Structures on G/H and K/T

Abstract: Let G be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all G-homogeneous (holomorphic) Poisson structures on G/H, where H⊂G is a Cartan subgroup, come from solutions to the Classical Dynamical Yang–Baxter equations which are classified by Etingof and Varchenko. A similar result holds for a maximal compact subgroup K, and we get a family of K-homogeneous Poisson structures on K/T, where T=K∩H is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on K/T up to isomorphisms. We study some Poisson geometrical properties of members of this family such as their symplectic leaves, their modular classes, and the moment maps for the T-action.

Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat-Poisson structure on G/B

For the flag manifoldX=G/B of a complex semi-simple Lie groupG, we make connections between the Kostant harmonic forms onG/B and the geometry of the Bruhat Poisson structure. We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure. We do this by using an explicit set of coordinates on the Schubert cell.

On Hopf algebras with positive bases

We show that if a finite dimensional Hopf algebra H over C has a basis with respect to which all the structure constants are nonnegative, then H is isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi and Majid from a finite group G and a unique factorization G = G+ G− of G into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way. Our results can be used to explain some results on Hopf algebras from the set-theoretical point of view.

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group G and a real form Go we define a Poisson structure on the variety of Borel subgroups of G with the property that all Go-orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a Go-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

Thompson's conjecture for real semi-simple Lie groups

Aproof of Thompson’s conjecture for real semisimple Lie groups has been given by Kapovich, Millson, and Leeb. In this paper, we give another proof of the conjecture by using a theorem of Alekseev, Meinrenken, and Woodward from symplectic geometry.

On a class of double cosets in reductive algebraic groups

We study a class of double coset spaces RAG1 × G2/RC, where G1 and G2 are connected reductive algebraic groups, and RA and RC are certain spherical sub- groups of G1×G2 obtained by identifying Levi factors of parabolic subgroups in G1 and G2. Such double cosets naturally appear in the symplectic leaf decompositions of Pois- son homogeneous spaces of complex reductive groups with the Belavin-Drinfeld Poisson structures. They also appear in orbit decompositions of the De Concini-Procesi compact- ifications of semi-simple groups of adjoint type. We find explicit parametrizations of the double coset spaces and describe the double cosets as homogeneous spaces of RA × RC. We further show that all such double cosets give rise to set-theoretical solutions to the quantum Yang-Baxter equation on unipotent algebraic groups. 1. The setup

DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

Let G be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure πst determined by a pair of opposite Borel subgroups (B, B_). We prove that for each υ in the Weyl group W of G, the double Bruhat cell Gυ,υ = BυB Ω B_υB_ in G, together with the Poisson structure πst, is naturally a Poisson groupoid over the Bruhat cell BυB/B in the flag variety G/B. Correspondingly, every symplectic leaf of πst in Gυ,υ is a symplectic groupoid over BυB/B. For u, υ ϵ W, we show that the double Bruhat cell (Gu,υ, πst) has a naturally defined left Poisson action by the Poisson groupoid (Gu,υ, πst) and a right Poisson action by the Poisson groupoid (Gu,υ, πst), and the two actions commute. Restricting to symplectic leaves of πst, one obtains commuting left and right Poisson actions on symplectic leaves in Gu,υ by symplectic leaves in Gu,u and Gυ,υ as symplectic groupoids.