Poisson algebras of block-upper-triangular bilinear forms and braid group action (original) (raw)

Poisson algebras and symmetries of block-upper-triangular matrices

Arxiv preprint arXiv:1012.5251, 2010

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on C N with the property that for any n, m ∈ N such that nm = N , the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size m × m is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-uppertriangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for on and for m = 2 is the twisted q-Yangian for sp 2n . We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2.

On a Poisson space of bilinear forms with a Poisson Lie action

2014

We consider the space A of bilinear forms on C N with defining matrix A endowed with the quadratic Poisson structure studied by the authors in . We classify all possible quadratic brackets on (B, A) ∈ GL N × A with the property that the natural action A → BAB T of the GL N Poisson-Lie group on the space A is a Poisson action thus endowing A with the structure of Poisson space. Beside the product Poisson structure on GL N × A we find two more (dual to each other) structures for which (in contrast to the product Poisson structure) we can implement the reduction to the space of bilinear forms with block upper triangular defining matrices by Dirac procedure. We consider the generalisation of the above construction to triples (B, C, A) ∈ GL N × GL N × A with the Poisson action A → BAC T and show that A then acquires the structure of Poisson symmetric space. We study also the generalisation to chains of transformations and to the quantum and quantum affine algebras and the relation between the construction of Poisson symmetric spaces and that of the Poisson groupoid.

Quantum and braided-Lie algebras

Journal of Geometry and Physics, 1994

We show that such an object has an enveloping braided-bialgebra U (L). We show that every generic R-matrix leads to such a braided Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U (L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations U q (g) are understood as the enveloping algebras of such underlying braided Lie algebras with [ , ] on L ⊂ U q (g) given by the quantum adjoint action.

Braiding structures on formal Poisson groups and classical solutions of the QYBE

Journal of Geometry and Physics, 2003

If g is a quasitriangular Lie bialgebra, the formal Poisson group F[[g∗]] can be given a braiding structure. This was achieved by Weinstein and Xu using purely geometrical means, and independently by the authors by means of quantum groups. In this paper we compare these two approaches. First, we show that the braidings they produce share several similar properties (in

Twisting Poisson algebras, coPoisson algebras and Quantization

The purpose of this paper is to study twistings of Poisson algebras or bialgebras, coPoisson algebras or bialgebras and star-products. We consider Hom-algebraic structures generalizing classical algebraic structures by twisting the identities by a linear self map. We summarize the results on Hom-Poisson algebras and introduce Hom-coPoisson algebras and bialgebras. We show that there exists a duality between Hom-Poisson bialgebras and Hom-coPoisson bialgebras. A relationship between enveloping Homalgebras endowed with Hom-coPoisson structures and corresponding Hom-Lie bialgebra structures is studied. Moreover we set quantization problems and generalize the notion of star-product. In particular, we characterize the twists for the Moyal-Weyl product for polynomials of several variables.

1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups

1999

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Communications in Mathematical Physics, 1993

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups U q (g). They have the same FRT generators l ± but a matrix braided-coproduct ∆L = L⊗L where L = l + Sl − , and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices BM q (2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double D(U q (sl 2)) (also known as the 'quantum Lorentz group') is the semidirect product as an algebra of two copies of U q (sl 2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.

Classical deformations, Poisson–Lie contractions, and quantization of dual Lie bialgebras

Journal of Mathematical Physics, 1995

A Poisson-Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincare, and Heisenberg groups by using a classical r-matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson-Lie quantization that deforms the standard Moyal-Weyl ah-product. 0 1995 American Institute of Physics.

Quantum groups and cylinder braiding

Forum Mathematicum, 1998

The purpose of this paper is to introduce a new structure into the representation theory of quantum groups. The structure is motivated by braid and knot theory. Represen¬ tations of quantum groups associated to classical Lie algebras have an additional symmetry which cannot be seen in the classical limit. We first explain the general formalism of these symmetries (called cylinder forms) in the context of comodules. Basic ingredients are tensor representations of braid groups of type B derived from standard R-matrices associated to socalled four braid pairs. These are applied to the Faddeev-Reshetikhin-Takhtadjian construc¬ tion of bialgebras from R-matrices. As a consequence one obtains four braid pairs on all representations of the quantum group. In the second part of the paper we study in detail the dual situation of modules over the quantum enveloping algebra Uq(sl2). The main result here is the computation of the universal cylinder twist.