Quantum Yang-Baxter module algebras

LetH be a quantum group over a commutative ringR. We introduce the concept of quantum Yang-BaxterH-module algebra, generalizing the notion ofH-dimodule algebra in the case whereH is commutative, cocommutative and faithfully projective. After discussing some examples, we introduceH-Azumaya algebras. The set of quivalence classes ofH-Azumaya algebras can be made into a group, called the Brauer group of the quantum groupH. This group is a generalization of the Brauer-Long group.This author wishes to thank the Department of Mathematics, UIA, for its hospitality and financial support during the time when most of this paper was written.     

Compact orthoalgebras

We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated is of finite height. We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated is determined by that of its space of atoms.

     

W-代数W(2,2)的单参数量子形变(英文)

众所周知,泛包络代数的量子形变所对应的量子群结构是依据所给代数的单根系给出的.我们构造了q-形变W代数Wq,并给出其非平凡的量子群结构.     

Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a finite-dimensional hereditary algebra over a finite field k, () and () be, respectively, the Hall algebra and the composition algebra of , be the isomorphism classes of finite dimensional -modules and I the isomorphism classes of simple -modules. We define and , in , to be the right and left derivations of () respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of -mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u as the combinations of elements u_i in the Hall algebra, where i I and in is any exceptional -module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.     

Remarks on the order for quantum observables

Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002, grant VEGA 2/3163/23 and Center of excellence SAS, CEPI I/2/2005.     

一类基于量子程序理论的序列效应代数

空间上的算子理论是量子力学的基本数学框架之一.Hilbert空间效应代数是指小于等于单位算子的正算子集合.我们引入了Hilbert空间效应代数的一类子序列效应代数,并讨论了其上序列积的基本运算性质.我们发现：由于代数结构的不同,这类新的序列效应代数与现有效应代数上的运算性质有很大差异.     

Asymptotic Decoherence in Infinite-Dimensional Quantum Systems with Quadratic Hamiltonians

Mathematical Notes -     

Representations of Extended Affine Lie Algebras Coordinatized by Certain Quantum Tori

An irreducible representation of the extended affine Lie algebra of type A _n-1 coordinatized by a quantum torus of variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra .     

Compactifications of Discrete Quantum Groups

Given a discrete quantum group we construct a Hopf -algebra which is a unital -subalgebra of the multiplier algebra of . The structure maps for are inherited from and thus the construction yields a compactification of which is analogous to the Bohr compactification of a locally compact group. This algebra has the expected universal property with respect to homomorphisms from multiplier Hopf algebras of compact type (and is therefore unique). This provides an easy proof of the fact that for a discrete quantum group with an infinite dimensional algebra the multiplier algebra is never a Hopf algebra.Partially supported by Komitet Badań Naukowych grants 2P03A04022 & 2P03A01324, the Foundation for Polish Science and Deutsche Forschungsgemeinschaft.     

紧矩阵量子群G的余表示与其对偶量子群(A)的关系

设 G=（A,△）为紧矩阵量子群,G为A的所有有限维光滑的、不可约余表示等价类的集合.本文通过（A,△）的一个余表示Vo构造了两个相互配对的集合,利用Hilbert C＊-模的理论证明它们分别为A和Baaj与Skandalis构造的量子群A,并且证明了对任意的α∈G,在A中都对应一个有限维投影算子Pα,满足 dim（α）=dim（pα）.     

Quantum symmetry groups of Hilbert modules equipped with orthogonal filtrations

We define and show the existence of the quantum symmetry group of a Hilbert module equipped with an orthogonal filtration. Our construction unifies the constructions of Banica–Skalski?s quantum symmetry group of a C^?$C^{?}$-algebra equipped with an orthogonal filtration and Goswami?s quantum isometry group of an admissible spectral triple.     

Two Exterior Algebras for Orthogonal and Symplectic Quantum Groups

Let be one of the N ²-dimensional bicovariant first-order differential calculi on the quantum groups O _q (N) or Sp _q (N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra _s is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars bi-invariant 1-form. Moreover, is central in _u and _u is an inner differential calculus.     

Coxeter Transformations in Quantum Groups

It is natural to define a Coxeter transformation in quantum groups by the product of Lusztig's symmetries in some order. In this note, we show that the Ringel–Hall algebra approach enables us to determine the behavior of its action in case of finite type.     

A 类量子群和 Hecke 代数的 Schur-Weyl 对偶

设Vm是量子群Uq（SLm)的标准表示,通过Hecke代数的作用,作者将Vm的张量积V^nm分解成了Uq(SLm)的不可约表示的直和,从而给出了Uq(SLm)与Hecke代数的H（q,n)Schur-Weyl对偶的完整证明,进一步得到,当q是实数时,在Hilbert空间H^n上,Uq(SL∞）和H（q,n)之间存在着Schur-Weyl对偶。     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0

We present representation theoretical interpretations ofquasi-symmetric functions and noncommutative symmetric functions in terms ofquantum linear groups and Hecke algebras at q = 0. We obtain inthis way a noncommutative realization of quasi-symmetric functions analogousto the plactic symmetric functions of Lascoux and Schützenberger. Thegeneric case leads to a notion of quantum Schur function.     

Various Notions of Orthogonality in Normed Spaces

In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also considered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.     

Differential Operators Over Quantum Spaces

The topic of this paper is the development of a differential calculus on a quantum space covariant with respect to the action of a quantum group. Quantized differential operators, jets, the de Rham and Spencer complexes, etc., are constructed. Also the integration over a quantum space, adjoint operators, integral forms, Greens formula are discussed. The Wess–Zumino de Rham complex and the algebra of differential operators are treated as a basic example.     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.