Notes on 2-parameter quantum groups I

A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An operator realization of the positive part is given. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10431040, 10728102), the TRAPOYT, the FUDP and the Priority Academic Discipline from the MOE of China, the SRSTP from the STCSM, the Shanghai Priority Academic Discipline from the SMEC     

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We describe the relations of the bialgebras associated with these solutions and the standard deformations of GL_n and of the supergroup GL(m|n). We also show how the existence of zero divisors in some of these algebras are related to the combinatorics of their related matrix, providing a necessary and sufficient condition for the bialgebras to be a domain. We consider their Poincaré series, and we provide a Hopf algebra structure to quotients of these bialgebras in an explicit way. We discuss the problems involved with the lift of the Hopf algebra structure, working only by localization.     

(Co)bordism Groups in Quantum PDEs

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prástaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A ^m _{1 1}×···×A ^m _{s s} , with AA ₁×···×A _s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0pm–1) are canonically associated to any QPDE, Ê_k ^k _m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0p3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

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.     

Induced Corepresentations of Locally Compact Quantum Groups

We introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition. We also establish a quantum analogue of the classical bijective correspondence between quasi-invariant measures and certain measures on the larger locally compact group.     

The Triangulated Hopf Category n +SL(2)

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.     

The Structure of Quantum Group Uq(osp(1, 2, f))

In this paper we construct a new quantum group Uq(osp(1,2, f)), which can be seen as a generalization of Uq(oSp(1, 2)). A necessary and sufficient condition for the algebra Uq(oSp(1,2, f)) to be a super Hopf algebra is obtained and the center Z(Uq(osp(1,2, f))) is given.     

Simple modules of the quantum double of the Nichols algebra of unidentified diagonal type 𝔲𝔣𝔬(7)

The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra 𝔲𝔣𝔬(7) are classified.     

A Generalization of a Two-Parameter Quantization for the Group GL 2(k)

We generalize a well-known two-parameter quantization for the group GL ₂(k) (over an arbitrary field k). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for GL ₂(k).     

(Co)Bordism Groups in PDEs

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. Within this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicit proof that the usual Thom–Pontryagin construction in (co)bordism theory can be generalized also to a singular integral (co)bordism on the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates, in a natural way, Hopf algebras (full p-Hopf algebras, 0 p n – 1), to any PDE, recently introduced, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier–Stokes equation (NS) and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of (NS) with a change in sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full p-Hopf algebras, 0 p 3, for the Navier–Stokes equation are determined.     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

Cartan calculus for quantum differentials on bicrossproducts

We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M) k G associated to finite group factorizations X = GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form which is a generator in the noncommutative de Rham cohomology H ¹. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble D ^*(S ₃)k(S ₃) k₆ and the quantum double D(S ₃) \triangleleft $$ " align="middle" border="0"> k S ₃, finding respectively a natural calculus and a unique calculus with H ⁰ = k.1.     

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.     

Hopf modules,Frobenius functors and (one-sided) Hopf algebras

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Quantum and Integral (Co)Bordisms in Partial Differential Equations

Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an application, we relate integral bordism to the spectral term that represents the space of conservation laws for PDEs. This also gives a general method to associate in a natural way a Hopf algebra to any PDE.     

Flag-Transitive 3-(v,k, 3) Designs and PSL(2, q) Groups

This article is a contribution to the study of the automorphism groups of 3-(v,k,3) designs.Let S =(P,B) be a non-trivial 3-(q+ 1,k,3) design.If a two-dimensional projective linear group PSL(2,q) acts flag-transitively on S,then S is a 3-(q + 1,4,3) or 3-(q + 1,5,3) design.     

量子群Uq(D4)上不可约模的Grobner-Shirshov对

首先,利用量子群Uq (D4)的已知的Grobner-Shirshov基和Chibrikov的双自由模方法来计算量子群Uq(D4)上不可约模Vq(λ)的一个Grobner-Shirshov对,然后在Uq(D4)的适当形式U'q(D4)中取q=1得到D4型单李代数的泛包络代数U(D4)上不可约模V(λ)的一个Grobner-Shirshov对.