Grothendieck Groups of a Class of Quantum Doubles

2008 ◽  
Vol 15 (03) ◽  
pp. 431-448 ◽  
Author(s):  
Yun Zhang ◽  
Feng Wu ◽  
Ling Liu ◽  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Grothendieck Group ◽  
Ring Structure ◽  
Quantum Double ◽  
Simple Modules ◽  
Loewy Length ◽  
Finite Dimensional ◽  
Grothendieck Groups

Let k be a field and An(ω) be the Taft n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. We have constructed an n4-dimensional Hopf algebra Hn(p,q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω-1, and studied the irreducible and finite-dimensional representations of Hn(1,q). In this paper, we continue our study of Hn(1,q), examine the Grothendieck group G0(Hn(1,q)) ≅ G0(D(An(ω)), and describe its ring structure. We also give the Loewy length of the tensor product of two simple modules over Hn(1,q).

Hom-L-R-smash biproduct and the category of Hom–Yetter–Drinfel’d–Long bimodules

2018 ◽  
Vol 17 (07) ◽  
pp. 1850133 ◽  
Author(s):  
Daowei Lu ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Monoidal Category ◽  
Admissible Pair ◽  
Smash Product ◽  
Monoidal Categories ◽  
Finite Dimensional

Let [Formula: see text] be a Hom-bialgebra. In this paper, we firstly introduce the notion of Hom-L-R smash coproduct [Formula: see text], where [Formula: see text] is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra [Formula: see text], we introduce the notion of Hom-L-R-admissible pair [Formula: see text]. We prove that [Formula: see text] becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category [Formula: see text] of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair [Formula: see text] actually corresponds to a bialgebra in the category [Formula: see text], when [Formula: see text] and [Formula: see text] are involutions. Finally, we prove that when [Formula: see text] is finite dimensional Hom-Hopf algebra, [Formula: see text] is isomorphic to the Yetter–Drinfel’d category [Formula: see text] as braid monoidal categories where [Formula: see text] is the tensor product Hom–Hopf algebra.

scholarly journals On finite dimensionality of mixed Tate motives

2008 ◽  
Vol 4 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Grothendieck Group ◽  
Ring Structure ◽  
Triangulated Category ◽  
Cohomology Groups ◽  
Weight Filtration ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Tate Motives

AbstractWe prove a few results concerning the notion of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology groups defined by means of the Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ–ring structure on K0(D).

The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

The Representation Ring and the Centre of a Hopf Algebra

1999 ◽  
Vol 51 (4) ◽  
pp. 881-896 ◽  
Author(s):  
Sarah J. Witherspoon
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Character Table ◽  
Quantum Double ◽  
Representation Ring ◽  
Finite Dimensional ◽  
Dual Character ◽  
Structure Constants ◽  
A Finite Group ◽  
Cocommutative Hopf Algebra

AbstractWhen H is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of H which generalizes the conjugacy class sums of a finite group, and express the class equation of H in terms of this basis. We show that the representation ring and the centre of H are dual character algebras (or signed hypergroups).

scholarly journals On the restricted Jordan plane in odd characteristic

2020 ◽  
pp. 2140012
Author(s):  
Nicolás Andruskiewitsch ◽  
Héctor Peña Pollastri
Keyword(s):  
Hopf Algebra ◽  
Positive Characteristic ◽  
Drinfeld Double ◽  
Nichols Algebra ◽  
Simple Modules ◽  
Restricted Version ◽  
Defining Relations ◽  
Finite Dimensional ◽  
Nichols Algebras

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND TENSOR PRODUCTS OF SYMMETRIC GROUP REPRESENTATIONS

1991 ◽  
Vol 01 (02) ◽  
pp. 207-221 ◽  
Author(s):  
JEAN-YVES THIBON
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Tensor Products ◽  
Ring Structure ◽  
Algebra Structure ◽  
Group Representations ◽  
Symmetric Group Representations

The Hopf algebra structure of the ring of symmetric functions is used to prove a new identity for the internal product, i.e., the operation corresponding to the tensor product of symmetric group representations. From this identity, or by similar techniques which can also involve the λ-ring structure, we derive easy proofs of most known results about this operation. Some of these results are generalized.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

Double-bosonization of braided groups and the construction of Uq(g)

1999 ◽  
Vol 125 (1) ◽  
pp. 151-192 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Cartan Subalgebra ◽  
Positive Root ◽  
Point Of View ◽  
Root Space ◽  
Quantum Double ◽  
Braided Group ◽  
Negative Root ◽  
Quasitriangular Hopf Algebra

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

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