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.

scholarly journals Finite-dimensional Nichols algebras over dual Radford algebras

2020 ◽  
pp. 2140001
Author(s):  
O. Márquez ◽  
D. Bagio ◽  
J. M. J. Giraldi ◽  
G. A. García
Keyword(s):  
Jorge Luis Borges ◽  
Drinfeld Modules ◽  
Indecomposable Modules ◽  
Simple Modules ◽  
Generators And Relations ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Nichols Algebras ◽  
The Way

For [Formula: see text], let [Formula: see text] be the dual of the Radford algebra of dimension [Formula: see text]. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over [Formula: see text]. Along the way, we describe the simple objects in [Formula: see text] and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case [Formula: see text]. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for [Formula: see text] and [Formula: see text], [Formula: see text], which recovers some results of the second and third author in the former case, and of Xiong in the latter. Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es. Jorge Luis Borges

scholarly journals Jones Type Basic Construction on Hopf Spin Models

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1547
Author(s):  
Cao Tianqing ◽  
Xin Qiaoling ◽  
Wei Xiaomin ◽  
Jiang Lining
Keyword(s):  
Hopf Algebra ◽  
Crossed Product ◽  
Spin Models ◽  
Drinfeld Double ◽  
Field Algebra ◽  
Basic Construction ◽  
Finite Dimensional ◽  
Observable Algebra

Let H be a finite dimensional C∗-Hopf algebra and A the observable algebra of Hopf spin models. For some coaction of the Drinfeld double D(H) on A, the crossed product A⋊D(H)^ can define the field algebra F of Hopf spin models. In the paper, we study C∗-basic construction for the inclusion A⊆F on Hopf spin models. To achieve this, we define the action α:D(H)×F→F, and then construct the resulting crossed product F⋊D(H), which is isomorphic A⊗End(D(H)^). Furthermore, we prove that the C∗-basic construction for A⊆F is consistent to F⋊D(H), which yields that the C∗-basic constructions for the inclusion A⊆F is independent of the choice of the coaction of D(H) on A.

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).

scholarly journals ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

2010 ◽  
Vol 09 (02) ◽  
pp. 195-208 ◽  
Author(s):  
SEBASTIÁN FREYRE ◽  
MATÍAS GRAÑA ◽  
LEANDRO VENDRAMIN
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Drinfeld Modules ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mathieu Groups ◽  
Nichols Algebras

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽q) and PSL(2, q) = PSL(2, 𝔽q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M20 or M21 = PSL(3, 4) is the group algebra.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

Sign in / Sign up

Export Citation Format

Share Document