scholarly journals Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic

2020 ◽  
Vol 559 ◽  
pp. 547-579 ◽  
Author(s):  
Jing Wang
Keyword(s):  
Positive Characteristic ◽  
Finite Dimensional ◽  
Nichols Algebras

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 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.

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

Finite dimensional Nichols algebras over certainp-groups

2016 ◽  
Vol 45 (9) ◽  
pp. 3691-3702
Author(s):  
Yibo Yang ◽  
Shenglin Zhu
Keyword(s):  
Finite Dimensional ◽  
Nichols Algebras

scholarly journals POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250066
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Groups ◽  
Drinfeld Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Necessary And Sufficient

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by 𝕊nare infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by A to be finite dimensional.

scholarly journals Fields with few types

2013 ◽  
Vol 78 (1) ◽  
pp. 72-84
Author(s):  
Cédric Milliet
Keyword(s):  
Division Ring ◽  
Positive Characteristic ◽  
Field Extension ◽  
Order Structure ◽  
Small Field ◽  
Finite Degree ◽  
Locally Finite ◽  
First Order ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractAccording to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.

scholarly journals A new approach to the Koszul property in representation theory using graded subalgebras

2012 ◽  
Vol 12 (1) ◽  
pp. 153-197 ◽  
Author(s):  
Brian J. Parshall ◽  
Leonard L. Scott
Keyword(s):  
Positive Characteristic ◽  
Hereditary Algebra ◽  
New Approach ◽  
Schur Algebras ◽  
Graded Modules ◽  
Finite Dimensional ◽  
New Information ◽  
Koszul Property ◽  
Semisimple Algebraic Groups ◽  
Finite Dimensional Algebras

AbstractGiven a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\mathrm{gr} \hspace{0.167em} B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A, \mathfrak{a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak{a}$. The algebra $B$ arises as a quotient $B= A/ J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have a structure as graded modules for $\mathfrak{a}$. These results are applied to obtain new information about the finite dimensional algebras (e.g., the $q$-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic $p$. These results require, at least at present, considerable restrictions on the size of $p$.

scholarly journals Central elements of the Jennings basis and certain Morita invariants

2019 ◽  
Vol 19 (08) ◽  
pp. 2050160 ◽  
Author(s):  
Taro Sakurai
Keyword(s):  
Group Algebra ◽  
Positive Characteristic ◽  
Finite Dimensional Algebra ◽  
Independent Subset ◽  
Dimensional Algebra ◽  
Ideal Formula ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Morita Invariant ◽  
Central Elements

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the [Formula: see text]th (right) socle [Formula: see text] of a finite-dimensional algebra [Formula: see text] is a Morita invariant; this is a generalization of important Morita invariants — the center [Formula: see text] and the Reynolds ideal [Formula: see text]. As an example, we also studied [Formula: see text] for the group algebra FG of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of positive characteristic [Formula: see text]. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of [Formula: see text]. In fact, such elements form a basis of [Formula: see text] for every integer [Formula: see text] if [Formula: see text] is powerful. As a corollary we have [Formula: see text] if [Formula: see text] is powerful.

scholarly journals Examples of finite-dimensional rank 2 Nichols algebras of diagonal type

2007 ◽  
Vol 143 (01) ◽  
pp. 165-190 ◽  
Author(s):  
I Heckenberger
Keyword(s):  
Finite Dimensional ◽  
Rank 2 ◽  
Nichols Algebras
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