scholarly journals A classification of Nichols algebras of semisimple Yetter–Drinfeld modules over non-abelian groups

2017 ◽  
Vol 19 (2) ◽  
pp. 299-356 ◽  
Author(s):  
István Heckenberger ◽  
L. Vendramin
Keyword(s):  
Abelian Groups ◽  
Drinfeld Modules ◽  
Nichols Algebras

scholarly journals On finite GK-dimensional Nichols algebras over abelian groups

2021 ◽  
Vol 271 (1329) ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Iván Angiono ◽  
István Heckenberger
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Jordan Block ◽  
Vector Spaces ◽  
Direct Sums ◽  
Content Type ◽  
Nichols Algebra ◽  
Nichols Algebras ◽  
Kirillov Dimension

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

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.

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

Some results on the classification of expansive automorphisms of compact abelian groups

1996 ◽  
Vol 16 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Fabio Fagnani
Keyword(s):  
Finite Group ◽  
Topological Entropy ◽  
Abelian Groups ◽  
Topological Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Abelian Groups ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

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

A Classification Of n-Abelian Groups

1969 ◽  
Vol 21 ◽  
pp. 1238-1244 ◽  
Author(s):  
J. L. Alperin
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Recent Result ◽  
Abelian Groups ◽  
The Way

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,(xy)n = xnynfor all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

EQUIVALENT AUTOMATIC STRUCTURES AND THEIR BOUNDARIES

1992 ◽  
Vol 02 (04) ◽  
pp. 443-469 ◽  
Author(s):  
WALTER D. NEUMANN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Equivalence Class ◽  
Abelian Groups ◽  
Automatic Structures ◽  
Automatic Structure

Two (synchronous, asynchronous, or non-deterministic asynchronous) automatic structures on a group G are “equivalent” if their union is a non-deterministic asynchronous automatic structure. We discuss this relation, giving a classification of structures up to equivalence for abelian groups and partial results in some other cases. We also discuss a “boundary” of an asynchronous automatic structure. We show that it is an invariant of the equivalence class of the structure, and describe other properties. We describe a “rehabilitated boundary” which yields Sn−1 for any automatic structure on ℤn.

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