On Nichols algebras of infinite rank with finite Gelfand–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.

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On finite GK-dimensional Nichols algebras over abelian groups

Memoirs of the American Mathematical Society ◽

10.1090/memo/1329 ◽

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.

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TOPOLOGICAL NOETHERIANITY FOR ALGEBRAIC REPRESENTATIONS OF INFINITE RANK CLASSICAL GROUPS

Transformation Groups ◽

10.1007/s00031-021-09656-x ◽

2021 ◽

Author(s):

R. H. EGGERMONT ◽

A. SNOWDEN

Keyword(s):

Classical Groups ◽

Infinite Rank ◽

Polynomial Representations ◽

Algebraic Representations

AbstractDraisma recently proved that polynomial representations of GL∞ are topologically noetherian. We generalize this result to algebraic representations of infinite rank classical groups.

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Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

Algebra Colloquium ◽

10.1142/s1005386716000596 ◽

2016 ◽

Vol 23 (04) ◽

pp. 701-720 ◽

Cited By ~ 2

Author(s):

Xiangui Zhao ◽

Yang Zhang

Keyword(s):

Computational Method ◽

Finitely Generated ◽

Finitely Generated Module ◽

Polynomial Algebras ◽

Ore Extensions ◽

Basis Method ◽

Finitely Generated Modules ◽

Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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Nichols algebras of group type with many quadratic relations

Advances in Mathematics ◽

10.1016/j.aim.2011.04.006 ◽

2011 ◽

Vol 227 (5) ◽

pp. 1956-1989 ◽

Cited By ~ 15

Author(s):

M. Graña ◽

I. Heckenberger ◽

L. Vendramin

Keyword(s):

Group Type ◽

Nichols Algebras

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EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501915 ◽

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.

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ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

International Journal of Number Theory ◽

10.1142/s1793042110003071 ◽

2010 ◽

Vol 06 (03) ◽

pp. 579-586 ◽

Cited By ~ 3

Author(s):

ARNO FEHM ◽

SEBASTIAN PETERSEN

Keyword(s):

Finite Field ◽

Abelian Variety ◽

Galois Extension ◽

Characteristic Zero ◽

Rational Point ◽

Abelian Varieties ◽

Finitely Generated ◽

Infinite Rank ◽

Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

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