Finite Gelfand Kirillov Dimension Research Articles

Coideal subalgebras of pointed and connected Hopf algebras

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T T . The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume R R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.

On ultragraph Leavitt path algebras with finite Gelfand-Kirillov dimension

In this article, we prove that every ultragraph Leavitt path algebra is a direct limit of Leavitt path algebras of finite graphs and determine the Gelfand-Kirillov dimension of an ultragraph Leavitt path algebra. We also characterize ultragraph Leavitt path algebras whose simple modules are finitely presented, and show that these algebras have finite Gelfand-Kirillov dimension. Moreover, we construct new classes of simple modules over ultragraph Leavitt path algebras associated with minimal infinite emitters and minimal sinks, which have not appeared in the context of Leavitt path algebras of graphs.

The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

Corrigendum: Liftings of Jordan and Super Jordan Planes

AbstractWe complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in [N. Andruskiewitsch, I. Angiono and I. Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc., II. Ser. 61(3) (2018), 661–672.].

Graphs with disjoint cycles classification via the talented monoid

We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid TE has a certain Jordan-Hölder composition series. These are graphs E whose associated Leavitt path algebras Lk(E) have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since TE is the positive cone of the graded Grothendieck group K0gr(Lk(E)), we conclude that for graphs E and F, if K0gr(Lk(E))≅K0gr(Lk(F)) then GKdimLk(E)=GKdimLk(F), thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.

On finite GK-dimensional Nichols algebras over abelian groups

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.

EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

AbstractWe 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.

The structure of connected (graded) Hopf algebras

In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.

On Nichols algebras of infinite rank with finite Gelfand–Kirillov dimension

We classify infinite-dimensional decomposable braided vector spaces arising from abelian groups whose components are either points or blocks such that the corresponding Nichols algebras have finite Gelfand–Kirillov dimension. In particular we exhibit examples where the Gelfand–Kirillov dimension attains any natural number greater than one.

Stably Noetherian Algebras of Polynomial Growth

Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb {N}$ -graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions.

Weighted Leavitt Path Algebras that are Isomorphic to Unweighted Leavitt Path Algebras

Let K be a field. We characterise the row-finite weighted graphs (E, w) such that the weighted Leavitt path algebra LK(E, w) is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if LK(E, w) is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then LK(E, w) is isomorphic to an unweighted Leavitt path algebra.

On Nichols algebras over basic Hopf algebras

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $$\mathbb {k}$$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter–Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand–Kirillov dimension.

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ⁡ ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M ( X , r ) M(X,r) . These results allow us to control the prime spectrum of the algebra K [ M ( X , r ) ] K[M(X,r)] and to describe the Jacobson radical and prime radical of K [ M ( X , r ) ] K[M(X,r)] . Finally, we give a matrix-type representation of the algebra K [ M ( X , r ) ] / P K[M(X,r)]/P for each prime ideal P P of K [ M ( X , r ) ] K[M(X,r)] . As a consequence, we show that if K [ M ( X , r ) ] K[M(X,r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G 1 , … , G m G_1,\dotsc ,G_m , each being the group of quotients of a cancellative subsemigroup of M ( X , r ) M(X,r) such that the algebra K [ M ( X , r ) ] K[M(X,r)] embeds into M v 1 ⁡ ( K [ G 1 ] ) × ⋯ × M v m ⁡ ( K [ G m ] ) \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m]) , a direct product of matrix algebras.

Ideals in the Enveloping Algebra of the Positive Witt Algebra

Let W+ be the positive Witt algebra, which has a mathcal {C}-basis {e_{n}: n in mathcal {Z}_{geq 1}}, with Lie bracket [ei,ej] = (j − i)ei+j. We study the two-sided ideal structure of the universal enveloping algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).

Hopf Ore Extensions

Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

Graph algebras and the Gelfand–Kirillov dimension

We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand–Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225–1250231.

Liftings of Jordan and Super Jordan Planes

AbstractWe classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

On finite dimensional Nichols algebras of diagonal type

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.

Separating Ore sets for prime ideals of quantum algebras

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus-invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand–Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac–Moody algebras.

The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type

Let Bq be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix q. We consider the graded dual Lq of the distinguished pre-Nichols algebra e Bq from (A3) and the divided powers algebra Uq, a suitable Drinfeld double of Lq#kZ � . We provide basis and presentations by generators and relations of Lq and Uq, and prove that they are noetherian and have finite Gelfand-Kirillov dimension.