DOCSLIB.ORG

On Finite-Dimensional Pointed Hopf Algebras Over Simple Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

On ¯nite-dimensional pointed Hopf algebras over simple groups Nicol¶asAndruskiewitsch Universidad de C¶ordoba,Argentina VII workshop in Lie theory and its appplications C¶ordoba,November 30th, 2009. Joint work with Fernando Fantino, Mat¶³asGra~naand Leandro Vendramin 1 Plan of the talk. I. The problem. II. Main results. III. The schemes of the proofs. References: Finite-dimensional pointed Hopf algebras with alternating groups are trivial. (N. A., F. Fantino, M. Gra~na,L. Vendramin). arXiv:00812.4628v4. 26 pages. Submitted. Pointed Hopf algebras over the sporadic simple groups. (N. A., F. Fantino, M. Gra~na,L. Vendramin). In preparation. 2 I. The problem. C alg. closed ¯eld char. 0 De¯nition. (H; m; ¢), Hopf algebra: ² (H; m) alg. with unit 1, ² ¢ : H ! H­H morphism of algebras (coproduct), ² ¢ coasociative with counit ", ² 9S : H ! H "antipode" s. t. m(S­ id)¢ = idH = m(id ­S)¢. 3 Examples. ² ¡ group, C¡ = group algebra = vector space with basis eg (g 2 ¡ ) and product egeh = egh. It becames a Hopf algebra with coproduct ¢(eg) = eg ­ eg and ¡1 antipode S(eg) = eg ² g Lie algebra, U(g) = universal algebra enveloping of g It becames a Hopf algebra with coproduct ¢(x) = x ­ 1 + 1 ­ x and antipode S(x) = ¡x, x 2 g 4 Suppose now that the group ¡ acts on a Lie algebra g by Lie algebra automorphisms. Let U(g)#C¡ = U(g)­C¡ as vector space, with tensor prod- uct comultiplication and semi-direct product multiplication. This is a Hopf algebra. Theorem. (Cartier, Kostant, Milnor-Moore). If H is a cocommutative Hopf algebra, H ' U(g)#C¡ . 5 De¯nition. A Hopf algebra H is pointed if any irreducible co- module (= representation of the dual Hopf algebra) has dimen- sion 1. For any Hopf algebra H, G(H) = fg 2 H ¡ 0 : ¢(g) = g­gg is a group. Then `pointed' means CG(H) ' the coradical of H. ² H = U(g)#C¡ is pointed with G(H) ' ¡ ; in particular, the group algebra C¡ is pointed; ² the quantum groups of Drinfeld-Jimbo and the ¯nite-dimensio- nal variations of Lusztig are pointed. 6 g simple Lie algebra, Cartan matrix (aij)1·i;j·θ; q root of 1, order N ¯nite quantum group uq(g) = Chk1; : : : ; kθ; e1; : : : ; eθ; f1; : : : ; fθi relations: N kikj = kjki; ki = 1; ¡1 diaij ¡1 ¡diaij kiejki = q ej; kifjki = q fj; 1¡a adc(ei) ij (ej) = 0; i 6= j 1¡a adc(fi) ij (fj) = 0; i 6= j ¡diaij 2 eifj ¡ q fjei = ±ij(1 ¡ ki ); i < j; i j N N e® = 0; f® = 0; ¢(g) = g­g; ¢(xi) = gi­xi + xi­1: dim g uq(g) is a pointed Hopf algebra, dim. N . Here adc(xi)(xj) = xixj ¡ qijxjxi: 7 An important part of the classi¯cation of all ¯nite-dimensional Hopf algebras over C is the following. Problem I. Classify all ¯nite-dimensional pointed Hopf algebras. Approach group-by-group. For a given ¯nite group ¡ , classify all ¯n.-dim. pointed Hopf algebras H such that G(H) ' ¡ . ² If ¡ is abelian and the prime divisors of ¡ are > 5, then the classi¯cation is known. N. A. and H.-J. Schneider, On the classi¯cation of ¯nite-dimensional pointed Hopf algebras, Ann. Math., to appear. The outcome is that all are variations of the Lusztig's small quantum groups. For small prime divisors, variations of small quantum supergroups appear. 8 In this talk we shall consider ¡ simple. A landmark in mathematics is the classi¯cation of ¯nite simple groups: any ¯nite simple group is isomorphic to one of ² a cyclic group Zp, of prime order p; ² an alternating group An, n ¸ 5; ² a ¯nite group of Lie type; ² a sporadic group { there are 27 of them (including the Tits group); the most prominent is the Monster. 9 II. Main results. We shall say that a ¯nite group ¡ collapses if for any ¯n.-dim. pointed Hopf algebra H, with G(H) ' ¡ , then H ' C¡ . Theorem I. If ¡ is either ² the alternating group An, n ¸ 5; ² or else a sporadic simple group, di®erent from the Fischer group F i22, the Baby Monster B, or the Monster M; then ¡ collapses. 10 Comments. The groups F i22, B and M do not admit so far any ¯n.-dim. pointed Hopf algebra (except the group algebra) but the com- putations are very hard. We are working in the same problem for ¯nite groups of Lie type. The proof lies on reductions to questions on conjugacy classes; then these questions are checked (for sporadic groups) using GAP. In order to state our other main results, and also to explain these reductions, we need to present the notion of Nichols algebra. 11 Nichols algebras. Suppose that the group ¡ acts linearly on a vector space V , hence on the free Lie algebra L(V ) by Lie algebra automorphisms. Since U(L(V )) ' T (V ), we have the Hopf algebra T (V )#C¡ . Variation. What else is needed to have Hopf algebra T (V )#C¡ = T (V )­C¡ as vector space, but with semi-direct product comultiplication and semi-direct product multiplication? 12 De¯nition. A Yetter-Drinfeld module over ¡ is a vector space V provided with ² a linear action of ¡ , ² a ¡ -grading V = ©g2¡ Vg, such that h ¢ Vg = Vhgh¡1, for all g; h 2 ¡ . Then T (V )#C¡ = T (V )­C¡ as vector space, but with semi- direct product comultiplication and semi-direct product mul- tiplication is a Hopf algebra. Here ¢(v) = v­1 + g­v, g 2 ¡ , v 2 Vg. 13 If V is a Yetter-Drinfeld module over ¡ , then H = T (V )#C¡ is pointed with G(H) ' ¡ . n Fact. There exists a homogeneous ideal J = ©n¸2J of T (V ) such that ² the quotient T (V )#C¡=J #C¡ is a Hopf algebra, ² J is maximal with respect to this property. The Nichols algebra is B(V ) = T (V )=J . If ¡ is ¯nite and dim B(V ) < 1, then T (V )#C¡=J #C¡ ' B(V )#C¡ is a ¯n.-dim. pointed Hopf algebra over ¡ . 14 Conversely, if H is a ¯n.-dim. pointed Hopf algebra over a ¯nite group ¡ , then there exists a Yetter-Drinfeld module V canonically attached to H s. t. dim B(V ) < 1. Problem II. For a given ¯nite group ¡ , classify all Yetter- Drinfeld modules V s. t. dim B(V ) < 1. Since any Yetter-Drinfeld module is semisimple, the question splits into two cases: (i) V irreducible, (ii) V direct sum of (at least 2) irreducibles. 15 It turns out that irreducible Yetter-Drinfeld modules are param- eterized by pairs (O; ½), where ² O a conjugacy class of G, ² ½ an irreducible repr. of the centralizer CG(σ) of σ 2 O ¯xed. If M(O; ½) denotes the irreducible Yetter-Drinfeld module cor- responding to a pair (O; ½) and V is the vector space a®ording the representation ½, then M(O; ½) ' IndG ½ with the grading CG(σ) given by the identi¯cation IndG ½ = CG ­ V ' CO ­ V . CG(σ) CG(σ) C 16 The Nichols algebra of M(O; ½) is denoted B(O; ½). Problem II bis. For a given ¯nite group ¡ , ¯nd all (O; ½) s. t. dim B(O; ½) < 1. Warning. Nichols algebras are very di±cult to compute; it is not known if the ideal of relations is ¯nitely-generated or not; in all known cases it is, but the degrees of the de¯ning relations can be arbitrarily high. 17 n n n Theorem II. Let m ¸ 5. Let σ 2 Sm be of type (1 1; 2 2; : : : ; m m), \ O the conjugacy class of σ; let ½ 2 CSm(σ). If dim B(O; ½) < 1, then the type of σ and ½ are in the following list: n ² (1 1; 2), ½1 = sgn or ², ½2 = sgn. ¡! ² (2; 3) in S5, ½2 = sgn, ½3 = Â0. 3 ¡! ¡! ² (2 ) in S6, ½2 = Â1 ­ ² or Â1 ­ sgn. 4 Actually, the orbit of type (1 ; 2) in S6 is isomorphic to that of 3 type (2 ), because of the outer automorphism of S6. The Nichols algebras corresponding to types (1n1; 2), and these representations, were considered by Fomin and Kirillov in relation with the quantum cohomology of the flag variety. They can not be treated by our methods. 18 Example: O = transpositions in G = Sn, s = (12), ½ = sgn n rk Relations dim B(V ) top 3 3 5 relations in degree 2 12 = 3.22 4 = 22 4 6 16 relations in degree 2 576 12 5 10 45 relations in degree 2 8294400 40 S3, S4: A. Milinski & H. Schneider, Contemp. Math. 267 (2000), 215{236. S. Fomin & K. Kirillov, Progr. Math. 172, Birkhauser, (1999), pp. 146{182. S5: [FK], plus web page of M. Gra~na. http://mate.dm.uba.ar/~matiasg/ Sn, n ¸ 6: open! 19 It is convenient to restate Problem II in terms of racks and co- cycles. For this, recall ¯rst that a braided vector space is a pair (V; c), where V is a vector space and c 2 GL(V ­ V ) is a solution of the braid equation: (c ­ id)(id ­c)(c ­ id) = (id ­c)(c ­ id)(id ­c): Any super vector space V = V0 © V1 is braided: c(v ­ w) = (¡1)jvjjwjw ­ v; for v; w 2 V . Any Yetter-Drinfeld module V is naturally a braided vector space: c(v ­ w) = g ¢ w ­ v; v 2 Vg; (g 2 G); w 2 V: 20 If V is a Yetter-Drinfeld module, then the Nichols algebra B(V ) depends only on the braiding c. This leads us to the consideration of a class of braided vector spaces where the study of the corresponding Nichols algebras is performed in a uni¯ed way.
Recommended publications
  • Braided Hopf Algebras of Triangular Type

    Braided Hopf Algebras of Triangular Type

    Stefan Ufer Braided Hopf algebras of triangular type RR RR = RR RR Braided Hopf algebras of triangular type Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig-Maximilians-Universit¨atM¨unchen eingereicht von Stefan Ufer im Mai 2004 Erster Gutachter: Prof. Dr. H.-J. Schneider, LMU M¨unchen Zweiter Gutachter: Priv. Doz. Dr. P. Schauenburg, LMU M¨unchen Dritter Gutachter: Prof. Ch. Kassel, Universit´eLouis Pasteur Strasbourg Tag der m¨undlichen Pr¨ufung:Freitag, 16. Juli 2004 Contents Contents 1 Introduction 3 1 Basic definitions 9 1.1 Lie algebras . 9 1.1.1 Definition and the universal enveloping algebra . 9 1.1.2 Root systems and Dynkin diagrams . 11 1.1.3 The classification of semi-simple Lie algebras . 14 1.2 Coalgebras, bialgebras and Hopf algebras . 15 1.2.1 Coalgebras . 15 1.2.2 Bialgebras and Hopf algebras . 17 1.2.3 Deformed enveloping algebras . 18 1.3 Yetter-Drinfeld modules and braidings . 21 1.3.1 Yetter-Drinfeld modules . 21 1.3.2 Braidings . 22 1.3.3 The braid group . 27 1.4 Braided Hopf algebras . 28 1.4.1 Definition and examples . 28 1.4.2 Radford biproducts and Hopf algebras with a projection 32 1.4.3 Braided Hopf algebras of triangular type . 34 2 Lyndon words and PBW bases 37 2.1 Lyndon words and braided commutators . 38 2.2 The PBW theorem . 40 2.3 Proof of the PBW theorem . 43 2.3.1 Braided commutators . 43 2.3.2 The comultiplication . 45 2.3.3 The PBW basis .
  • Pointed Hopf Algebras Over the Sporadic Simple Groups ✩ ∗ N

    Pointed Hopf Algebras Over the Sporadic Simple Groups ✩ ∗ N

    Journal of Algebra 325 (2011) 305–320 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Pointed Hopf algebras over the sporadic simple groups ✩ ∗ N. Andruskiewitsch a,F.Fantinoa,M.Grañab, L. Vendramin b,c, a Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM – CONICET, Medina Allende s/n (5000), Ciudad Universitaria, Córdoba, Argentina b Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pab. I, Ciudad Universitaria (1428), Buenos Aires, Argentina c Instituto de Ciencias, Universidad de Gral. Sarmiento, J.M. Gutierrez 1150, Los Polvorines (1653), Buenos Aires, Argentina article info abstract Article history: We show that every finite-dimensional complex pointed Hopf Received 7 January 2010 algebra with group of group-likes isomorphic to a sporadic group is Available online 5 November 2010 a group algebra, except for the Fischer group Fi22, the Baby Monster Communicated by Gerhard Hiss and the Monster. For these three groups, we give a short list of irreducible Yetter–Drinfeld modules whose Nichols algebra is not MSC: 16W30 known to be finite-dimensional. 17B37 © 2010 Elsevier Inc. All rights reserved. Keywords: Pointed Hopf algebras Nichols algebras Racks Introduction This paper contributes to the classification of finite-dimensional pointed Hopf algebras over an al- gebraically closed field k of characteristic 0. There are different possible approaches to this problem; oneofthemistofixafinitegroupG and to address the classification of finite-dimensional pointed Hopf algebras H such that G(H) G. Due to the intrinsic difficulty of this problem, it is natural to consider separately different classes of finite groups. A considerable progress in the case when G is abelian was achieved in [AS2].
  • Nichols Algebras and Weyl Groupoids

    Nichols Algebras and Weyl Groupoids

    Mathematisches Forschungsinstitut Oberwolfach Report No. 47/2012 DOI: 10.4171/OWR/2012/47 Mini-Workshop: Nichols Algebras and Weyl Groupoids Organised by Nicolas Andruskiewitsch, Cordoba Michael Cuntz, Kaiserslautern Istvan Heckenberger, Marburg Sarah Witherspoon, College Station 30th September – 6th October 2012 Abstract. Nichols algebras are graded braided Hopf algebras satisfying a universal property. Many structural results of a Nichols algebra can be ob- tained by studying its Weyl groupoid and its homology. In the mini-workshop, introductions to and recent developments on these structures were presented and open problems were discussed. Mathematics Subject Classification (2000): 17B37, 16S40, 16TXX, 81R50, 58B32, 16E40, 33D80, 20G42, 20F55, 20L05. Introduction by the Organisers Since its introduction in 1998, the Lifting Method grew to be one of the most powerful and fruitful methods to study Hopf algebras. Over the years, the method showed strong relationship with other areas of mathematics such as quantum groups, non-commutative differential geometry, knot theory, combinatorics of root systems and Weyl groups, Lyndon words, cohomology of flag varieties, projective representations, conformal field theory. The influence of the method was corrobo- rated by the “New Hot Paper” award of Essential Science IndicatorsSM, Thomson Reuters, May 2011, for N. Andruskiewitsch and H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. 171 (2010), 375–417. The heart of the Lifting Method is formed by the structure theory of Nichols algebras. Nichols algebras are connected graded braided Hopf algebras gener- ated by primitive elements, all primitive elements having degree one. They were 2880 Oberwolfach Report 47/2012 first studied by Nichols and were re-discovered by Lusztig, Woronowicz and oth- ers.
  • On Fomin--Kirillov Algebras for Complex Reflection Groups

    On Fomin--Kirillov Algebras for Complex Reflection Groups

    ON FOMIN–KIRILLOV ALGEBRAS FOR COMPLEX REFLECTION GROUPS ROBERT LAUGWITZ Abstract. This note is an application of classification results for finite-dimensional Nichols algebras over groups. We apply these results to generalizations of Fomin–Kirillov algebras to complex reflec- tion groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger– Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification. 1. Introduction The quadratic algebras En were introduced by S. Fomin and A. N. Kirillov as an algebraic tool to study Schubert calculus and the cohomology of flag varieties in [FK99]. This class of algebras attracted attention in the study of pointed Hopf algebras as they are quadratic covers of certain Nichols algebras Bn (and hence are braided Hopf algebras generated by primitive elements). It this framework, the Nichols algebras Bn appeared independently in [MS00, Section 5] in larger generality. The Hopf algebra structure on the bosonization of En with the group algebra was also observed in [FP00]. It was conjectured in [Maj05, 3.1] that En is infinite-dimensional for n ě 6. This question, as well as the question whether Bn is quadratic (and hence equals En), remain open (cf. [AFGV11, Introduction]).
  • From Racks to Pointed Hopf Algebras$

    From Racks to Pointed Hopf Algebras$

    ARTICLE IN PRESS Advances in Mathematics 178 (2003) 177–243 http://www.elsevier.com/locate/aim From racks to pointed Hopf algebras$ Nicola´ s Andruskiewitscha,* and Matı´ as Gran˜ ab,c a Facultad de Matema´tica, Astronomı´ayFı´sica, Universidad Nacional de Co´rdoba, CIEM-CONICET, Ciudad Universitaria, 5000 Co´rdoba, Argentina b MIT, Mathematics Department, 77 Mass. Ave., 02139 Cambridge, MA, USA c FCEyN-UBA, Pab. I-Ciudad Universitaria, 1428 Buenos Aires, Argentina Received 9 February 2002; accepted 22 May 2002 Communicated by Pavel Etingof Abstract A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arisingfrom groups.The most important class of braided vector spaces arisingfrom groups is the class of braided vector spaces ðCX; cqÞ; where X is a rack and q is a 2-cocycle on X with values in CÂ: Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containingproperly the existingones. We introduce a ‘‘Fourier transform’’ on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras. r 2003 Elsevier Inc. All rights reserved. MSC: Primary 16W30; Secondary 17B37; 52M27 Keywords: Pointed Hopf algebras; Racks; Quandles $This work was partially supported by ANPCyT, CONICET, Agencia Co´ rdoba Ciencia, Fundacio´ n Antorchas, Secyt (UBA) and Secyt (UNC).
  • Pointed Hopf Algebras

    Pointed Hopf Algebras

    New Directions in Hopf Algebras MSRI Publications Volume 43, 2002 Pointed Hopf Algebras NICOLAS´ ANDRUSKIEWITSCH AND HANS-JURGEN¨ SCHNEIDER Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf al- gebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0-coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the corad- ical A0 = |Γ, Γ the group of group-like elements of A, and gr A ' R#A0. We call the braiding of the primitive elements of R the infinitesimal braid- ing of A. If this braiding is of Cartan type [AS2], then it is often possible to determine R, to show that R is generated as an algebra by its prim- itive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that gr A ' R#|Γ. In the last chapter, as a con- crete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras A with abelian group of group-likes G(A) and in- finitesimal braiding of type An (up to some exceptional cases). In other words, we compute all the liftings of type An; this result is our main new contribution in this paper. Contents Introduction 2 1. Braided Hopf Algebras 5 2. Nichols Algebras 16 3. Types of Nichols Algebras 24 4. Nichols Algebras of Cartan Type 34 5. Classification of Pointed Hopf Algebras by the Lifting Method 37 6.
  • On Finite-Dimensional Hopf Algebras

    On Finite-Dimensional Hopf Algebras

    1 On finite-dimensional Hopf algebras 2 Dedicado a Biblioco 34 3 4 Nicolás Andruskiewitsch 5 Abstract. This is a survey on the state-of-the-art of the classification of finite-dimensional complex 6 Hopf algebras. This general question is addressed through the consideration of different classes of such 7 Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those 8 with abelian group is expected to be completed soon and there is substantial progress in the non-abelian 9 case. 10 Mathematics Subject Classification (2010). 16T05, 16T20, 17B37, 16T25, 20G42. 11 Keywords. Hopf algebras, quantum groups, Nichols algebras. 12 1. Introduction Hopf algebras were introduced in the 1950’s from three different perspectives: algebraic groups in positive characteristic, cohomology rings of Lie groups, and group objects in the category of von Neumann algebras. The study of non-commutative non-cocommutative Hopf algebras started in the 1960’s. The fundamental breakthrough is Drinfeld’s report [25]. Among many contributions and ideas, a systematic construction of solutions of the quantum Yang-Baxter equation (qYBE) was presented. Let V be a vector space. The qYBE is equivalent to the braid equation: (c ⊗ id)(id ⊗c)(c ⊗ id) = (id ⊗c)(c ⊗ id)(id ⊗c); c 2 GL(V ⊗ V ): (1.1) 13 If c satisfies (1.1), then (V; c) is called a braided vector space; this is a down-to-the-earth 14 version of a braided tensor category [54]. Drinfeld introduced the notion of quasi-triangular 15 Hopf algebra, meaning a pair (H; R) where H is a Hopf algebra and R 2 H ⊗H is invertible 16 and satisfies the approppriate conditions, so that every H-module V becomes a braided vec- 17 tor space, with c given by the action of R composed with the usual flip.
  • Factorization of Graded Traces on Nichols Algebras

    Factorization of Graded Traces on Nichols Algebras

    axioms Article Factorization of Graded Traces on Nichols Algebras Simon Lentner 1,∗ and Andreas Lochmann 2 1 Department of Mathematics, University Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany 2 Faculty of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany; [email protected] * Correspondence: [email protected] Received: 27 October 2017; Accepted: 25 November 2017; Published: 4 December 2017 Abstract: A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize. Keywords: Nichols algebra; Hilbert series; graded traces 1. Introduction A Nichols algebra B(M) over a finite group is a certain graded algebra associated with a given Yetter–Drinfel’d module over this group. For example, the Borel subalgebras uq±(g) of the Frobenius–Lusztig kernels uq(g) are finite-dimensional Nichols algebras over quotients of the root lattice of g as an abelian group.

  • Factorization of Graded Traces on Nichols Algebras

    axioms Article Factorization of Graded Traces on Nichols Algebras Simon Lentner 1,∗ and Andreas Lochmann 2 1 Department of Mathematics, University Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany 2 Faculty of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany; [email protected] * Correspondence: [email protected] Received: 27 October 2017; Accepted: 25 November 2017; Published: 4 December 2017 Abstract: A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize. Keywords: Nichols algebra; Hilbert series; graded traces 1. Introduction A Nichols algebra B(M) over a finite group is a certain graded algebra associated with a given Yetter–Drinfel’d module over this group. For example, the Borel subalgebras uq±(g) of the Frobenius–Lusztig kernels uq(g) are finite-dimensional Nichols algebras over quotients of the root lattice of g as an abelian group.
  • View This Volume's Front and Back Matter

    View This Volume's Front and Back Matter

    Mathematical Surveys and Monographs Volume 247 Hopf Algebras and Root Systems István Heckenberger Hans-Jürgen Schneider 10.1090/surv/247 Hopf Algebras and Root Systems Mathematical Surveys and Monographs Volume 247 Hopf Algebras and Root Systems István Heckenberger Hans-Jürgen Schneider EDITORIAL COMMITTEE Robert Guralnick, Chair Natasa Sesum Bryna Kra Melanie Matchett Wood 2010 Mathematics Subject Classification. Primary 16Txx. For additional information and updates on this book, visit www.ams.org/bookpages/surv-247 Library of Congress Cataloging-in-Publication Data Names: Heckenberger, Istv´an, 1969– author. | Schneider, Hans-J¨urgen, 1944– author. Title: Hopf algebras and root systems / Istv´an Heckenberger, Hans-J¨urgen Schneider. Description: Providence, Rhode Island: American Mathematical Society, [2020] | Series: Mathe- matical surveys and monographs, 0076-5376; volume 247 | Includes bibliographical references and index. Identifiers: LCCN 2019059299 | ISBN 9781470452322 (hardcover) | ISBN 9781470456801 (ebook) Subjects: LCSH: Hopf algebras. | Root systems (Algebra) | AMS: Associative rings and algebras – Hopf algebras, quantum groups and related topics. Classification: LCC QA613.8 .H43 2020 | DDC 512/.55–dc23 LC record available at https://lccn.loc.gov/2019059299 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center.
  • Arxiv:Math/0202084V3 [Math.QA] 9 Jul 2002 Ey UA N Ey (UNC)

    Arxiv:Math/0202084V3 [Math.QA] 9 Jul 2002 Ey UA N Ey (UNC)

    FROM RACKS TO POINTED HOPF ALGEBRAS NICOLAS´ ANDRUSKIEWITSCH AND MAT´IAS GRANA˜ Abstract. A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX,cq), where X is a rack and q is a 2-cocycle on X with values in C×. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group- theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras. Contents Introduction 1 1. Preliminaries 4 2. Extensions 13 3. Simple racks 21 4. Cohomology 26 5. Braided vector spaces 31 6. Nichols algebras and pointed Hopf algebras 40 References 53 Introduction 1. This paper is about braided vector spaces arising from pointed Hopf algebras, and their arXiv:math/0202084v3 [math.QA] 9 Jul 2002 Nichols algebras. Our general reference for pointed Hopf algebras is [AS3]. We shall work over the field C of complex numbers; many results below are valid over more general fields. We denote by G∞ the group of roots of unity of C.
  • An Introduction to Nichols Algebras

    An Introduction to Nichols Algebras

    An Introduction to Nichols Algebras Nicolas´ Andruskiewitsch Abstract Nichols algebras, Hopf algebras in braided categories with distinguished properties, were discovered several times. They appeared for the first time in the thesis of W. Nichols [72], aimed to construct new examples of Hopf algebras. In this same paper, the small quantum group uq(sl3), with q a primitive cubic root of one, was introduced. Independently they arose in the paper [84] by Woronowicz as the invariant part of his non-commutative differential calculus. Later there were two + unrelated attempts to characterize abstractly the positive part Uq (g) of the quan- tized enveloping algebra of a simple finite-dimensional Lie algebra g at a generic + parameter q. First, Lusztig showed in [64] that Uq (g) can be defined through the + radical of a suitable invariant bilinear form. Second, Rosso interpreted Uq (g) in [74, 75] via quantum shuffles. This two viewpoints were conciliated later, as alter- native definitions of the same notion of Nichols algebra. Other early appearances of Nichols algebras are in [65, 77]. As observed in [17, 18], Nichols algebras are basic invariants of pointed Hopf algebras, their study being crucial in the classification program of Hopf algebras; see also [10]. More recently, they are the subject of an intriguing proposal in Conformal Field Theory [79]. This is an introduction from scratch to the notion of Nichols algebra. I was invited to give a mini-course of two lessons, 90 minutes each, at the Geometric, Algebraic and Topological Methods for Quantum Field Theory, Villa de Leyva, Colombia, in July 2015.