On finite dimensional Nichols algebras of diagonal type

Bulletin of Mathematical Sciences, Dec 2017

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.

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On finite dimensional Nichols algebras of diagonal type

On finite dimensional Nichols algebras of diagonal type Nicolás Andruskiewitsch 0 1 Iván Angiono 0 1 Iván Angiono 0 1 0 FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba , Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba , Argentina 1 Communicated by Efim Zelmanov 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 220:59124, 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 work of Nicolás Andruskiewitsch and Iván Angiono was partially supported by CONICET, Secyt (UNC), the MathAmSud Project GR2HOPF. The work of Iván Angiono was partially supported by ANPCyT (Foncyt). The work of Nicolás Andruskiewitsch, respectively Iván Angiono, was partially done during a visit to the University of Hamburg, respectively the MPI (Bonn), supported by the Alexander von Humboldt Foundation. - 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. Mathematics Subject Classification 16T05 · 16T20 · 17B22 · 17B37 · 17B50 Alles Gescheidte ist schon gedacht worden, man muß nur versuchen es noch einmal zu denken. Goethe Contents Introduction What is a Nichols algebra? 1. Let k be a field, V a vector space and c ∈ GL(V ⊗ V ). The braid equation on c is (c ⊗ id)(id ⊗c)(c ⊗ id) = (id ⊗c)(c ⊗ id)(id ⊗c). (0.1) If c satisfies (0.1), then the pair (V , c) is a braided vector space. The braid equation, or the closely related quantum Yang–Baxter equation, is the key to many developments in the last 50 years in several areas in mathematics and theoretical physics. Ultimately these applications come from the representations n of the braid groups Bn on T n(V ) induced by (0.1), for n ≥ 2. Indeed, let In := {1, 2, . . . , n}, where n is a natural number. Recall that Bn is presented by generators (σ j ) j∈In−1 with relations σ j σk = σk σ j , | j − k| ≥ 2, σ j σk σ j = σk σ j σk , | j − k| = 1. (0.2) Thus n applies σ j → idV ⊗( j−1) ⊗c ⊗ idV ⊗(n− j−1) . 2. Assume that char k = 2. Let c be a symmetry, i.e. a solution of (0.1) such that c2 = id. Then n factorizes through the representation n of the symmetric group Sn given by s j := ( j j + 1) → idV ⊗( j−1) ⊗c ⊗ idV ⊗(n− j−1) . The symmetric algebra of (V , c) is the quadratic algebra Sc(V ) = T (V )/ ker(c + id) = ⊕n∈N0 Scn (V ). For instance, if c = τ is the usual transposition, then Sc(V ) = S(V ), the classical symmetric algebra; while if V = V0 ⊕ V1 is a super vector space and c is the super transposition, then Sc(V ) S(V0)⊗ (V1), the super symmetric algebra. The adequate setting for such symmetries is that of symmetric tensor categories, advocated by Mac Lane in 1963. In this context, the symmetric algebra satisfies the same universal property as in the classical definition. In particular, symmetric algebras are Hopf algebras in symmetric tensor categories. Assume that char k = 0. Then, as vector spaces, Scn(V ) T n(V )Sn = Im n T n(V )/ ker n, (0.3) where n = s∈Sn n(s): T n(V ) → T n(V ). 3. The adequate setting for braided vector spaces is that of braided tensor categories [ 56 ]; there is a natural notion of Hopf algebra in such categories. Let H be a Hopf algebra (with bijective antipode). Then H gives rise to a braided tensor category HH YD [ 39 ], and consequently is a source of examples of braided vector spaces. Namely, an object M ∈ HH YD, called a Yetter–Drinfeld module over H , is simultan (...truncated)


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Nicolás Andruskiewitsch, Iván Angiono. On finite dimensional Nichols algebras of diagonal type, Bulletin of Mathematical Sciences, 2017, pp. 1-221, DOI: 10.1007/s13373-017-0113-x