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)