On finite dimensional Nichols algebras of diagonal type

Abstract

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) Communicated by Efim Zelmanov.

On actions of Drinfel'd doubles on finite dimensional algebras

“…Now (3.14) follows since when a = i + k, we have a k q = (a)q! (2) follows similarly. We also have by (3.11) that X a Y b , 1 = δ a,0 = ǫ(X a Y b ) and 1, x i y j = δ 0,i = ǫ(x i y j ).…”

“…We will denote the comultiplication and counit maps of a coalgebra by ∆ and ǫ, respectively. The set of grouplike elements of a coalgebra G(C) are the nonzero elements c such that ∆(c) = c ⊗ c. For g, h ∈ G(C), the space of (g, h)-skew primitive elements P g,h (C) is the set of elements c ∈ C such that ∆(c) = g ⊗ c + c ⊗ h. The symbol S will be used for the antipode of a Hopf algebra H. Sweedler notation will also be used for the comultiplication throughout: we will write a (1) ⊗ a (2) for ∆(a). For an algebra A and a left A-module M , we will denote the action of a ∈ A on m ∈ M by a · m. For a coalgebra C and a left C-comodule M , we will signify the coaction by ρ : M → C ⊗ M , and use the modified Sweedler notation ρ(m) = m (−1) ⊗ m (0) for m ∈ M .…”

“…When H is finite-dimensional, H • = H * is a Hopf algebra with multiplication given by ∆ * and comultiplication given by m * . Thus, for p ∈ H * , p, ab = p, m(a ⊗ b) = m * (p), a ⊗ b = p (1) , a p (2) , b . Therefore, a ≻ p = p (2) , a p (1) and p ≺ a = p (1) , a p (2) .…”

“…Thus, for p ∈ H * , p, ab = p, m(a ⊗ b) = m * (p), a ⊗ b = p (1) , a p (2) , b . Therefore, a ≻ p = p (2) , a p (1) and p ≺ a = p (1) , a p (2) . Combining these two facts gives…”

“…Thus, to achieve an algebra presentation of D(H), it remains to show how elements of H move past those of H * . We will compute relations giving this "commutation" between elements of H and H * using the following consequence of (1.9): For any p ∈ H * and a ∈ H, we have in D(H): (2) .…”