From racks to pointed Hopf algebras

By N. Andruskiewitsch, M. Graña. 2002

392 citations recorded by the source corpus.

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 ( C X,c q ) , 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00fb7d966821a850a2eed5e604ebe9189e65527e

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Citations in the corpus

Cohomology of Categorical Self-Distributivity (2006, neutral): “Generalizations have been discovered [1, 8, 9].”
Extensions of Quandles and Cocycle Knot Invariants (2001, methodology): “The cohomology theory has found applications to the classification of Nichols algebras [2].”
Homology Groups of Symmetric Quandles and Cocycle Invariants of Links and Surface-links (2009, background): “For details on quandle homologies and invariants, refer to [1, 3, 4, 5, 10, 11, 16, 20, 21].”
The structure of medial quandles (2014, background): “See [1, 13] for an introduction to the algebraic theory of quandles, and [4, 16] for a knot-theoretical perspective.”
Extensions, matched products, and simple braces (2015, background): “Besides its relation with theoretical physics, this class of solutions is important for its connections with many branches of algebra, for instance: semigroups of I-type and Bieberbach groups [12], bijective 1-cocycles [10], radical rings [17], triply factorized groups [19], construction of semisimp”
Leibniz Algebras, Lie Racks, and Digroups (2004, neutral): “As noted in §1, the notion of digroup has recently been introduced independently three times [5, 8, 12], but one should really think of it as being already implicit in Loday’s work.”
Classification of arithmetic root systems (2006, background): “Their structure was enlightened, mainly in the case when H0 is the group algebra of a finite group, among others in [9], [5], [18], [3].”
Nichols algebras of group type with many quadratic relations (2010, background): “The following result was proved in [2] for k = C.”
A SURVEY OF QUANDLE IDEAS (2010, neutral): “Letting [0] = 0, [1] = 1, [2] = t, and [3] = t+ 1, we have the following quandle table:”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “Important advances in knot-theoretic and Hopf-algebraic applications of self-distributivity are due to the homological approach, initiated by Fenn– Rourke–Sanderson [20] and Carter–Jelsovsky–Kamada–Langford–Saito [6], and further developed by Andruskiewitsch–Graña [1].”
Cocycle knot invariants from quandle modules and generalized quandle homology (2003, methodology): “When is not necessarily abelian, the 2 -cocycles are called (constant) 2 -cocyclesin [1].”
Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks (2016, background): “them are their relation with triangular semisimple and cosemisimple Hopf algebras explained in [24, 9], and the relation with ﬁnite dimensional pointed Hopf algebras through rack theory, explained in [1] (we clarify the connection with rack theory in Section 5). These two classes of Hopf algebras ha”
Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2008, background): “rms of the conjugation. A rack is a set with a binary operation satisfying the basic properties of the conjugation in a group (see §2.3 below). Then Question 1 is equivalent to the following one, see [AG]. Question 2. For any ﬁnite rack X, for any n ∈ N, and for any nonprincipal 2-cocycle q as in pa”
On finite-dimensional Hopf algebras (2014, background): “[10] A 2-cocycle of degree n = (nk)k∈I , associated to X, is a family q = (qk)k∈I of maps qk : X ×Xk → GL(nk,C) such that”