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39
Pointed Hopf algebras
 In “New directions in Hopf algebras”, MSRI series Cambridge Univ
, 2002
"... Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf alg ..."
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Cited by 53 (4 self)
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Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the coradical A0 = Γ, Γ the group of grouplike elements of A, and gr A ≃ R#A0. We call the braiding of the primitive elements of R the infinitesimal braiding 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 primitive 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 concrete illustration of the method, we describe explicitly all finitedimensional pointed Hopf algebras A with abelian group of grouplikes G(A) and infinitesimal 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
On the classification of finitedimensional pointed Hopf algebras, preprint
 2005. PARAMETERIZED BY A PAIR OF CHARACTERS 33
"... Abstract. We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to ..."
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Cited by 45 (1 self)
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Abstract. We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.
From racks to pointed Hopf algebras
 Adv. Math
"... Abstract. A fundamental step in the classification of finitedimensional complex pointed Hopf algebras is the determination of all finitedimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of b ..."
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Cited by 44 (7 self)
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Abstract. A fundamental step in the classification of finitedimensional complex pointed Hopf algebras is the determination of all finitedimensional 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, c q), where X is a rack and q is a 2cocycle 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 grouptheoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks contaninig properly the existing ones. We introduce a “Fourier transform ” on racks
On pointed Hopf algebras associated to some conjugacy classes
 in Sn, Proc. Amer. Math. Soc
"... Abstract. We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ Sn is infinitedimensional, if either the order of π is odd, or π is a product of disjoint cycles of odd order except for exactly two transpositions. ..."
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Cited by 22 (13 self)
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Abstract. We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ Sn is infinitedimensional, if either the order of π is odd, or π is a product of disjoint cycles of odd order except for exactly two transpositions.
Classification of arithmetic root systems
, 2006
"... Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the constr ..."
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Cited by 18 (3 self)
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Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.
THE NICHOLS ALGEBRA OF A SEMISIMPLE YETTERDRINFELD MODULE
, 2008
"... We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s ..."
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Cited by 16 (8 self)
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We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s automorphisms of quantized KacMoody algebras to the nilpotent part. As a direct application we complete the classifications of finitedimensional pointed Hopf algebras over S3, and of finitedimensional Nichols algebras over S4. This theory has led to surprising new results in the classification of finitedimensional pointed Hopf algebras with a nonabelian group of grouplike elements.
Pointed Hopf algebras of dimension 32
 Commun. Algebra
"... We give a complete classification of the 32dimensional pointed Hopf algebras over an algebraically closed field k with chark ̸ = 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterex ..."
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Cited by 12 (0 self)
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We give a complete classification of the 32dimensional pointed Hopf algebras over an algebraically closed field k with chark ̸ = 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails. 1.
PBW and duality theorems for quantum groups and quantum current algebras
"... Abstract. We give proofs of the PBW and duality theorems for the quantum KacMoody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebr ..."
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Cited by 11 (3 self)
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Abstract. We give proofs of the PBW and duality theorems for the quantum KacMoody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the A (1) 1 case. 1. Outline of results 1.1. Quantum KacMoody algebras. Let A = (aij)1≤i,j≤n be a symmetrizable Cartan matrix. Let (di)1≤i≤n be the coprime positive integers such that the matrix (diaij)1≤i,j≤n is symmetric. Let r be the rank of A; we assume that the matrix (aij)n−r+1≤i,j≤n is nondegenerate. Let g be the KacMoody Lie algebra associated with A; let n+ be its positive pronilpotent subalgebra and (ēi)1≤i≤n be the generators of n+ corresponding to the simple roots of g. Let C[[�]] be the formal series ring in �. Let U�n+ be the quotient of the free algebra with n generators C[[�]]〈ei, i = 1,..., n 〉 by the twosided ideal generated by the quantum Serre relations
Lifting of Nichols algebras of type B2
, 2001
"... We compute liftings of the Nichols algebra of a YetterDrinfeld module of Cartan type B2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n ̸ = 5. As well, we compute the liftings of a Nichols algebra of Cartan type A2 if the di ..."
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Cited by 10 (1 self)
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We compute liftings of the Nichols algebra of a YetterDrinfeld module of Cartan type B2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n ̸ = 5. As well, we compute the liftings of a Nichols algebra of Cartan type A2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasiisomorphic. The Appendix (with I. Rutherford) contains a generalization of the quantum binomial formula. This formula was used in the computation of liftings of type B2 but is also of interest independent of these results. 1 Introduction and
Finite quantum groups over abelian groups of prime exponent
"... Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebra ..."
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Cited by 10 (2 self)
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Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit