We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}... more We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$ as algebras, where ${\mathfrak{g}}={\textrm{Lie}}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.
We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for f... more We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).
We generalize the definition of an exact sequence of tensor categories due to Bruguières and Nata... more We generalize the definition of an exact sequence of tensor categories due to Bruguières and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms.
A fundamental problem in the theory of Hopf algebras is the classification and construction of fi... more A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne's theorem on Tannakian categories and on Movshev's theory in an essential way. We explain Movshev's theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property.
In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started ... more In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started in [EG] and [G]. First, we completely describe the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has prime codimension. As a corollary we obtain that if p > 2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have Frobenius-Perron dimension p N , N = 1, 2, 3, 4, 5 or 7. Second, we construct new examples of finite dimensional quasi-Hopf algebras which are not twist equivalent to a Hopf algebra. For instance, to every finite dimensional simple Lie algebra g and an odd integer n, coprime to 3 if g = G 2 , we attach a quasi-Hopf algebra of dimension n dimg .
We introduce and study the new notion of an exact factorization B = A • C of a fusion category B ... more We introduce and study the new notion of an exact factorization B = A • C of a fusion category B into a product of two fusion subcategories A, C ⊆ B of B. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied in [EG]. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.
Let k be an algebraically closed field of characteristic p ≥ 0. Let G be an affine group scheme o... more Let k be an algebraically closed field of characteristic p ≥ 0. Let G be an affine group scheme over k. We classify the indecomposable exact module categories over the rigid tensor category Coh f (G) of coherent sheaves of finite dimensional k−vector spaces on G, in terms of (H, ψ)−equivariant coherent sheaves on G. We deduce from it the classification of indecomposable geometrical module categories over Rep(G). When G is finite, this yields the classification of all indecomposable exact module categories over the finite tensor category Rep(G). In particular, we obtain a classification of twists for the group algebra k[G] of a finite group scheme G. Applying this to u(g), where g is a finite dimensional p−Lie algebra over k with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.
We construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n 3 , n > 2, para... more We construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n 3 , n > 2, parametrized by primitive roots of unity q of order n 2 , with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in [3]. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By [4], our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with H [0] = (k[Z/nZ],), where n is prime and is a nontrivial associator, such that H [1] is a free left module over H [0] of rank 1 (it is always free), then H is isomorphic to A(q).
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2−cocycles J for th... more We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2−cocycles J for the function algebra O(G)), which was started in [EG1, EG2], and initiate the study of the associated one-sided twisted function algebras O(G) J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G) J is finitely generated with center O(G/H). We then use it to study the structure of O(G) J for connected nilpotent G. We show that in this case O(G) J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2−cocycles for connected nilpotent G, hence of fiber functors Rep(G) → Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G) J for arbitrary G.
In this paper we determine when Lusztig's Uq(sln)′ has all the desired properties necessary t... more In this paper we determine when Lusztig's Uq(sln)′ has all the desired properties necessary to define invariants of knots, links and 3-manifolds. Specifically, we determine when it is ribbon, unimodular and factorizable. We also compute the integrals and distinguished elements involved.
We characterize a natural class of modular categories of prime power Frobenius-Perron dimension a... more We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple
Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.Q... more Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not on...
We introduce a new notion of the core of a braided fusion category. It allows to separate the par... more We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them premodular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.
In formulating a generalized framework to study certain noncommutative algebras naturally arising... more In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}... more We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$ as algebras, where ${\mathfrak{g}}={\textrm{Lie}}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.
We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for f... more We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).
We generalize the definition of an exact sequence of tensor categories due to Bruguières and Nata... more We generalize the definition of an exact sequence of tensor categories due to Bruguières and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms.
A fundamental problem in the theory of Hopf algebras is the classification and construction of fi... more A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne's theorem on Tannakian categories and on Movshev's theory in an essential way. We explain Movshev's theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property.
In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started ... more In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started in [EG] and [G]. First, we completely describe the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has prime codimension. As a corollary we obtain that if p > 2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have Frobenius-Perron dimension p N , N = 1, 2, 3, 4, 5 or 7. Second, we construct new examples of finite dimensional quasi-Hopf algebras which are not twist equivalent to a Hopf algebra. For instance, to every finite dimensional simple Lie algebra g and an odd integer n, coprime to 3 if g = G 2 , we attach a quasi-Hopf algebra of dimension n dimg .
We introduce and study the new notion of an exact factorization B = A • C of a fusion category B ... more We introduce and study the new notion of an exact factorization B = A • C of a fusion category B into a product of two fusion subcategories A, C ⊆ B of B. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied in [EG]. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.
Let k be an algebraically closed field of characteristic p ≥ 0. Let G be an affine group scheme o... more Let k be an algebraically closed field of characteristic p ≥ 0. Let G be an affine group scheme over k. We classify the indecomposable exact module categories over the rigid tensor category Coh f (G) of coherent sheaves of finite dimensional k−vector spaces on G, in terms of (H, ψ)−equivariant coherent sheaves on G. We deduce from it the classification of indecomposable geometrical module categories over Rep(G). When G is finite, this yields the classification of all indecomposable exact module categories over the finite tensor category Rep(G). In particular, we obtain a classification of twists for the group algebra k[G] of a finite group scheme G. Applying this to u(g), where g is a finite dimensional p−Lie algebra over k with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.
We construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n 3 , n > 2, para... more We construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n 3 , n > 2, parametrized by primitive roots of unity q of order n 2 , with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in [3]. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By [4], our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with H [0] = (k[Z/nZ],), where n is prime and is a nontrivial associator, such that H [1] is a free left module over H [0] of rank 1 (it is always free), then H is isomorphic to A(q).
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2−cocycles J for th... more We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2−cocycles J for the function algebra O(G)), which was started in [EG1, EG2], and initiate the study of the associated one-sided twisted function algebras O(G) J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G) J is finitely generated with center O(G/H). We then use it to study the structure of O(G) J for connected nilpotent G. We show that in this case O(G) J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2−cocycles for connected nilpotent G, hence of fiber functors Rep(G) → Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G) J for arbitrary G.
In this paper we determine when Lusztig's Uq(sln)′ has all the desired properties necessary t... more In this paper we determine when Lusztig's Uq(sln)′ has all the desired properties necessary to define invariants of knots, links and 3-manifolds. Specifically, we determine when it is ribbon, unimodular and factorizable. We also compute the integrals and distinguished elements involved.
We characterize a natural class of modular categories of prime power Frobenius-Perron dimension a... more We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple
Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.Q... more Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not on...
We introduce a new notion of the core of a braided fusion category. It allows to separate the par... more We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them premodular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.
In formulating a generalized framework to study certain noncommutative algebras naturally arising... more In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
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