Finite Dimensional quasi-Hopf Algebra Research Articles

Categorical aspects of cointegrals on quasi-Hopf algebras

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra H, we give an explicit description of categorical cointegrals of the category MH of left H-modules in terms of cointegrals on H. Provided that H is unimodular, we also express the Frobenius structure of the ‘adjoint algebra’ in the Yetter-Drinfeld category YHHD by using an integral in H and a cointegral on H. Finally, we give a description of the twisted module trace for projective H-modules in terms of cointegrals on H.

Frobenius-Perron Dimensions of Integral $\mathbb {Z}_{+}$-rings and Applications

We introduce the notion of the Frobenius-Perron dimension of an integral \(\mathbb {Z}_{+}\)-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension p.

Module categories over finite pointed tensor categories

We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence, we classify exact module categories over some families of pointed tensor categories with cyclic group of invertible objets of order p, where p is a prime number.

Freeness of Quasi-Hopf Algebras Over Right Coideal Subalgebras

We will generalize Skryabin's Freeness Theorem [11]to quasi-Hopf algebras. We will show that for a finite dimensional quasi-Hopf algebra H and a right coideal subalgebra K ⊂ H all (H, K)-quasi Hopf bimodules are free K-modules, in particular, H is a free right and left K-module.

The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra A q ( g ) of dimension n dim g attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra g and a primitive root of unity q of order n 2 is equivalent to Lusztig's small quantum group u q ( g ) (under some conditions on n). We also give a conceptual construction of A q ( g ) using the notion of de-equivariantization of tensor categories.

A Morita Context and Galois Extensions for Quasi-Hopf Algebras

If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we show that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A H . We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.

On Quasi-Hopf Smash Products and Twisted Tensor Products of Quasialgebras

We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products.

The representation-theoretic rank of the doubles of quasi-quantum groups

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D ( H ) , within the rigid braided category of finite dimensional left D ( H ) -modules.

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG ( p ) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C , whose radical has codimension p . The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p ; that is, A with gr ( A ) in RG ( p ) , where gr ( A ) is the associated graded algebra taken with respect to the radical filtration on A . The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p . Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H ( 2 ) , H ± ( p ) , A ( q ) , or H ( 32 ) , constructed in [5,8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.

On Radically Graded Finite-Dimensional Quasi-Hopf Algebras

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional quasi-Hopf algebras started in math.QA/0310253 (p=2) and math.QA/0402159 (p>2). More specifically, we completely describe the class RG(p) for p>2. Namely, we show that if H\in RG(p) has a nontrivial associator, then the rank of H[1] over H[0] is \le 1. This yields the following classification of H\in RG(p), p>2, up to twist equivalence: (a) Duals of pointed Hopf algebras with p grouplike elements, classified in math.QA/9806074. (b) Group algebra of Z_p with associator defined by a 3-cocycle. (c) The algebras A(q), introduced in math.QA/0402159. This result implies, in particular, 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. In the second half of the paper we construct new examples of finite dimensional quasi-Hopf algebras H, which are not twist equivalent to a Hopf algebra. They are radically graded, and H/Rad(H)=C[Z_n^m], with a nontrivial associator. 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^{dim(g)}.

Quasi-Hopf algebras and representations of octonions and other quasialgebras

Modules over a quasialgebra (here, by quasialgebra we mean a left H-module algebra, where H is a quasi-Hopf algebra), as defined by Albuquerque and Majid, coincide with modules over a certain associative algebra, a quasi-Hopf smash product. As a consequence of this, we get that the category of modules over the octonions is isomorphic to the category of modules over the algebra of 8×8 real matrices. We provide a new approach to the endomorphism quasialgebra associated to a left H-module, which in the finite dimensional case yields the same results as the one of Albuquerque and Majid. We discuss possible definitions as endomorphism quasialgebras for Heisenberg doubles of a finite dimensional quasi-Hopf algebra.

Factorizable quasi-Hopf algebras—applications

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D( H) of any finite dimensional quasi-Hopf algebra H is factorizable, and we characterize D( H) when H itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double D( H) is a unimodular quasi-Hopf algebra.

Finite dimensional quasi-Hopf algebras with radical of codimension 2

It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known that these are exactly the Nichols Hopf algebras H_{2^n} of dimension 2^n, n\ge 1 (one for each value of n). The main result of this paper is that if H is a finite dimensional quasi-Hopf algebra over C with radical of codimension 2, then H is twist equivalent to a Nichols Hopf algebra H_{2^n}, n\ge 1, or to a lifting of one of the four special quasi-Hopf algebras H(2), H_+(8), H_-(8), H(32) of dimensions 2, 8, 8, and 32, defined in Section 3. As a corollary we obtain that any finite tensor category which has two invertible objects and no other simple object is equivalent to \Rep(H_{2^n}) for a unique n\ge 1, or to a deformation of the representation category of H(2), H_+(8), H_-(8), or H(32). As another corollary we prove that any nonsemisimple quasi-Hopf algebra of dimension 4 is twist equivalent to H_4.

Two-sided two-cosided Hopf modules and Doi–Hopf modules for quasi-Hopf algebras

Let H be a finite-dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra. We introduce the category of two-sided Hopf modules, and prove that it is isomorphic to a module category. We also show that two-sided Hopf modules are coalgebra over a certain comonad. We introduce Doi–Hopf modules, and show that they are comodules over a certain coring. If the underlying H-module coalgebra is finite-dimensional, then Doi–Hopf modules are modules over a certain smash products. A similar result holds for two-sided two-cosided Hopf modules.

A quasi-Hopf algebra freeness theorem

We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

Integrals for (dual) quasi-Hopf algebras. Applications

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ⩽1, and for a finite-dimensional Hopf algebra, this dimension is exactly one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite-dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.

The Quantum Double for Quasitriangular Quasi-Hopf Algebras

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

Existence of integrals for finite dimensional quasi-Hopf algebras

If A is a finite dimensional Hopf algebra and ∫ ⊆ A is the space of integrals in A, it is well known that dim( ∫ ) = 1. The proof given in [12] actually shows the existence and uniqueness of integrals in A∗ and it relies on the structure of Hopf modules over A, namely one has to prove that A∗ is a right A-Hopf module and then the result follows from the fundamental theorem for Hopf modules (see [12] for details). It is very natural to ask if the result remains true if A is not a Hopf algebra, but a quasi-Hopf algebra (this question arose in [9], where the following version of Maschke’s theorem for quasi-Hopf algebras was proved: A is semisimple if and only if e( ∫ ) 6= 0). The answer is positive for some particular quasi-Hopf algebras, for instance for Dijkgraaf-Pasquier-Roche’s quasi-Hopf algebras D(G) (where G is a finite group and ω is a normalized 3-cocycle on G) and for their generalizations D(H) introduced in [1] (where H is a finite dimensional cocommutative Hopf algebra and ω : H⊗H⊗H → k is a normalized 3-cocycle in Sweedler’s cohomology). But if one tries to generalize the proof given in [12] to quasi-Hopf algebras some problems occur, for example it is not clear which could be the appropriate definition for a Hopf module over a quasi-Hopf algebra. The existence and uniqueness of integrals for finite dimensional Hopf algebras have been reproved in [11], [8] by avoiding the use of Hopf modules. In this note we shall prove the existence of integrals for finite dimensional quasi-Hopf algebras, by generalizing the short and direct proof given by A. Van Daele in [11] for the Hopf algebra case. It seems that the method in [11] does not yield a proof for the uniqueness property.