Infinite Dimensional Hopf Algebras Research Articles

Hopfological Algebra for Infinite Dimensional Hopf Algebras

We consider “Hopfological” techniques as in Khovanov, M., J. Knot Theory Ramificat 25(3), 26 (2016) but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, H = k[ℤ]#k[x]/x2 is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the “Homology functor” (we prove it is homological) for any co-Frobenius algebra, with coefficients in H-comodules, that recover usual homology of a complex when H = k[ℤ]#k[x]/x2. Another easy example of co-Frobenius Hopf algebra gives the category of “mixed complexes” and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its K0 is a ring, and we prove a “last part of a localization exact sequence” for K0 that allows us to compute -or describe- K0 of some family of examples, giving light of what kind of rings can be categorified using this techniques.

Connected (graded) Hopf algebras

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

Homological integral of Hopf algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke’s theorem for infinite dimensional Hopf algebras. The generalization of Maschke’s theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

The symplectic geometry of the Gel'fand--Cetlin--Molev basis for representations of $Sp(2n,\C)$

Gelfand and Cetlin constructed in the 1950s a canonical basis for a finite-dimensional represen- tation V (�) of U(n, C) by successive decompositions of the representation by a chain of subgroups (4, 5). Guillemin and Sternberg constructed in the 1980s the Gelfand-Cetlin integrable system on the coadjoint or- bits of U(n, C), which is the symplectic geometric version, via geometric quantization, of the Gelfand- Cetlin construction. (Much the same construction works for representations of SO(n, R).) A. Molev (11) in 1999 found a Gelfand-Cetlin-type basis for representations of the symplectic group, using essentially new ideas. An important new role is played by the Yangian Y (2), an infinite-dimensional Hopf algebra, and a subalgebra of Y (2) called the twisted Yangian Y − (2). In this paper we use deformation theory to give the analogous symplectic-geometric results for the case of U(n, H), i.e. we construct a completely integrable system on the coadjoint orbits of U(n, H). We call this the Gelfand-Cetlin-Molev integrable system.

Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group H L 2 ( A ) associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac–Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of H L 2 ( A ) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.

Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product

We prove that the category of Hopf bimodules over any Hopf algebra has enough injectives, which enables us to extend some results on the unification of Hopf bimodule cohomologies of [R. Taillefer, PhD thesis, 2001; arXiv preprint math.QA/0005019] to the infinite dimensional case. We also prove that the cup-product defined on these cohomologies is graded-commutative. Unlike the algebra case (see [S. Schwede, J. Reine Angew. Math. 498 (1998) 153–172]), these methods do not give a non-trivial Gerstenhaber algebra structure on the cohomology we consider. We also comment that the other approach to finding such a structure that we know of (see [M. Farinati, A. Solotar, arXiv preprint math.KT/0207243]) also gives a trivial Gerstenhaber algebra structure.

Skew derivations of finite-dimensional algebras and actions of the double of the Taft Hopf algebra

g e G(H), and A(x) =xRl+gRx, s(x) = 0. As an application, we show that actions of uq{sh) on A are determined by the action of a Borel subalgebra. This result gives a quantum analog of classical work of Jacobson on the Witt algebra in characteristic p # 0. It also gives an analog for q a root of 1 of [MSm] on actions of Uq{sl2) in the generic case. We then determine the actions of the Drinfel'd double D{H) on A, and as a consequence, those actions of H on A which give Yetter-Drinfel'd structures, answering a question from [CFM]. More specifically, in Section 2 we study the actions of H. First note that H is the (Hopf) homomorphic image of the infinite-dimensional Hopf algebra

Double Quantum Groups

The construction of the Drinfeld double D(H) of a finite dimensional Hopf algebra H was one of the first examples of a quasitriangular Hopf algebra whose category of modules MD(H) is braided. The braided category of Yetter–Drinfeld modules YDHH is the analogue for infinite dimensional Hopf algebras. It uses a strong dependence between the H-module and the H-comodule structures. We generalize this construction to the category MCA(ψ) of entwined modules, that is, A-modules and C-comodules over Hopf algebras A and C where the structures are only related by an entwining map ψ: C⊗A→A⊗C. We show that MCA(ψ) is braided iff there is an r-map r:C⊗C→A⊗A satisfying suitable axioms that generalize the axioms of an R-matrix. For finite dimensional C there is a quasitriangular Hopf algebra structure on Hom(C,A), the quantum group double, generalizing the construction of the Drinfeld double.

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( $$\mathfrak{g}$$ ) containing u( $$\mathfrak{g}$$ ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G r ) u( $$\mathfrak{g}$$ ) have also been constructed, and the results are stated in this setting.

Good filtrations for representations of Lie algebras of Cartan type

In this paper we define induced and “Weyl” modules for an infinite-dimensional Hopf algebra D( G) u( g ). This algebra is important because its representation theory encompasses the representation theory of a given Lie algebra of Cartan type. A cohomological criterion is provided which states precisely when a D( G) u( g )-module admits a filtration with sections of induced modules or “Weyl” modules.

The coinduced functor for infinite dimensional Hopf algebras

Let H be a Hopf algebra, and A an H-comodule algebra. It is well known that the functor sending a relative ( A, H)-module to its module of coinvariants has a left adjoint, the induction functor. If H is a cosemisimple Hopf algebra over a field k, then we show that the coinvariant functor also has a right adjoint. Applications of this coinduction functor are given.

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

The antipode of a finite-dimensional Hopf algebra over a field has finite order

The purpose of this note is to indicate a proof of the fact that the order of the antipode of a finite-dimensional Hopf algebra over a field has finite order. It has been shown [2], [5] that the order of the antipode of an infinite-dimensional Hopf algebra is not necessarily finite; and the order of the antipode is finite in the finite-dimensional case if the Hopf algebra is unimodular [1] or pointed and the ground field has prime characteristic [6] . Using the bilinear form introduced and studied in [3] we prove that the order is finite for any finite-dimensional Hopf algebra over a field. The bilinear form, integrals, grouplike elements, one-dimensional ideals, and the antipode are all related in rather intriguing ways. The proof of the finite order theorem is based on an explicit formula describing the fourth power of the antipode, as suggested by Theorem 5.5 of [1]. Full details will appear elsewhere [4].

On antipodes in pointed Hopf algebras

If S is the antipode of a Hopf algebra H, the order of S is defined to be the smallest positive integer n such that S n = I (in case such integers exist) or ∞ (if no such integers exist). Although in most familiar examples of Hopf algebras the antipode has order 1 or 2, examples are known of infinite dimensional Hopf algebras in which the antipode has infinite order or arbitrary even order [1, 4, 6] and also of finite dimensional Hopf algebras in which the antipode has arbitrary even order [3, 5]. Some sufficient conditions for the antipode to have order ⩽4 are known [2, 4], but the following questions remain open: Does the antipode of a finite dimensional Hopf algebra necessarily have finite order? If the antipode S of a Hopf algebra H has finite order is that order bounded by some function of dim H? In this paper, by constructing a certain basis for an arbitrary pointed coalgebra and studying the action of the antipode on the elements of such a basis for a pointed Hopf algebra, we obtain affirmative answers to the second question in case H is pointed and to the first question in case H is pointed over a field of prime characteristic. We use freely the definitions, notation, and results of [4].

Characters of Hopf algebras

In the development of the character theory of finite groups, one of the key facts used in proving the orthogonality relations is the invariance property of cg E kG. (See, e.g., the proofs of Lemma 5.1.3 in [2] and of Eq. (31.8) in [l].) Since finite-dimensional Hopf algebras [5] and the dual algebras of certain infinite-dimensional Hopf algebras [7] contain elements with properties analogous to those of Cg, it is reasonable to expect that an orthogonality relation will hold for characters of such Hopf algebras. In Section 2 of this paper we prove such an orthogonality relation. In order to include a character theory for infinite dimensional Hopf algebras, we must consider characters as elements of the Hopf algebra which are associated with comodules over the Hopf algebra, rather than as functionals on the Hopf algebra which are associated with modules over the Hopf algebra. This point of view allows a simultaneous treatment of the characters of compact Lie groups and completely reducible affine algebraic groups (taking as the Hopf algebra the algebra of representative functions), of the characters of semisimple Lie algebras over an algebraically closed field of characteristic 0 (taking U(L)” as the Hopf algebra; the characters studied here differ from those of Expose 18 of [6] by a scalar multiple), and of the characters of finite groups (taking as the Hopf algebra the dual Hopf algebra to the group algebra). We then use our results to prove that the dimension of a simple comodule of an involutory cosemisimple Hopf algebra over an algebraically closed field is not divisible by the characteristic of the field. In Section 3 we prove that the antipode y of a cosemisimple Hopf algebra is bijective, and that y2 maps each simple subcoalgebra onto itself. In Section 4 we give a generalization of Maschke’s Theorem: If the characteristic of the field does not divide the dimension of a finite dimensional involutory Hopf algebra, then the Hopf algebra and its dual are semisimple; if the characteristic does divide the