A Hopf algebra having a separable Galois extension is finite dimensional

It is shown that a Hopf algebra over a field admitting a Galois extension separable over its subalgebra of coinvariants is of finite dimension. This answers in the affirmative a question posed by Beattie et al. in [Proc. Amer. Math. Soc. 128, No. 11 (2000), 3201-3203]. It is also proven that this result holds true if has bijective antipode and the extension is Frobenius.

     

A co-Frobenius Hopf algebra with a separable Galois extension is finite

If is a co-Frobenius Hopf algebra over a field, having a Galois -object which is separable over , its ring of coinvariants, then is finite dimensional.

     

The extension of a quantized Borcherds superalgebra by a Hopf algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b_(ik), c_(ik), g_(ik), h_(ik)(i ∈I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra U_q(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A =(aij)i,j∈I. Therefore, we define an extended Hopf superalgebra HU_q(G). We also discuss the basis and the grouplike elements of HU_q(G).     

The structure of bialgebra with a Hopf module

Let H be a Hopf algebra, B a bialgebra, and (B, ?, ρ) a right H-Hopf module. Assume that (B, ρ) is a right H-comodule algebra, (B, ?) is a right H-module coalgebra, and let A = B ^{co H} = {a ∈ B | ρ(a) = a ? 1}. Then we prove that B has a factorization of A□ _ρ ^? (the underlying space is A ? H) as a bialgebra, which generalizes Radford’s factorization of bialgebras with projection [12].     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

A Hopf algebra for counting cycles

Cycles, also known as self-avoiding polygons, elementary circuits or simple cycles, are closed walks which are not allowed to visit any vertex more than once. We present an exact formula for enumerating such cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from a Hopf algebra, which we construct explicitly, and which provides further means of counting cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate cycles.     

The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

Computing the connes spectrum of a Hopf algebra

LetH be a finite-dimensional Hopf algebra over the fieldK and letA be anH-module algebra. In a previous paper, we defined the Connes spectrum CS(A, H) for the action ofH onA to be a certain subset of the set Irr(H) of irreducible representations ofH. In this paper, we compute a number of examples; specifically, we consider certain inner and outer actions and we take a closer look at the cocommutative situation. We discover that the information encoded in the Connes spectrum is rather subtle and elusive. Research supported in part by NSF Grant DMS-8900405.     

Transmutation theory of a coquasitriangular weak Hopf algebra

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.     

Free Rota-Baxter systems and a Hopf algebra structure

In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gröbner-Shirshov bases method and then we obtain a Hopf algebra structure on a free Rota-Baxter system.     

An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra

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 , provides a noetherian prime counterexample.

     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

On the central charge of a factorizable Hopf algebra

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel’d element differs from the value that it takes on the Drinfel’d element itself by at most a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ by at most a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.     

The quantum double of a factorizable weak Hopf algebra

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k. We first construct a weak Hopf algebra [Δ(1)(H?H)Δ(1)]^R, which is based on the subalgebra of the tensor product algebra H?H. Next we verify that if H is factorizable, then the Drinfeld’s quantum double of H is isomorphic to [Δ(1)(H?H)Δ(1)]^R.     

Poisson Hopf algebra related to a twisted quantum group

A Poisson algebra ?[G] considered as a Poisson version of the twisted quantized coordinate ring ?_q,p[G], constructed by Hodges et al. [11 Hodges, T. J., Levasseur, T., Toro, M. (1997). Algebraic structure of multi-parameter quantum groups. Adv. Math. 126:52–92.[Crossref], [Web of Science ®] , [Google Scholar]], is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of ?[G] are characterized. Further it is shown that ?[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ?[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ?[G] onto the Poisson primitive ideal space.