Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula for all and . In this paper we consider multiplier Hopf algebras B (over ) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double D ^π where the coproduct and the product depend on the action π. We also treat the ^*-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair . On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra. Presented by K. Goodearl.     

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ? A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5 Delvaux , L. ( 2004 ). Twisted tensor coproduct of multiplier Hopf (*)-algebras . J. Algebra 274 : 751 – 771 . [Google Scholar]]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ? _R  B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ? _R  B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.     

A Generalized Drinfeld Quantum Double Construction Based on Weak Hopf Algebras

Let B and H be weak Hopf algebras with bijective antipodes S _B and S _H , respectively. Based on a compatible weak Hopf dual pairing (B, H, σ), we construct a generalized Drinfeld quantum double 𝔻(B, H) which is a weak T-coalgebra over a twisted semi-direct square of groups. In particular, when B and H are finite dimensional and the above pairing map σ is nondegenerate, 𝔻(B, H) admits a nontrivial quasitriangular structure. Some explicit examples are given as an application of our theory.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

When is a Cleft Extension H-Azumaya?

We show which H ^op -cleft extensions of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).Presented by A. Verschoren     

Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).     

The Representations of Cyclotomic BMW Algebras,II

In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras over an integral domain. As a by-product, we determine explicitly when is semisimple over a field. This generalizes our previous result on Birman-Murakami-Wenzl algebras in Rui and Si (J Reine Angew Math 631:153–180, 2009).     

Universal Coefficient Theorem in Triangulated Categories

We consider a homology theory on a triangulated category with values in an abelian category. If the functor h reflects isomorphisms, is full and is such that for any object x in there is an object X in with an isomorphism between h(X) and x, we prove that is a hereditary abelian category, all idempotents in split and the kernel of h is a square zero ideal which as a bifunctor on is isomorphic to. The second author is a researcher from CONICET, Argentina.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

The quasitriangular structures for a class of<Emphasis Type="Italic">T</Emphasis>-smash product Hopf algebras

We study the quasitriangular structures ofT-smash product Hopf algebrasB _⋈ H which are constructed by Caenepeel, Ion, Militaru and Zhu. The necessary and sufficient conditions for a class ofT-smash product Hopf algebras to be quasitriangular Hopf algebras are given. As applications of our results, some corollaries and examples are given as well. This work was partially supported by the NSF grant of Henan Province, P. R. China.     

The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Given a projective scheme X over a field k, an automorphism σ: X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.     

Orderings and *-Orderings on Cocommutative Hopf Algebras

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings G and smash products U(L) G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for and L finite–dimensional, we characterize the orderability of U(L) G; for , we give a necessary and a sufficient condition for ∗-orderability of G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L) G to the problem of characterizing the ∗-orderability of G. The latter remains open. The research of the first author was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia under grant P1-0222 (Algebraic methods in operator theory). The research of the second and third author was supported by the Natural Sciences and Engineering Research Council of Canada.     

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ¹ over a Hopf algebra A which are quotients of the augmentation ideal A ⁺ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation Open image in new window provides a bicovariant differential graded algebra on A. We introduce Λ _{m a x} providing the maximal prolongation, while the canonical braided-exterior algebra Λ _min = B _?(Λ¹) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ? by super-braided Fourier transform on B _?(Λ¹) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S ₃] with its 3D calculus and obeying the q-Hecke relation ?² = 1 + (q ? q ^?1)? in middle degree on k _q [S L ₂] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A ⁺ → Λ¹.     

Strongly approximative similarity of operators

Two operators A, B ∈ B（H） are said to be strongly approximatively similar, denoted by A -sas B, if （i） given ε 〉 0, there exist Ki ∈ B（H） compact with ｜｜Ki｜｜ 〈ε（i = 1,2） such that A＋K1 and B ＋ K2 are similar; （ii） σ0（A） = σ0（B） and dim H（λ; A） = dim H（λ; B） for each λ ∈ σ0（A）. In this paper, we prove the following result. Let S,T ∈ B（H） be quasitriangular satisfying： （i） σ（T） = σ（S） = σw（S） is connected and σe（S） = σlre（S）; （ii） ρs-F（S） ∩ σ（S） consists of at most finite components and each component Ω satisfies that Ω = int Ω, where int Ω is the interior of Ω. Then, S -sas T if and only if S and T are essentially similar.