Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

On the Guionnet-Jones-Shlyakhtenko construction for graphs

Using an analogue of the Guionnet-Jones-Shlyakhtenko construction for graphs we show that their construction applied to any subfactor planar algebra of finite depth yields an inclusion of interpolated free group factors with finite parameter, thereby giving another proof of their universality for finite depth planar algebras.     

Tracial algebras and an embedding theorem

We prove that every positive trace on a countably generated ∗-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial ∗-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial ∗-algebras, which as ∗-algebras embed into a matrix-ring over a commutative algebra.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Temperley-Lieb planar algebra modules arising from the ADE planar algebras

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley-Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley-Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A_n (n?3), D_n (n?4), E₆, E₇, and E₈.     

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and ^∗〈A^∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ^∗〈A^∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(^∗〈A^∗A〉) of ^∗〈A^∗A〉 are obtained, and a characterization of Zt(^∗〈A^∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(^∗〈A^∗A〉) are also answered.     

对偶的Hom-型拟Hopf代数的Monoidal范畴解释（英文）

In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain a 2-Hom-associative algebra. On the other hand, we introduce the dual Hom-quasi-Hopf algebra and show that any dual Homquasi-Hopf algebras can be viewed as a 2-Hom-associative algebra.     

Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of A_p(G) algebras split strongly. Furthermore, each extension of A_p(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GL_n(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of A_p(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of A_p(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of B_p(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between A_p(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.     

A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak^∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401-2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak^∗-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak^∗ Morita equivalence bimodule. We also develop the theory of the W^∗-dilation, which connects the non-selfadjoint dual operator algebra with the W^∗-algebraic framework. In the case of weak^∗ Morita equivalence, this W^∗-dilation is a W^∗-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W^∗-dilation is a key part of the proof of our main theorem.     

Representations of group planar algebras

We explicitly find out the irreducible representations of the planar algebra corresponding to the subfactor arising from the action of a finite group. We also answer the question posed by Vaughan Jones on the radius of convergence of the dimension of a representation in the affirmative for the case of group planar algebras.     

Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras. Received December 20, 1996; in final form September 15, 1997     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

On density of transitive operator algebras

In this paper we study the transitive algebra question by considering the invariant subspace problem relative to von Neumann algebras. We prove that the algebra (not necessarily ∗) generated by a pair of sums of two unitary generators of L(F_∞) and its commutant is strong-operator dense in B(H). The relations between the transitive algebra question and the invariant subspace problem relative to some von Neumann algebras are discussed.     

Lie isomorphisms of reflexive algebras

A Lie isomorphism ? between algebras is called trivial if ?=ψ+τ, where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.     

Cocommutative Hopf Algebras of Permutations and Trees

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra. Aguiar supported in part by NSF grant DMS-0302423. Sottile supported in part by NSF CAREER grant DMS-0134860, the Clay Mathematics Institute, and MSRI.