Guionnet–Jones–Shlyakhtenko subfactors associated to finite-dimensional Kac algebras

We analyse the Guionnet–Jones–Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.     

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.     

A complete set of numerical invariants for a subfactor

We show that certain numerical invariants associated naturally to a subfactor planar algebra constitute a complete family in the sense of determining the isomorphism class of the subfactor planar algebra.In the course of the proof, we show also that planar algebra isomorphisms of subfactor planar algebras can always be chosen to be ∗-preserving. This latter statement generalises the fact that ‘Hopf algebra isomorphisms of finite-dimensional Kac algebras can be chosen to be ∗-preserving’.     

Quantum automorphism groups of homogeneous graphs

Associated to a finite graph X is its quantum automorphism group G. The main problem is to compute the Poincaré series of G, meaning the series f(z)=1+c₁z+c₂z²+? whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives f for all homogeneous graphs having 8 vertices or less.     

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.     

On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

On cyclically orientable graphs

Graph G is called cyclically orientable (CO) if it admits an orientation in which every simple chordless cycle is cyclically oriented. This family of graphs was introduced by Barot et al. [Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (3) 73 (2006) 545-564]. The authors obtained several nice characterizations of CO-graphs, being motivated primarily by their applications in cluster algebras. Here we obtain several new characterizations that provide algorithms for recognizing CO-graphs and obtaining their cyclic orientations in linear time. We show that the edge maximal CO-graphs are 2-trees; that is, G=(V,E) is a 2-tree if and only if G is CO and G^′=(V,E^′) is not CO whenever E is a proper subset of E^′.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

On stable cutsets in claw-free graphs and planar graphs

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let $K_4$ and $K_{1,3}$ (claw) denote the complete (bipartite) graph on 4 and $1+3$ vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a $K_4$-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, $K_4$)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for $K_4$-free planar graphs with maximum degree five.     

A relation between certain interpolated Cuntz algebras and interpolated free group factors

We investigate von Neumann algebras generated by the real parts of generators of Toeplitz extensions of interpolated Cuntz algebras on sub-Fock spaces. We show that some of them are isomorphic to interpolated free group factors . For example, in case of the golden number the corresponding number is .

     

On 3-colorable planar graphs without prescribed cycles

In this paper we prove that every planar graph without cycles of length 4, 5, 6 and 8 is 3-colorable.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.     

The planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.     

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane.     

Branching graphs for finite unitary groups in nondefining characteristic

We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favorable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.     

Nonlinearity of the Automorphism Groups of Some Free Algebras

We prove that the groups of tame automorphisms of a free Lie algebra (free associative algebra, absolutely free algebra, algebra of polynomials) of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field.