Narrow Results

We present several equivalent conditions for C*-finitely correlated states defined on the UHF algebras to be factor states and consider the types of factors generated by them. Subfactors generated by generalized quantum Markov chains defined on the gauge-invariant parts of the UHF algebras are also discussed.

Let M₀ ⊂ M₁ be a finite index infinite depth hyperfinite II₁ subfactor and ω a free ultrafilter of the natural numbers. We show that if this subfactor is constructed from a commuting square then the central sequence inclusion has infinite Pimsner–Popa index. We will also demonstrate this result for certain infinite depth hyperfinite subfactors coming from groups.

Based on the fact that, for a subfactor N of a II₁ factor M, the first nontrivial Jones index is two and then M is decomposed as the crossed product of N by an outer action of ℤ₂, we study pairs {N, uNu*} from the view-point of entropy for two subalgebras of M in connection with the entropy for automorphisms, where the inclusion of II₁ factors N ⊂ M is given with M being the crossed product of N by a finite group of outer automorphisms and u is a unitary in M.

In recent joint work with Jones and Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper, we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans–Kawahigashi.

We consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite-dimensional *-algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also show how parametric families of (not necessarily) twisted commuting squares yield associative deformations of the matrix multiplication.

In this paper, we show in a combinatorial way that the 0-box space of the E₈ subfactor planar algebra is 1-dimensional. In the proof, we improve on Bigelow's relations for the E₈ subfactor planar algebra and give an efficient algorithm to reduce any planar diagram to the empty diagram.

Given any quadruple $(N,P,Q,M)$ of ${II}_1$-factors with finite index, the notions of interior and exterior angles between $P$ and $Q$ were introduced in [An angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc.371(8) (2019) 5973–5991]. We determine the possible values of these angles in terms of the cardinalities of the Weyl groups of the intermediate subfactors when $(N,P,Q,M)$ is an irreducible quadrilateral and the subfactors $N\subset P$ and $N\subset Q$ are both regular. For an arbitrary irreducible quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained when $N\subset P$ is regular, which enforce some restrictions on the index of $N\subset Q$ in terms of that of $N\subset P$.

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that ${\bigcup }_{k\ge 1}{M}_k(D)$ admits a factor trace if and only if $D\in \{2cos(\pi /n)+1|n\ge 3\}\cup [3,\infty )$ and the higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over these factors is constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.

For a finite-index ${\mathrm{\text{II}}}_1$ subfactor $N\subset M$, we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,$q$)), $q$ prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings $({{ℛ}}_q)$ interpolates to all integers $q\ge 2$, providing (when $q$ is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring ${{ℛ}}_q$ (with $q$ non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

To a planar algebra in the sense of Jones we associate a natural non-commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by . We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special symmetries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra . This talk is based on joint work with A. Guionnet and V. Jones.