1,273 research outputs found

    Hopf algebras and subfactors associated to vertex models

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    If H is a Hopf algebra whose square of the antipode is the identity, v\in\l (V)\otimes H is a corepresentation, and \pi :H\to\l (W) is a representation, then u=(idπ)vu=(id\otimes\pi)v satisfies the equation (tid)u1=((tid)u)1(t\otimes id)u^{-1}=((t\otimes id)u)^{-1} of the vertex models for subfactors. A universal construction shows that any solution uu of this equatio n arises in this way. A more elaborate construction shows that there exists a ``minimal'' triple (H,v,π)(H,v,\pi) satisfying (idπ)v=u(id\otimes\pi)v=u. This paper is devoted to the study of this latter construction of Hopf algebras. If uu is unitary we construct a \c^*-norm on HH and we find a new description of the standard invariant of the subfactor associated to uu. We discuss also the ``twisted'' (i.e. S2idS^2\neq id) case.Comment: 25 pages, Late

    Construction of Curtis-Phan-Tits system in black box classical groups

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    We present a polynomial time Monte-Carlo algorithm for finite simple black box classical groups of odd characteristic which constructs all root SL2(q){\rm{SL}}_2(q)-subgroups associated with the nodes of the extended Dynkin diagram of the corresponding algebraic group.Comment: 35 page

    Automorphisms of the bipartite graph planar algebra

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    For any abstract subfactor planar algebra PP, there exists a finite index extremal subfactor M0M1M_0 \subset M_1 with PP as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and obtain subfactor planar subalgebras by taking fixed points under groups of automorphisms. This construction provides both new examples of subfactors and new descriptions of the planar algebras of previously known examples.Comment: 22 pp

    The non-commuting, non-generating graph of a finite simple group

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    Let GG be a group such that G/Z(G)G/Z(G) is finite and simple. The non-commuting, non-generating graph Ξ(G)\Xi(G) of GG has vertex set GZ(G)G \setminus Z(G), with edges corresponding to pairs of elements that do not commute and do not generate GG. We show that Ξ(G)\Xi(G) is connected with diameter at most 55, with smaller upper bounds for certain families of groups. When GG itself is simple, we prove that the diameter of the complement of the generating graph of GG has a tight upper bound of 44. In the companion paper arXiv:2211.08869, we consider Ξ(G)\Xi(G) when G/Z(G)G/Z(G) is not simple.Comment: 20 page

    Quantitative K-Theory Related to Spin Chern Numbers

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    We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant

    CC^*-algebras from planar algebras I: canonical CC^*-algebras associated to a planar algebra

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    From a planar algebra, we give a functorial construction to produce numerous associated CC^*-algebras. Our main construction is a Hilbert CC^*-bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular CC^*-algebras. By compressing this system, we obtain various canonical CC^*-algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-)Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical CC^*-algebras associated to a planar algebra.Comment: 47 pages, many figure

    Isotropic quantum walks on lattices and the Weyl equation

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    We present a thorough classification of the isotropic quantum walks on lattices of dimension d=1,2,3d=1,2,3 for cell dimension s=2s=2. For d=3d=3 there exist two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.Comment: 16 pages, 1 figur
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