Braided Endomorphisms of Cuntz Algebras

Abstract

We discuss sufficient conditions ensuring that certain endomorphisms of infinite factors arising from Cuntz algebras are braided. We analyze some explicit non-trivial examples associated to unitary solutions of the quantum Yang-Baxter equation on a Hilbert space of dimension 2X In particular we show the existence of endomorphisms of index 2 associated to representations of Hecke algebras at a primitive fourth root of unity. In this case we compute the associated fusion rules. These fusion rules define a finitely generated à -semiring which is not finite. Such a picture seems to be closely related to the description of (the dual of) a deformation, at a fourth root of unity, of some compact matrix group. This could be of some interest for the investigation of quantum symmetries naturally appearing in low-dimensional Quantum Field Theory.

Computing the Jones index of quadratic permutation endomorphisms of O2

“…2 These values are plotted in the last column of Table 1 (below), modelled on those in [18,19,26], where for multiindices α and β we denote s α s * β by s α,β . 3 Notice that by the results in [18,19] (24) and ρ (14) (23) , which clearly give index 1.…”

Yang–Baxter endomorphisms

“…The R-matrix R 4 (8.2) is special from various points of view: Up to applying quasi-free automorphisms, R 4 is the unique non-trivial R-matrix in R (2) for which λ R is not ergodic, and the unique R-matrix in R(2) with index 2. We also mention that R 4 generates a representation of the Temperley-Lieb algebra at loop parameter δ = 1 2 , and satisfies R 4 4 ∈ C. Furthermore, λ R4 (O 2 ) is the fixed point algebra of an explicit order two automorphism α ∈ Aut O 2 [11]. The images of the braid group representations ρ R (B n ) are described in [25] in terms of extraspecial 2-groups, and its relevance for topological quantum computing is discussed in [43].…”

“…We also mention that generates a representation of the Temperley–Lieb algebra at loop parameter , and satisfies . Furthermore, is the fixed point algebra of an explicit order two automorphism [11]. The images of the braid group representations are described in [25] in terms of extraspecial 2‐groups, and its relevance for topological quantum computing is discussed in [43].…”

“…We use our previous results to analyze the properties of the corresponding endomorphisms in detail. In particular, we discuss the special case, an R‐matrix that has appeared in various places in the literature (see, for example, [11, 25, 54]), explain why it is also special from the point of view of endomorphisms, and compute its (infinite dimensional) fixed‐point algebra .…”

On Invariant MASAs for Endomorphisms of the Cuntz Algebras

“…Thirty years later, endomorphisms of O n still constitute an active area of research offering a number of challenging problems and interesting connections with other fields, including index theory, noncommutative entropy, quantum groups and classical dynamical systems (see for example [3,5,11,14,19]). In particular, significant progress has been achieved recently in the study of endomorphisms corresponding to permutation unitaries, [4,8,7,15,16,24,25].…”