Quantum symmetric pairs and representations of double affine Hecke algebras of type C∨CnC^\vee C_n

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic

Some Nearly Quantum Theories

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes with an extra classical bit. In both of these categories, states are morphisms from systems to the tensor unit, which helps give the categorical structure a clear operational interpretation. A no-go result shows that the first of these categories, at least, cannot be extended to include spin factors other than the (real, complex, and quaternionic) quantum bits, while preserving the representation of states as morphisms. The same is true for attempts to extend the second category to even-dimensional spin-factors. Interesting phenomena exhibited by some composites in these categories include failure of local tomography, supermultiplicativity of the maximal number of mutually distinguishable states, and mixed states whose marginals are pure.Comment: In Proceedings QPL 2015, arXiv:1511.0118