Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer $n$, the number of elements of order $n$ is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.Comment: 25 pages; To appear in J. of Algebra. Main modifications are the following: (i) Verbose parts of the paper were summarized. (ii) Theorem 6.3 is added. (iii) The relation between Theorem 1.1 and works of Ng and Schauenburg is adde

Drinfeld center and representation theory for monoidal categories

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category C and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of C. This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories C of Hilbert bimodules over a II$_1$-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II$_1$-factor obtained by the Longo-Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor $N \subset M$ is equivalent to Popa's property (T) for the corresponding SE-inclusion of II$_1$-factors. In the last part of the paper we study M\"uger's notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.Comment: v3: minor corrections, to appear in Comm. Math. Phys.; v2: 37 pages, with a new section; v1: 24 page

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

Realization of rigid C∗^*-tensor categories via Tomita bimodules

Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_\lambda$, $\lambda\in (0,1]$. The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction. Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of these algebras, with finite Jones index, by exhibiting a fully faithful unitary tensor functor $F:\mathscr{C} \hookrightarrow End_0(\Phi)$ where $\Phi$ is a factor (of type II or III). The construction relies on methods from free probability (full Fock space, amalgamated free products), it does not depend on amenability assumptions, and it can be applied to categories with uncountable spectrum (hence it provides an alternative answer to a conjecture of Yamagami \cite{Y3}). Even in the case of uncountably generated categories, we can refine the previous equivalence to obtain realizations on $\sigma$-finite factors as endomorphisms (in the type III case) and as bimodules (in the type II case). In the case of trivial Tomita structure, we recover the same algebra obtained in \cite{PopaS} and \cite{AMD}, namely the (countably generated) free group factor $L(F_\infty)$ if the given category has denumerable spectrum, while we get the free group factor with uncountably many generators if the spectrum is infinite and non-denumerable.Comment: 39 page

Quantum gauge symmetries in Noncommutative Geometry

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M_n(R), M_n(C) and M_n(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A_F=C+H+M_3(C) and A^{ev}=H+H+M_4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).Comment: 31 pages, no figures; v2: minor changes, added reference