A Von Neumann Algebra Approach To Quantum Metrics Quantum Relations

In A von Neumann Algebra Approach to Quantum Metrics, Kuperberg and Weaver propose a new definition of quantum metric spaces, or W*-metric spaces, in the setting of von Neumann algebras. Their definition effectively reduces to the classical notion in the atomic abelian case, has both concrete and intrinsic characterizations, and admits a wide variety of tractable examples. A natural application and motivation of their theory is a mutual generalization of the standard models of classical and quantum error correction. In Quantum Relations Weaver defines a ``quantum relation'' on a von Neumann algebra $\mathcal{M}\subseteq\mathcal{B}(H)$ to be a weak* closed operator bimodule over its commutant $\mathcal{M}'$. Although this definition is framed in terms of a particular representation of $\mathcal{M}$, it is effectively representation independent. Quantum relations on $l^\infty(X)$ exactly correspond to subsets of $X^2$, i.e., relations on $X$. There is also a good definition of a ``measurable relation'' on a measure space, to which quantum relations partially reduce in the general abelian case. By analogy with the classical setting, Weaver can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and he can generalize Arveson's fundamental work on weak* closed operator algebras containing a masa to these cases. He is also able to intrinsically characterize the quantum relations on $\mathcal{M}$ in terms of families of projections in $\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)$.

We define a "quantum relation" on a von Neumann algebra M⊆B(H) to be a weak* closed operator bimodule over its commutant M′. Although this definition is framed in terms of a particular representation of M, it is effectively representation independent. Quantum relations on l∞(X) exactly correspond to subsets of X2, i.e., relations on X. There is also a good definition of a "measurable relation" on a measure space, to which quantum relations partially reduce in the general abelian case. By analogy with the classical setting, we can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and we can generalize Arveson's fundamental work on weak* closed operator algebras containing a masa to these cases. We are also able to intrinsically characterize the quantum relations on M in terms of families of projections in M⊗ ̄B(l2).

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements.

These formulas have integral coefficients and imply that one of the main results of Lauda's paper--identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)--also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).

The authors prove that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, they show that the projective kernel is a congruence subgroup. To do this, they introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras.

The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras.

The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.

C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables. The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models. It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding. The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).

"November 2012, volume 220, Number 1033 (first of 4 numbers)."