Quantum structures - WS 2018/19


Lecture and Seminar:

Monday 9:45 - 11:15 Uhr
Thursday 9.45 - 11.15 Uhr
Begin: Monday, 22 October 2018
Seminar-Room 7.527

The seminar centers around the ubiquity of L-algebras. As L-algebras are present in the theory of Artin groups, knot theory, topology, functional analysis, von Neumann algebras, etc., our aim is to amplify that algebraic logic is a vital force right at the heart of mathematics, widely ignored and misconstrued by preoccupation, but nevertheless essential with a lot to tell to the working mathematician.

From a logical point of view, L-algebras provide the semantics of the three main generalizations of classical logic, namely, the Intuitionistic Propositional Calculus (IPC), the many-valued calculus of Lukasiewicz (MVC), and the logic of quantum mechanics, as introduced in 1936 by Birkhoff and von Neumann. The semantics of IPC is essentially given by Hilbert algebras, which form the essence of topology; MVC is interpreted by MV-algebras, the essence of measure theory, while quantum logic is formalized by OMLs, the essential structure of von Neumann algebras, which can also be seen as non-commutative measure theory. All these algebras are L-algebras.

We will touch cycle sets, and their associated L-algebras, showing that the (full) structure group of this L-algebra is just Etingof's structure group of a cycle set (or rather its corresponding solution to the Yang-Baxter equation).

Dense, regular, and prime elements are introduced for arbitrary L-algebras, and applied to the context of Hilbert algebras, Heyting algebras, and locales. We determine the structure group of a bounded Hilbert algebra, reminiscent of Glivenko's fundamental theorem.

Measures on an MV-algebra are group homomorphisms between the structure groups, which will lead us to a covering theory, so that the universal covering represents any MV-algebra by a measure on a Stone space, invariant under the fundamental group. Together with the Stone space, this group provides a complete invariant for MV-algebras. A similar situation will be found in Esakia's representation of Heyting algebras, which points to a general principle that has yet to be explored.

Handwritten Notes:

Page 1-9.
Page 9a-18.
Page 19-24.
Page 25-31.
Page 32-35.