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
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
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.