10 research outputs found
Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond
The recollement approach to the representation theory of sequences of
algebras is extended to pass basis information directly through the
globalisation functor. The method is hence adapted to treat sequences that are
not necessarily towers by inclusion, such as symplectic blob algebras (diagram
algebra quotients of the type-\hati{C} Hecke algebras).
By carefully reviewing the diagram algebra construction, we find a new set of
functors interrelating module categories of ordinary blob algebras (diagram
algebra quotients of the type- Hecke algebras) at {\em different} values
of the algebra parameters. We show that these functors generalise to determine
the structure of symplectic blob algebras, and hence of certain two-boundary
Temperley-Lieb algebras arising in Statistical Mechanics.
We identify the diagram basis with a cellular basis for each symplectic blob
algebra, and prove that these algebras are quasihereditary over a field for
almost all parameter choices, and generically semisimple. (That is, we give
bases for all cell and standard modules.)Comment: 61 page
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Representation theory of algebras related to the partition algebra
The main objective of this thesis is to determine the complex generic representation theory of the Juyumaya algebra. We do this by showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by Martin (the representation theory of which is computable by a combination of classical and category theoretic techniques). We then use this result and general arguments of Cline, Parshall and Scott to prove that the Juyumaya algebra En(x) over the complex field is generically semisimple for all n 2 N. The theoretical background which will facilitate an understanding of the construction process is developed in suitable detail. We also review a result of Martin on the representation theory of the small ramified partition algebra, and fill in some gaps in the proof of this result by providing proofs to results leading to it
Noncommutative Independence from the Braid Group B∞
We introduce `braidability' as a new symmetry for (infinite) sequences of
noncommutative random variables related to representations of the braid group
. It provides an extension of exchangeability which is tied to the
symmetric group . Our key result is that braidability implies
spreadability and thus conditional independence, according to the
noncommutative extended de Finetti theorem (of C. K\"{o}stler). This endows the
braid groups with a new intrinsic (quantum) probabilistic interpretation.
We underline this interpretation by a braided extension of the Hewitt-Savage
Zero-One Law.
Furthermore we use the concept of product representations of endomorphisms
(of R. Gohm) with respect to certain Galois type towers of fixed point algebras
to show that braidability produces triangular towers of commuting squares and
noncommutative Bernoulli shifts. As a specific case we study the left regular
representation of and the irreducible subfactor with infinite Jones
index in the non-hyperfinite -factor related to it. Our
investigations reveal a new presentation of the braid group , the
`square root of free generator presentation' . These new
generators give rise to braidability while the squares of them yield a free
family. Hence our results provide another facet of the strong connection
between subfactors and free probability theory and we speculate about
braidability as an extension of (amalgamated) freeness on the combinatorial
level.Comment: minor changes, added 3.3-3.6, version to be published in
Comm.Math.Phys. (47 pages
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page