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November 29, 2007
On BV Quantization, Part VIII
Posted by Urs Schreiber
A good way to understand the relevance and meaning of the trinity of
in the BV-formulation of quantum field theory (part I, II, III, IV, V, VI, VII) is to interpret the story of the charged -particle in the world of Lie -algebroids (as opposed to Lie -groupoids), which have a restriction on the sign of the degrees of their morphisms, but then form the configuration space of fields by taking the internal hom in the category of arbitrarily graded things.ghosts – fields – anti-fields
This is, to my mind, how to interpret one of the crucial points of AKSZ: The Geometry of the Master Equation and Topological Quantum Field Theory.
Here I start to talk about that, but it will probably take me more than one installment. Progress will be documented in On Lie -modules and the BV complex.
Geometric Representation Theory (Lecture 14)
Posted by John Baez
This time in the Geometric Representation Theory seminar I tried to state the Fundamental Theorem of Hecke Operators… and I screwed up. Luckily, I screwed up in an instructive way!
Pick a group . We’ve seen that every -invariant relation between finite -sets gives an intertwining operator between the resulting permutation representations of . These operators are called Hecke operators.
Any category theorist worth their salt should want to make this into a functor. Since there’s a category with
- finite -sets as objects
- -invariant relations as morphisms
and a category with
- finite-dimensional representations of on complex vector spaces as objects
- -invariant operators (= intertwining operators) as morphisms
one might hope the Hecke operator trick gave a functor
But, it doesn’t!
November 27, 2007
Rejecta Mathematica
Posted by John Baez
Sick of getting your papers rejected?
Tired of useless arguments with referees, like this?
In conclusion, although a good paper of this kind would be interesting, I don’t believe that the present manuscript meets the required standards for acceptance in the Journal.
You are right here. It is way too good for it! I kick myself for being nice and submitting this paper to your narrow-minded pretentious journal! It won’t happen again.
Then publish your rejected papers in Rejecta Mathematica!
November 26, 2007
This Week’s Finds in Mathematical Physics (Week 258)
Posted by John Baez
In week258 of This Week’s Finds, learn what happens when sand grains flow. Read more about dust in the Red Rectangle:
and discover why some of it looks like this:
Then, find out about Deligne’s conjecture concerning Hochschild cohomology — and get a nice workout in homological algebra, categories and operads.
Poncelet’s Porism
Posted by John Baez
Gavin Wraith is a mathematician with wide-ranging interests and a fondness for mysteries. To get a sense of this, try his article on ‘Ptolemy and non-Archimedes’.
Let me whet your appetite….
Categories, Logic and Physics in London
Posted by John Baez
Category theory and logic seem to be finding more connections with physics. Bob Coecke and Andreas Döring have decided to run a series of workshops on these connections, starting with this:
- Categories, Logic and Foundations of Physics, January 9, 2008, Imperial College, London.
It’s too bad I can’t make it! I hope someone here does, and tells us what happened. Maybe Jamie Vicary? Maybe even David?
More details follow…
November 23, 2007
Concordance
Posted by Urs Schreiber
I am thinking about the notion of concordance of 2-coycles used by Baas, Bökstedt and Kro.
It seems more or less obvious how it is related to transformations of the ana-2-functors involved, but it also seems that there is a big interesting issue lurking here, once we try to get serious about thinking about -anafunctors.
Influenced by the discussion about Sjoerd Crans’s generalization of the Gray tensor product to -categories with Todd Trimble in Extended Worldvolumes, I had the following thoughts (unfinished, need to catch a train right now)
Homotopy, Concordance and Natural Transformation
Abstract. Transformations between (-)functors are like homotopies between maps of topological spaces. This statement can be given a precise meaning using the closed structure of in terms of the extension of the Gray tensor product from 2-categories to -categories given by Sjoerd Crans. The analogous construction is familar in homological algebra from categories of chain complexes.
After recalling the basics, we turn to “anafunctors” (certain spans of functors) and highlight how the general relation between transformations, homotopies and concordances appears in the study of nonabelian -cocycles classifying -bundles.
MA in Reasoning
Posted by David Corfield
- Logical reasoning, fallacies and paradoxes
- Causal and probabilistic reasoning
- Scientific and mathematical reasoning
- The advent of scientific reasoning: Galileo and Descartes to Newton and Kant
November 22, 2007
Geometric Representation Theory (Lecture 13)
Posted by John Baez
Happy Thanksgiving! Some blogs may quiet down during the holidays, but not this one. We know your desire for math, physics and philosophy doesn’t slack off just because some Americans are snoozing as they recover from gorging on turkey.
In this episode of the Geometric Representation Theory seminar, James Dolan begins to explain how the story so far is connected to knot theory — or for now, the theory of braids. Namely, he shows how decorated braids can be used to describe Hecke operators between flag representations of — or its version, the symmetric group .
He’s already described these operators using certain matrices. The braid picture is equivalent. But, I think it’s the tip of a bigger iceberg: a categorified version of the theory of quantum groups and their associated braid group representations! We’ll take a serious stab at that next quarter, once we get some more machinery in place.
November 20, 2007
Something Like Lie-Rinehart ∞-pairs and the BV-complex (BV, part VII)
Posted by Urs Schreiber
In the context of our previous discussion in Modules for Lie -algebras I am preparing some notes
On Lie -modules and the BV complex
Abstract. Some tentative remarks on generalizations of Chevalley-Eilenberg algebras from Lie algebras and their modules to Lie -algebras and their modules, with an eye towards understanding the Batalin-Vilkovisky complex.
While not finished, I thought that I would share this now as kind of a reply to some of the things Jim Stasheff, Todd Trimble, Johannes Huebschmann and others said.
Apart from getting the generalities right, like formulating everything as nicely as possible internal to the category of chain complexes, the point is to discuss the BV complex “for the (-1)-brane” which I talked about in BV for Dummies and in On Noether’s second and take it as the motivating and guiding example to obtain and check the right definition of a Lie -algebroid, such that its dual is an arbitrarily graded dg-algebra of sorts.
All comments are welcome. Especially corrections.
Previous entries in this BV series are I II III IV V VI.
By the way, a pretty good set of slides summarizing BV is
Glenn Barnich
Algebraic structure of gauge systems:
Theory and Applications
(pdf)
The toy example that I was talking about in BV for Dummies is essentially the one on slide 3 there.
November 18, 2007
Geometric Representation Theory (Lecture 12)
Posted by John Baez
In 1925, Werner Heisenberg came up with a radical new approach to physics in which processes were described using matrices of complex numbers. What makes this especially remarkable is that Heisenberg, like most physicists of his day, had not heard of matrices!
It’s hard to tell what Heisenberg was thinking, but in retrospect we might say his idea was that given a system with some set of states, say , a process would be described by a bunch of complex numbers specifying the ‘amplitude’ for any state to turn into any state . He composed processes by summing over all possible intermediate states: Later he discussed his theory with his thesis advisor, Max Born, who informed him that he had reinvented matrix multiplication!
November 16, 2007
Geometric Representation Theory (Lecture 11)
Posted by John Baez
This time in the Geometric Representation Theory seminar, Jim Dolan recalls how to describe Hecke operators between flag representations using certain matrices.
Does composition of the Hecke operators correspond to multiplying these matrices? No! — and yet, in a certain limit it does resemble matrix multiplication.
November 15, 2007
Category Theory and Biology
Posted by David Corfield
Some of us at the Centre for Reasoning here in Kent are thinking about joining forces with a bioinformatics group. Over the years I’ve caught glimpses of people trying out category theoretic ideas in biology, so naturally I’ve wanted to take a closer look. An initial foray has revealed some intriguing work: André Ehresmann and Jean-Paul Vanbremeersch on Memory Evolutive Systems and Gerhard Mack (somewhere near Urs in Hamburg) on Universal Dynamics, a Unified Theory of Complex Systems: Emergence, Life and Death. Climbing the n-category ladder, Nils Baas who has ideas on abstract matter, has worked with Ehresmann and Vanbremeersch on ‘Hyperstructures and memory evolutive systems’, and with Torbjorn Helvik on higher-order cellular automata.
November 13, 2007
Modules for Lie infinity-Algebras
Posted by Urs Schreiber
This here is mainly a question to Jim Stasheff – and possibly to his former student Lars Kjeseth in case he is reading this – concerning the general issue addressed in the article
Lars Kjeseth
Homotopy Lie Rinehard cohomology of homotopy Lie-Rinehart pairs
HHA 3, Number 1 (2001), 139-163.
which we were discussing in BV for Dummies.
The question is
What is the right -categorification of a Lie-Rinehart pair?
A Lie-Rinehart pair is a pair consisting of an associative algebra and a Lie algebra , such that acts on and acts on in a compatible way, where the two compatibility conditions are the obvious ones you find when looking at the archetypical example of the Lie-Rinehart pair obtained from the smooth functions on a smooth manifold and the vector fields on acting on these.
This example clearly encodes the same information as the tangent Lie algebroid of , and in fact it is rather manifest that whenever for some space , a Lie-Rinehart pair is precisely a Lie algebroid structure over , and vice versa.
We have discussed that people are thinking that a Lie -algebroid, whatever it is in direct terms, is dually encoded precisely in non-negatively graded dg-manifolds.
I found that disturbing. In light of the fact that non-negatively graded dg-algebras beautifully and neatly capture everything about semistrict Lie -algebras, with the latter being a very natural categorical concept, I am not prepared to accept that there should be no equally nice categorical picture for arbitrarily graded dg-manifolds.
My conjecture therefore:
- non-negatively graded dg-manifolds appear when in a Lie-Rinehart pair you categorify only the , not the .
- a