Research Interests:

My research is in a field known as `higher-dimensional algebra’. My work blends Lie theory with elements of category theory and has connections to knot theory and Lie algebra cohomology.  In my dissertation, I defined and explored generalized Lie algebras, called `Lie 2-algebras’, and classified them up to equivalence in terms of Lie algebra cohomology.  A Lie 2-algebra is a category equipped with algebraic structure much like that of a Lie algebra, but where the laws involving the bracket only hold up to isomorphism.  In my dissertation, I concentrated on semistrict Lie 2-algebras, that is, those where only the Jacobi identity fails to hold as an equation.   I showed that just as any Lie algebra gives a solution of the Yang-Baxter equation, a (semistrct) Lie 2-algebra gives a solution of the Zamolodchikov tetrahedron equation, the higher dimensional analog of the Yang-Baxter equation.  Furthermore, I investigated the relationship between Lie algebras and algebraic structures known as quandles, and described a novel means of passing from a Lie group to its Lie algebra. 

John Baez and I recently finished a paper, Higher Dimensional Algebra VI: Lie 2-Algebras, (see below) where we introduce (semistrict) Lie 2-algebras.  The previous paper in this ``Higher Dimensional Algebra" series, Higher Dimensional Algebra V: 2-Groups, by John Baez and Aaron Lauda, gives a description of a generalized group. Aaron, John and I are currently trying to show that every Lie 2-group has a Lie 2-algebra. To learn more about my work, please read (pdf versions of) my:

      Research Summary

      Dissertation Abstract

And if you’re feeling ambitious:

          My Dissertation:  Lie 2-Algebras  

This summer I visited the Institute for Mathematics and its Applications and attended the program: n-Categories: Foundations and Applications .  At the conference I presented a lecture entitled Higher Linear Algebra .  In October I visited the Banff Research Center and attended the conference: Braid Groups and Applications.  Here I gave a talk titled Solutions of the Yang-Baxter and Zamolodchikov Tetrahedron Equations. 

Publications:

Crans, A.; Fallat, S.; and Johnson, C. ``The Hadamard core of the totally nonnegative matrices.”  Linear Algebra and its Applications. 328 (2001), 203—222.

Crans, A.; Weinhold, R. ``What to Do on Your Summer Vacation.” Math Horizons.  February 2001, 23—26.

Baez, J.; Crans A. ``Higher Dimensional Algebra IV:  Lie 2-Algebras.”  Theory and Applications of Categories.  12(2004), 492-538.  (also available as math.QA/0307263)

 

In Preparation:

Crans, A. The Lie Algebra of a Pointed, Smooth Quandle,  to be submitted

            (Preliminary) Abstract:  In addition to providing interesting invariants of braids, quandles offer a conceptual explanation of the passage from a Lie group to its Lie algebra.  This results from the fact that the bracket in a Lie algebra arises from differentiating conjugation in the Lie group.  Our new description of this process captures this key aspect.  Since the theory of conjugation can be regarded as the theory of quandles, we begin by describing the means by which we can treat our Lie groups as quandles in Diff*, the category of pointed, smooth manifolds.  We continue by using the language of cojets to introduce a functor from Diff* to C, the category of `special coalgebras.’  Finally, we extract the underlying vector space from our special coalgebra and show that it is the Lie algebra of the Lie group we started with.  The Jacobi identity for the bracket follows from the self-distributive law for the quandle operation, while the antisymmetry of the bracket arises from the idempotence law satisfied by the quandle operation. 

The content of this paper is taken from the second chapter of my dissertation.  My goal is to categorify the results in this work to describe the passage from a Lie 2-group to its Lie 2-algebra via 2-quandles!

Crans, A. Quandles and the Braid Groupoid,  to be submitted

            (Preliminary) Abstract:  We introduce two new concepts related to racks and quandles:  `shelves’ and `spindles’.  Roughly speaking, a rack results when two shelves fit together nicely, and a quandle consists of two spindles which fit together nicely.  The braid and framed braid groupoids and monoids possess a contravariant relationship with the categories of shelves, spindles, racks and quandles.

The content of this paper is taken from the second chapter of my dissertation.  My goal is to categorify the results in this work to illustrate the relationship between higher algebra and higher braid theory.

 

Current Projects: 

Representation Theory of Semistrict Lie 2-algebras:

            John Baez, Danny Stevenson and I are investigating the representation theory of semistrict Lie 2-algebras.  Since 2-vector spaces and semistrict Lie 2-algebras are equivalent to skeletal versions, we suspect that we need only to determine the representations of skeletal Lie 2-algebras on skeletal 2-vector spaces.  We completely understand representations of discrete Lie 2-algebras on discrete and skeletal 2-vector spaces, as well as representations of skeletal Lie 2-algebras on discrete 2-vector spaces.  We are also looking at intertwiners between representations and 2-intertwiners between them.  Our preliminary work suggests that we will be able to describe representations of skeletal Lie 2-algebras on skeletal 2-vector spaces using Lie algebra cohomology.  

2-Quandles:

            By the end of the year, I hope to have a definition of a categorified quandle, or 2-quandle.  I have recently submitted an internal grant (Loyola Marymount University) to receive funding for this project during the summer.   My goals include classifying 2-quandles using quandle cohomology, just as we were able to classify Lie 2-algebras using Lie algebra cohomology in Higher-Dimensional Algebra VI, showing that 2-quandles give invariants of Carter and Saito’s 2-braids ,just as quandles provide invariants of braids, and finally, demonstrating that Lie 2-groups have Lie 2-algebras.  My dissertation contains all of the work I have done thus far with regards to this project and contains my guesses for what the definition of a 2-quandle should be like. 

Lie 2-algebras of Lie 2-groups:

            John Baez, Aaron Lauda and I have been attempting to show that Lie 2-groups have Lie 2-algebras.  This project, however, is fraught with difficulties, since we are unsure as to which Lie 2-groups and Lie 2-algebras we should attempt to use.  Naturally, it seems that a weaker notion of Lie 2-algebra is needed (see below) to match up with the notion of coherent 2-group, found in Higher-Dimensional Algebra V:  2-groups.  At the same time, it seems that a less weak version of a coherent 2-group should be used.  Some of these concerns are explained in greater detail in HDAV.  Our current goal is to categorify the passage from Lie group to Lie algebra described in my dissertation.  Further work on this project will require a definition of 2-quandle and better understanding of the concerns already mentioned.

Weak Lie 2-algebras:

            The semistrict Lie 2-algebras defined in Higher-Dimensional Algebra VI have a bracket functor for which antisymmetry holds on the nose.  A weak Lie 2-algebra, then, will be a 2-vector space L equipped with a bilinear bracket functor satisfying the Jacobi identity up to a trilinear natural transformation called the ‘Jacobiator’ and also a bilinear natural transformation called the ‘Antisymmetrizer’, each of which must satisfy new laws of its own.  These laws are depicted here.  In order to continue this project, I must test this definition for its correctness. 

My other mathematical interests include algebraic coding theory, knot theory, the history of mathematics, and the relationship between music and mathematics. Here's an excellent site I found for math history.

And...who doesn't love a fun math comic?


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