Math papers I’ve recently looked at

I’ve been very busy with work in the last four weeks and so have only skimmed over a few maths papers, in addition to doing my own research (unsurprisingly related to Hecke algebras, as mentioned in the first post).

First of all, Nathan Geer, of the Georgia Institute of Tech, and whose web-page is http://www.math.gatech.edu/~geer/ , has recently published four papers (or put them on the archive: http://xxx.lanl.gov/ ) which seem quite interesting. Nathan works in low-dimensional topology (ie, knot theory) and Lie (super-)algebras and quantum (super-)algebras. I hadn’t known of Nathan or his work before seeing his papers on the archive – and I’d worked in similar areas for 5-6 years! Anyway, his papers:

Some remarks on quantized Lie superalgebras of Classical type, at http://arxiv.org/abs/math.QA/0508440 . I don’t fully understand the abstract, but a phrase I do understand is

“…we show that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the Drinfeld-Jimbo type superalgebra.”

 Ah ha! This tells us what some of the representations of the Drinfeld-Jimbo quantum superalgebras are like – they are “deformations” of the reps of the corresponding Lie superalgebras. I like this.

The Kontsevich integral and quantized Lie superalgebras, Algebraic and Geometric Topology 5 (2005), paper no. 45, pages 1111-1139. This paper also looks very interesting – it discusses the Links-Gould knot invariant that Jon Links and Mark Gould discovered in the early 1990s, up at the Maths Dept at Queensland Uni. Mark was one of my thesis examiners 🙂 

According to the abstract of this paper, “Le and Murakami showed that the quantum group knot invariants derived using R-matrices and the Kontsevich universal link invariant followed by the Lie algebra based weight system are the same”. In this paper, Nathan shows a a similar result of Lie superalgebras of type A-G and he investigates the Links-Gould invariant.I don’t know much about the Kontsevich universal link invariant (it’s one of the things I want to read up on some day) – but it’s interesting that this paper apparently shows more links between different areas of mathematics.

Louis Kauffman and David De Wit deserve a mention when discussing the Links-Gould polynomial – David did his PhD thesis on using Mathematica to automatically calculate Links-Gould “tangle-invariants” and has done some subsequent work on this – I understand that Louis Kauffman helped out as well.

Nathan also wrote these two papers:

Multivariable link invariants arising from sl(2,1) and the Alexander polynomial. (Joint work with Bertrand Paturear-Mirand.)  math.GT/0601291
Etingof-Kazhdan quantization of Lie superbialgebras.  To appear in Advances in Mathematics.  math.QA/0409563

I’m too tired to discuss these, so I won’t even begin!

2 Responses to Math papers I’ve recently looked at

  1. Tel says:

    Hi Sach, four very interesting papers indeed. I’m quite interested in “Etingof-Kazhdan quantization of Lie superbialgebras”, there is a section where he extends the Drinfeld costruction of the quasi-Hopf algebra associated to a Lie algebra to the super case, I’m going to reread Drinfeld’s paper on QHA.

    It’s nice to see that the LG invariant is getting some interest. I went to some talks that Atsushi Ishii gave on the LG when he was out at UQ last year (or the year before) – he found connections to the Alexander-Conway ploynomial and some nice skein relations. Some of his papers are here

    http://www4.ocn.ne.jp/~xyz/AtsushiISHII.html

    (Looks from Geer’s paper that the LG is related to the a generalisation of the HOMFLY.)

  2. Sacha says:

    I wonder if people will investigate the link invariants arising from many different representations of quantum algebras and quantum superalgebras? There are infinite non-isomorphic representations, and the study of the link invariants that havn’t been studied much could prove fruitful.

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