gage
发表文章数: 466
武功等级: 空明拳
(第三重)
内力值: 415/415
Atiyah对Penrose扭量代数的一个观点
http://www.math.columbia.edu/~woit/wordpress/?p=402
====
Lusztig Birthday Conference, by Peter Woit
I was up in Boston for a few days, and managed to attend a few of the talks at the conference in honor of George Lusztig’s 60th birthday. Lusztig started out his career in geometry and topology; his thesis was in the area of index theory, working with Michael Atiyah and using the families version of the index theorem. He soon turned his attention to representation theory, which is the field that he has worked in for most of his career, often from a quite algebraic point of view. His papers are dense and can be difficult to read, especially for someone like me who is not so algebraically inclined, but many speakers at the conference remarked on how their work had drawn important inspiration from one or another of these papers.
Among the things he is famous for are his work on quantum groups, on the representation theory of reductive groups over finite fields (called Deligne-Lusztig theory, for an introduction, see here), on a whole new field in Lie theory known as Kazhdan-Lusztig theory (for an introduction, see the article by Deodhar in the proceedings of the 1991 AMS summer institute on algebraic groups), and many other things.
Of the few talks I heard at the conference, two were really exceptional. One of these was by Michael Atiyah, with the title “Quaternions in Geometry, Analysis and Physics”. He began by explaining that not only was Lusztig 60, but, if he were alive, the Irish mathematician Hamilton would be 200. There’s a famous story about Hamilton’s discovery of the quaternions: this took place in a flash of insight on October 16, 1843, after which he supposedly engraved the defining relations of the quaternion algebra into a Dublin bridge. Atiyah described a piece of history I didn’t know, showing an extract from a 1846 paper of Hamilton’s in which he takes a square root of the Laplacian and essentially writes down the Dirac equation (in Euclidean signature, this was long before special relativity…).
Hamilton was very taken with quaternions as a generalization of complex numbers, and wanted to develop a “quaternionic analysis” that would be a generalization of complex analysis, a project he thought would take him at least ten years. It turns out that you can’t simply generalize the beautiful subject of complex analysis and algebraic geometry over the complex numbers to the quaternionic case. Because of non-commutativity, polynomials behave very differently. Atiyah explained that in his view the correct generalization of complex analysis to the quaternionic case was Penrose’s twistor theory. Here one considers all possible ways of identifying R4 with C2, forming a 3 complex dimensional “twistor space”. Complex analysis on this twistor space is what Atiyah claimed should be thought of as the quaternionic analog of complex analysis (on the complex plane).
He reviewed the story of how solutions to various linear equations are related to sheaf cohomology groups on the twistor space, then went on to the non-linear case, where solutions of the anti-self-dual Yang-Mills equations correspond to holomorphic bundles on the twistor space. One can generalize twistor theory to what Atiyah claimed should be thought of as quaternionic analogs of Riemann surfaces: 4d Riemannian manifolds with holonomy in Sp(1)=SU(2), these are self-dual Einstein manifolds, what Penrose would call a “non-linear graviton” (although this is the Riemannian, not pseudo-Riemannian case). The twistor space of these 4d manifolds is a 3d complex manifold, and Atiyah considers complex analysis on this to be the quaternionic analog of complex analysis on a Riemann surface.
The quaternionic analog of higher dimensional complex manifolds are manifolds of dimension 4k, with holonomy Sp(k). Unlike in the complex case, there are few compact examples. Atiyah went on to discuss how examples (mostly non-compact) could be generated as quotients using the quaternionic analog of symplectic reduction. He described several different classes of examples, noting that this construction first appeared in work with physicists studying supersymmetric non-linear sigma models. While I was a post-doc at Stony Brook, Nigel Hitchin was visiting there and working with Martin Rocek and others on this, leading to the 1987 paper in CMP by Hitchin, Karlhede, Lindstrom and Rocek. Atiyah said that he wouldn’t try and describe the relation to supersymmetry, since “I don’t know much about supersymmetry, and if I tried to explain it, you would understand even less”. That Atiyah, after many years of working in this area, still finds supersymmetry to be something he can’t quite understand, is an interesting comment, reflecting the way the subject is still very imperfectly integrated into mathematician’s traditional ways of thinking about geometry and algebra.
Atiyah also commented that off and on over the years he had pursued the idea that quantum groups (which aren’t quite groups), are in some sense the quaternionification of a Lie group (which doesn’t quite exist). He said he hadn’t been successful with this idea, but still thought there was something to it, and hoped that someone else would take up the challenge of trying to make sense of it.
繁星满目的夜晚，我举头四望，却发现众星都离我远去。
一只小小的温度计，却透露了宇宙那无比的寒冷和荒凉。
多普勒说，你们都是红眼病。
阿基米德说，给个支点，你就要和整个地球上的人抬杠。
| 发表时间：2006-08-01, 05:42:29 |
 作者资料
|
gage
发表文章数: 466
武功等级: 空明拳
(第三重)
内力值: 415/415
Re: Atiyah对Penrose扭量代数的一个观点
稍加解释。
为庆祝Luszitg 60寿辰而开了个会议，M.Atiyah 作第一个报告。本文为听众之一的 Peter Woit 根据其记录写成，在此文后面的讨论中作者指出了这一点，也就是说其记录可能不准确。下面是Peter Woit 记录的大意。
M.Atiyah首先回顾了Hamilton的四元数，然后谈到对复数的推广问题。因为四元数非交换，所以四元数上的多项式并没有很好的性质。Atiyah 认为 Penrose 扭量对应的代数而不是四元数才是复数的自然推广。其余略。
繁星满目的夜晚，我举头四望，却发现众星都离我远去。
一只小小的温度计，却透露了宇宙那无比的寒冷和荒凉。
多普勒说，你们都是红眼病。
阿基米德说，给个支点，你就要和整个地球上的人抬杠。
| 发表时间：2006-08-01, 06:19:48 |
作者资料
|
星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Atiyah对Penrose扭量代数的一个观点
Atiyah 认为 Penrose 扭量对应的代数而不是四元数才是复数的自然推广。
－－－－－－－－－－－－－－－－－－－－－－－－－－－－
这种观点可能很符合Penrose的本意，只是Penrose稍微谦虚一点，意图把扭量作为四元数之后的推广，而不是取代四元数直接作为复数的推广。
如果我没有理解错，记得扭量的描述中，把四维时空坐标与四维动量置于平等地位，放在一起构成所谓“八维空间”，其八个自由度来自于“四维坐标”和“四维动量”，不过可能将它们混在一起进行某种线性组合。不管怎样组合，总自由度总是八。于是在这一点上有点类似于推广的相空间描述。
记得是2005还是2004在“科学的美国人”上有篇文章，专门科普loop量子引力，里面谈到了扭量在该理论中的重要应用。
One may view the world with the p-eye and one may view it with the q-eye but if one opens both eyes simultaneously then one gets crazy
| 发表时间：2006-08-01, 07:19:30 |
作者资料
|
gage
发表文章数: 466
武功等级: 空明拳
(第三重)
内力值: 415/415
Re: Atiyah对Penrose扭量代数的一个观点
请星空注意，这个只是Peter Woit的笔记，不一定准确。而事实证明这个记录确实不准确。因为Penrose的扭量对应的代数可以说就是四元数，当然不是通常的四元数，而是以复数为系数的四元数，用代数的话来说就是复数 C 和四元数 H 在实数域上的张量积。
繁星满目的夜晚，我举头四望，却发现众星都离我远去。
一只小小的温度计，却透露了宇宙那无比的寒冷和荒凉。
多普勒说，你们都是红眼病。
阿基米德说，给个支点，你就要和整个地球上的人抬杠。
| 发表时间：2006-08-01, 08:38:26 |
作者资料
|
星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Atiyah对Penrose扭量代数的一个观点
请星空注意，这个只是Peter Woit的笔记，不一定准确。
－－－－－－－－－－－－－－－－－－－－－－－－－－－
哦～～
One may view the world with the p-eye and one may view it with the q-eye but if one opens both eyes simultaneously then one gets crazy
| 发表时间：2006-08-01, 09:04:58 |
作者资料
|
轩轩
发表文章数: 1352
武功等级: 易筋经
(第二重)
内力值: 567/567
Re: Atiyah对Penrose扭量代数的一个观点
gage 兄对扭量感兴趣吗？
您是在北京吗？
《相对论通俗演义》
i will love you till the null infinity.
| 发表时间：2006-08-01, 22:22:35 |
作者资料
|
gage
发表文章数: 466
武功等级: 空明拳
(第三重)
内力值: 415/415
Re: Atiyah对Penrose扭量代数的一个观点
gage 兄对扭量感兴趣吗？
您是在北京吗？
================
不在北京。对扭量了解一点，主要是数学方面的，总结了一下。
繁星满目的夜晚，我举头四望，却发现众星都离我远去。
一只小小的温度计，却透露了宇宙那无比的寒冷和荒凉。
多普勒说，你们都是红眼病。
阿基米德说，给个支点，你就要和整个地球上的人抬杠。
| 发表时间：2006-08-01, 22:48:05 |
作者资料
|
星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Atiyah对Penrose扭量代数的一个观点
等将来条件机会合适的时候，我跟gage兄见面是很容易的。但目前还不行，还想处于保密状态。
One may view the world with the p-eye and one may view it with the q-eye but if one opens both eyes simultaneously then one gets crazy
| 发表时间：2006-08-01, 23:30:40 |
作者资料
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