SEMINAR

Homogeneous Differential Geometry / Based on Examples

(with Dr. Alex Levichev)

Meets on Wednesdays 4.30pm, MCS 180 Subject: Re: Seminar Wed 3/1

at the end of the last meeting we have learned about local H_0 and global H Hamiltonians, being, essentially, images of generators of local (Minkowskian) and global (Segal's) times. We are thus introduced to a promising field of open problems related to the physical notion of mass (of a particle). H, H_0 do not have common eigenvectors. I. Segal has only indicated how the interplay of the two Hamiltonians should result (in accordance with Mach's principle) in the known (approximately exact) values for elementary particles. Current theoretical physics HAS NO EXPLANATION of mass which is treated as an exact quantum number. Nobody has yet started mathematical calculations along the guidelines indicated by Segal. More details follow tomorrow. We meet on Wed 3/15 r.180, 4.30pm

It happened so on the 3/1 meeting that we have been discussing metric involving issues (in the context of a relationship between L and G). The purpose of the file below is to "polish" my Wednesday's instant response. Important are operators ad_v, v is from a Lie algebra L = L(G), they are introduce on p.13 (CP). The Killing form (= metric = "dot product") on L is introduced as = trace ((ad_v)(ad_w)^t). This always gives birth to a bi-invariant metric on the group G (just apply left or right translations in G to inroduce the dot product at each tangent space). The properties (both of L and of G) which can be expressed in terms of that metric are INTRINSIC properties. Say, < , > is non-degenerate iff it is a case of a SEMI-SIMPLE Lie algebra (and a Lie group). If < , > has eigenvalues of one sign only, then L is COMPACT (those are important classes of Lie algebras which we have THUS introduced). In terms of < , > one easily writes down an important element of the L's envelopping algebra (= the quadratic Casimir element), etc. Amanda presented an OUTER metric w.r.t. which the L = sl(2,R) turned out to be a 3-space orthogonal to the neutral element of G. This was a PARTICULAR example. It is my belief that the example does not make too much in understanding of how, IN GENERAL, L relates to G. Even within that construction, the most important was that L was a tangent 3-space to G rather than being orthogonal to that unit element.

Subject: Seminar Wed 3/1

I have left this year's substitute for the CP's p.24 in your mailboxes. There is nothing wrong about the original version but this year I have made it to match the Segal's choice of basic vectors in su(1,1) (see the "Pilot Model..."). Part of the forthcoming meeting will be to work with these new formulas. Last time we've discussed linear representations of Lie groups and Lie algebras (targeting the concepts which are important in theoretical physics).

Date: Tue, 1 Feb 2000 we are somewhere in L-2 (from now on, this refers to the CP which most of you have purchased). Welcome to try exercises from that L-2. My plan for tomorrow is to finish L-2 and to stay in L-3 (which introduces Lie algebras realized as certain totalities of vector fields). Those of you, who will be going through CP ahead of me, might be interested to understand (and to try to solve) the following open question:

OQ-1. Let E be the bundle over a circle S induced from a scalar representation of the subgroup P of upper-triangular matrices in G = SL(2,R). Is it a trivial (= cylindrical) bundle over S or is it a non-trivial one ?

Comments. I do not think that the solution is extremely difficult. It is rather that nobody (hopefully) has posed such a question for there is a few people only working on Chronometry. Please, be aware that a fiber is a complex dim=1 linear space. Check with the CP's last lectures regarding the definitions involved. Certain sections of such a bundle are states of scalar (= spinless) particles.

Subject: Seminar 2/9

On Wed I did mention that I've spent a few hours of the weekend on calculations. Those were to find left- and right-invariant vector fields on SL(2,R) starting from a particular parametrization of it. I did find the respective 3x3 matrix but, afterwards, I've realized that I should have used one other parametrization (the one t,x,theta which is introduced in the CP on pp.24,25) which matches the notation of the Segal's "Pilot Model for..." (see the references in CP). Volunteers (two, at least, in order to compare independent calculations) welcome ! I will be willing to give my scratch (mentioned above) and/or help with related issues. Results definitely will find their place in the next issue of the CP. I believe that they will lead to certain improvements in presenting the Chronometry to people. This might be your start in the chronometric research !

Lie groups, Lie algebras. Riemannian and Lorentzian spaces. Isometries. Lie algebras of vector fields. Lie algebra of a Lie group. Representations of Lie groups and Lie algebras. Induced vector bundles. Imporant Lie group actions.

All are invited. Topics for original research are provided. Prereqs: MA 242 (Linear Algebra), MA 225 (Multivariate Calculus), MA 226 (Differential Equations).

For more information contact Alex Levichev , 353-1487, BU Dept.of Math. room 236

The purchase of the textbook is not required.

TEXTS USED: R. Sachs, H. Wu "General Relativity for Mathematicians", 1977. S. Kobauashi, K. Nomizu "Foundations of Differential Geometry", 1969. A. Barut, R. Raczka "Representations of Lie groups and Lie algebras", 1977.

It is advised that a ($ 5) BU coursepack be purchased (the content of a similar Seminar which has been run a year ago) directly from A. Levichev .

New Graduate Study Opportunities

Advisors: Mark Kon & Alex Levichev

Examples of PhD and Masters themes which can be chosen:

"Non-canonical Parallelizations of Homogeneous Vector Bundles"

"Conformal Extensions of Arbitrary Spin Fields on Minkowski Space"

"Actions of the Conformal and Poincare Groups in Induced Bundles"

"Causal Structure of (certain) Homogeneous Lorentzian Manifolds"

"A Group-Invariant Imbedding of Two-Dimensional Minkowski Space into the Torus"

"Derivation of Equations (describing certain elementary particles) from Group-Theoretic Considerations"

(including such equations as the Dirac, Klein-Gordon, Maxwell, Rarita-Schwinger equations)

Linear algebra and multivariate calculus are a necessary prerequisite. Other required skills may vary depending on the problem chosen and can be developed while working on the problem itself. They might include topics from functional analysis, differential geometry, Lie algebras and group representation theory, C-* algebras, equations of mathematical physics, harmonic analysis, etc.

Beside their mathematical value, the problems are important in General Relativity and Theoretical Physics. Unsolved mathematical problems (of different levels of complexity) wait for the interested student.

Other related publications and/or references can be provided on request.

Mark Kon, mkon@math.bu.edu Dept. of Math., room 259

Alex Levichev, levit@math.bu.edu Dept. of Math., room 236