TQFT Club Meeting 29-Jun-2000

12.30 Lunch party meets at Restaurante "Rota do Colombo" (previously called "O Mimo"). All welcome.

Afternoon Session

Room 3.10 Mathematics Department, IST

14.00 - 15.10 Helena Albuquerque (Universidade de Coimbra)

Summary: Depois de fazer uma breve introdução à teoria das álgebras de Hopf apresentam-se as álgebras de Cayley (das quais os exemplos mais conhecidos são os complexos, os quaterniões e os octoniões) como deformações das álgebras de grupo de $Z_2^n\times Z_2^n\times...\times Z_2^n$ na categoria quasi-tensorial das representações de uma quasiHopf algebra associada a um 3-cociclo de grupo. Como exemplo de álgebras quasiassociativas apresentamos o conjunto das matrizes quadradas $n\times n$ com um certo produto deformado. Com este conjunto fazemos uma abordagem breve de alguns resultados da Álgebra quasilinear. Sabendo que uma álgebra quasiassociativa graduada sobre $Z_2$ só pode ser uma associativa ou uma antiassociativa apresentamos ainda como exemplo da teoria anterior a classificação das álgebras antiassociativas com parte par separável.

References:

1. V.G.Drinfeld, "Quasi Hopf Alfgebras" Lenin. Math. Journ. 1, 1419-1457, (1990)
2. S.Majid, "Foundations of Quantum Group Theory" Cambridge University Press (1995)
3. M.E. Sweedler, "Hopf Algebras", Benjamim, (1969)
4. H. Albuquerque e S. Majid, "QuasiAlgebra Structure of the Octonions", Journ. of Algebra, 220,188-224 (1999). math/9801116.
5. H. Albuquerque e S. Majid "New Approach to octonions and Cayley Algebras" L.N.in Pure and Ap.Math. 221, 1-7. math/9810037.
6. H.Albuquerque, A.Elduque e José Perez "On $Z_2$-quasialgebras"-submitted

15.15 - 16.25 Miguel Abreu (Instituto Superior Técnico)

Summary: I will present and discuss some joint results with Dusa McDuff on the rational cohomology ring of the symplectomorphism groups of rational ruled surfaces. I will also show how these results imply the existence of some nontrivial families of symplectic forms, a phenomenon first detected by Kronheimer using Seiberg-Witten invariants.

References:

1. M.Abreu, "Topology of symplectomorphism groups of S^2xS^2", Inv. Math., 131 (1998), 1-23. ps file.
2. M.Abreu and D.McDuff, "Topology of symplectomorphism groups of rational ruled surfaces", IST-preprint (1999), to appear in Jour. Amer. Math. Soc. ps file. .

16.25 Tea & chocolate biscuits!

17.00 - 18.10 Marco Mackaay (Unidade de Ciências Exactas e Humanas, Universidade do Algarve)

Summary: Last time I explained the basics of gerbes and their connections. This time I want to define the gerbe holonomy, H(G,A), for a gerbe, G, with gerbe connection, A, on a simply-connected manifold, M. First I will give the "standard" definition of the gerbe holonomy, H(G,A), as used by Brylinski [2] and Chatterjee [5]. Some easy observations show that H(G,A) defines a smooth group homomorphism from the thin second homotopy group of M, denoted by \pi_2^2(M), which was defined by Caetano and Picken in [4], to S^1. After that I show how to obtain a more concrete formula for $H(G,A)$, which uses a grid on the unit square. With this concrete formula for the gerbe holonomy Roger and I have been able to show that there is a bijective correspondence between equivalence classes of gerbes with connections on M and smooth group homomorphisms from \pi_2^2(M) to S^1. Our guess that there would be such a correspondence was inspired by the analogous results for bundles and connections obtained by Barrett [1] and by Caetano and Picken [3]. Our result for gerbes leads to a better understanding of gerbe holonomy and it answers the main open question in [4] about the geometrical interpretation of group homomorphisms from \pi_2^2(M) to S^1. I plan to explain this result in as much detail as time will allow me.

References:

1. J. Barrett. Holonomy and path structures in general relativity and Yang-Mills theory, Int. J. Theoret. Phys. 30 (1991), 1171-1215.
2. J-L. Brylinski. Loop Spaces, characteristic classes and geometric quantization, Progress in Mathematics 107, Birkhauser, Boston, 1993.
3. A. Caetano and R. Picken. An axiomatic definition of holonomy, Int. J. of Math. 5, No. 6 (1994), 835-848.
4. A. Caetano and R. Picken. On a family of topological invariants similar to homotopy groups, Rend. Ist. Mat. Univ. Trieste 30, No.1-2,81-90, 1998.
5. D. Chatterjee. On gerbs, PhD thesis, Cambridge, 1998.

DATE: Thursday, 29/06/2000

VENUE: Mathematics Department, Room 3.10

URL: http://www.math.ist.utl.pt/~rpicken/tqft