Groupoids and Lie bigebras in gauge and string theories. (English) Zbl 0661.22011
Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 149-164 (1988).
[For the entire collection see Zbl 0646.00009.]
This is a lecture on two mathematically rather different topics in order to make physicists acquainted with the use of groupoids and algebroids on the one hand and of Hopf and Lie Hopf algebras on the other hand, and to convince them, that these notions have several applications in modern physics as in quantum field theory, gauge theory, and string theory.
From the Introduction: “Part 1 of the lectures deals with some aspects of groupoid theory, as applied to gauge and string physics. It contains a brief review of the necessary mathematics.... Part 2 consists of a review of some of Drinfel’d’s definitions and an outline of the way I hope LHA-s will enter quantum field theory and conformal field theory in the near future.” (With respect to part 2 compare V. G. Drinfel’d [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 18-49 (1986; Zbl 0617.16004) (English version: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987)]\(\}\).
The lecture is organized as follows: 1. Groupoids, Gauge Theory, and Strings. 1.1. Introduction, 1.2 What are groupoids? 1.3 Open (bosonic) string interactions as groupoid operations, 1.4 Lie groupoids and algebroids. Connections and holonomy, 1.5 A groupoid formulation of gauge theory; 2. Lie Bigebras and Quantum groups. 2.1 What are bigebras? 2.2 Lie bigebras and their applications; 3. Outlook. There is a bibliography with selected references to both topics.
This is a lecture on two mathematically rather different topics in order to make physicists acquainted with the use of groupoids and algebroids on the one hand and of Hopf and Lie Hopf algebras on the other hand, and to convince them, that these notions have several applications in modern physics as in quantum field theory, gauge theory, and string theory.
From the Introduction: “Part 1 of the lectures deals with some aspects of groupoid theory, as applied to gauge and string physics. It contains a brief review of the necessary mathematics.... Part 2 consists of a review of some of Drinfel’d’s definitions and an outline of the way I hope LHA-s will enter quantum field theory and conformal field theory in the near future.” (With respect to part 2 compare V. G. Drinfel’d [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 18-49 (1986; Zbl 0617.16004) (English version: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987)]\(\}\).
The lecture is organized as follows: 1. Groupoids, Gauge Theory, and Strings. 1.1. Introduction, 1.2 What are groupoids? 1.3 Open (bosonic) string interactions as groupoid operations, 1.4 Lie groupoids and algebroids. Connections and holonomy, 1.5 A groupoid formulation of gauge theory; 2. Lie Bigebras and Quantum groups. 2.1 What are bigebras? 2.2 Lie bigebras and their applications; 3. Outlook. There is a bibliography with selected references to both topics.
Reviewer: H.Boseck
MSC:
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
| 58D30 | Applications of manifolds of mappings to the sciences |
| 22A30 | Other topological algebraic systems and their representations |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 81V99 | Applications of quantum theory to specific physical systems |
