Thursday, December 18, 2014
15:40, IUM, conference hall
Speaker: Mikhail Borovoi (Tel-Aviv University)
Title: Labelings of Dynkin diagrams and Galois cohomology
of simply connected real groups
A labeling of a finite graph D is a family (a_i) of numerical labels a_i, taking values 0 and 1, where i runs over the set of vertices of D. We say that vertices i and j of D are neighbors if they are connected by an edge. We define the elementary transformation T_i on the set of labeling (a_i) as follows: T_i does not change a_j for vertices j different from i, and it adds (modulo 2) to a_i the sum of the labels a_k for all neighbors k of i. We say that two labelings of D are equivalent, if we can get one of them from the other by a finite chain of elementary transformations. In the first part of my talk I shall describe the equivalence classes of labelings for an important class of graphs: Dynkin diagrams.
In the second part of my talk I shall discuss the problem of computing the Galois cohomology set of a simply connected simple real algebraic group. The cardinality of this pointed finite set was computed by Jeffrey Adams in a preprint of 2013, who used results the speaker"s note of 1988. However, the cardinalities only are not sufficient for certain applications. It turns out that if G is a compact, simply connected, simple algebraic group over the field R of real numbers, then its Galois cohomology set is in a canonical bijection with the set of equivalence classes of labelings of the Dynkin diagram D of G. Thus using labelings of Dynkin diagrams we obtain an explicit functorial description of the pointed set H^1(R,G).
Galois cohomology naturally appears in the problem of classification of tensors of given type over R (e.g. pairs of quadratic forms) up to a change of basis. The set of real tensors that are equivalent to a given real tensor t over the field of complex number C is a finite union of equivalence classes over R, and these equivalence classes correspond to the elements of the kernel of the map
H^1(R, H) ---> H^1(R, G),
where H is an R-subgroup of an R-group G. If the groups G and H are simply connected, we can compute this kernel using labelings of Dynkin diagrams.
This is a joint work with Zachi Evenor. No preliminary knowledge of algebraic groups and Galois cohomology is assumed.
The talk will be held in Russian.
| 16.12.2014 | Leonid Petrov |