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Globus Seminar
December 18, 2014 15:40, Moscow, IUM (Bolshoi Vlas'evskii per., 11)

Labelings of Dynkin diagrams and Galois cohomology of simply connected real groups

M. V. Borovoi

Tel Aviv University, School of Mathematical Sciences

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Abstract: 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.

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