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Cohomology in algebra, geometry, physicsand statistics
Structure functions and Spencer cohomology in zero and positive characteristics
Speaker’s name:
Pasha Zusmanovich
Speaker’s affiliation:
University of Ostrava
Place:
in IM building, ground floor
Date:
Wednesday, 29. May 2019 - 11:30 to 12:30
Abstract:
Structure functions are obstructions to integrability of G-structures, where G is a Lie group,
on a real or complex manifold. These well known invariants are expressed in terms of Spencer
cohomology of the corresponding graded Lie algebra. Many important particular cases of structure
functions are known under the names of Riemann tensor, Nijenhuis tensor, etc.
Structure functions can be defined, via Spencer cohomology, also over arbitrary fields, including the case of positive characteristic, though immediate
connection with the corresponding geometric situation is then lost.
Nevertheless, these invariants are important in a purely algebraic situation,
namely, structure theory of Lie algebras in positive characteristic.
While vanishing of certain important structure functions is a classical result
due to Serre and others, the positive characteristic case is, somewhat
surprisingly, was not systematically treated in the literature. We present a
unifying approach to computation of the relevant Spencer cohomology.
Our approach clearly demonstrates the difference between zero and
positive characteristics: the (non)vanishing of Spencer cohomology is controlled by
(non)vanishing of coinvariants of the derivatives of the corresponding
polynomial algebras.
on a real or complex manifold. These well known invariants are expressed in terms of Spencer
cohomology of the corresponding graded Lie algebra. Many important particular cases of structure
functions are known under the names of Riemann tensor, Nijenhuis tensor, etc.
Structure functions can be defined, via Spencer cohomology, also over arbitrary fields, including the case of positive characteristic, though immediate
connection with the corresponding geometric situation is then lost.
Nevertheless, these invariants are important in a purely algebraic situation,
namely, structure theory of Lie algebras in positive characteristic.
While vanishing of certain important structure functions is a classical result
due to Serre and others, the positive characteristic case is, somewhat
surprisingly, was not systematically treated in the literature. We present a
unifying approach to computation of the relevant Spencer cohomology.
Our approach clearly demonstrates the difference between zero and
positive characteristics: the (non)vanishing of Spencer cohomology is controlled by
(non)vanishing of coinvariants of the derivatives of the corresponding
polynomial algebras.