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EP/J009342/1 - The Conjecture of Dixmier

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Professor V Bavula EP/J009342/1 - The Conjecture of Dixmier

Principal Investigator - Pure Mathematics, University of Sheffield

Scheme

Standard Research

Research Areas

Algebra Algebra

Geometry & Topology Geometry & Topology

Start Date

04/2012

End Date

10/2014

Value

£55,282

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Grant Description

Summary and Description of the grant

In Mathematics there are two old open problems: the Jacobian
Conjecture (open since 1938) for the polynomial algebras
in n variables and the
Conjecture of Dixmier (open since 1968) for the algebras A(n) of
polynomial differential operators, the so-called Weyl algebras,
that claims that
the Weyl algebras behave like the finite fields. More precisely,
every algebra endomorphism of the Weyl algebra is an
automorphism. In 1982, Bass, Connell and Wright proved that the
Conjecture of Dixmier implies the Jacobian Conjecture. In
2005-07, Tsuchimoto, Belov-Kanel and Kontsevich proved that these
two conjectures are equivalent. The Weyl algebra A(n) is a
subalgebra of the algebra I(n) of polynomial integro-differential
operators. At the end of 2010, I proved that an an analogue of the
Conjecture of Dixmier holds for the algebra I(1) (V. Bavula, ``An
analogue of the Conjecture of Dixmier is true for the algebra of
polynomial integro-differential operators,'' Arxiv:math.RA:
1011.3009), and conjectured that the same result is true for all
algebras I(n). The aim of this project is to prove this conjecture
and as a result to have a progress on the Conjecture of Dixmier.
Another goal of the project is to find the K-groups for the
algebras I(n) and to answer the question of whether or not the
Bott periodicity holds. The most interesting (and difficult) is
the case of the K(1)-groups for the algebras I(n) since it leads
to finding explicit generators for the automorphism groups of the
algebras I(n). The groups of automorphisms of the algebras I(n)
are infinite dimensional algebraic groups. Little is known about
their structure in general. In the polynomial case there are
several papers by Shafarevich (1966, 1981) and more recently by
Kambayashi (1996, 2003, 2004). We are going to obtain
generalizations of these results for the Weyl algebras A(n) and
I(n).

Structured Data / Microdata


Grant Event Details:
Name: The Conjecture of Dixmier - EP/J009342/1
Start Date: 2012-04-09T00:00:00+00:00
End Date: 2014-10-08T00:00:00+00:00

Organization: University of Sheffield

Description: In Mathematics there are two old open problems: the Jacobian Conjecture (open since 1938) for the polynomial algebras in n variables and the Conjecture of Dixmier (open since 1968) for the algebras A(n) of polynomial differential operators, the so-called ...