# EP/J009342/1 - The Conjecture of Dixmier

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Principal Investigator - Pure Mathematics, University of Sheffield

Start Date

04/2012

End Date

10/2014

Value

£55,282

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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).

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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 ...