algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

of a

Weyl algebra In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. ...

is an

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

. Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the

Jacobian conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero con ...

References

Abstract algebra Conjectures Unsolved problems in mathematics {{abstract-algebra-stub