**The Dixmier conjecture** in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra (the algebra of polynomial differential operators in $n$ variables) is invertible.

See ["The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture"][1]
Alexei Belov-Kanel, Maxim Kontsevich

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A conjecture (Kontsevich???) which says that the automorphism group
of the Weyl algebra in characteristic zero is canonically isomorphic to the
automorphism group of the corresponding Poisson algebra of classical polynomial symbols.

See ["Automorphisms of the Weyl algebra"][2]
Alexei Belov-Kanel, Maxim Kontsevich

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**Kaplansky's 6th conjecture**: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

See http://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alg

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**Kaplansky zero-divisor conjecture**
Let K be a field and G a group. The so called zero-divisor conjecture for group rings asserts that the group ring K[G] is a domain if and only if G is a torsion-free group.

See
[What is the current status of the Kaplansky zero-divisor conjecture for group rings?][3]

[Zero divisor conjecture and idempotent conjecture][4]



  [1]: http://arxiv.org/abs/math/0512171
  [2]: http://arxiv.org/abs/math/0512169
  [3]: http://mathoverflow.net/questions/79559/what-is-the-current-status-of-the-kaplansky-zero-divisor-conjecture-for-group-ri
  [4]: http://mathoverflow.net/questions/33460/zero-divisor-conjecture-and-idempotent-conjecture