Symmetric Functions, Conjugacy Classes and the Flag Variety.
The almost isomorphism relation for simple regular rings.
A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K. C. O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring.
The Automorphism Group of the Free Algebra of Rank Two
The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group...
The automorphism group of the rank 2 Lie algebra free in the variety .
The Boolean algebra and central Galois algebras.
The Boolean algebra of Galois algebras.
The coefficient ring of the skew group ring
The existence of envelopes
The existence of Galois sets
The Galois algebras and the Azumaya Galois extensions.
The Galois extensions induced by idempotents in a Galois algebra.
The general Ikehata theorem for -separable crossed products.
The group of automorphisms of a commutative algebra.
The group of automorphisms of the coordinate ring of quantum symplectic space.
The intersection of distinct Galois subrings is not necessarily Galois
The invariant subrings of DeMeyer-Kanzaki Galois extensions.
The normal elements of an algebra of general matrices are central.
The order of algebras with nontrivial fixed point subalgebras.
The Strong Anick Conjecture is true
Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra over a field of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of . In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a...
Twisted Hopf Algebras.