Classical rings

On Classical Quotient Rings of Skew Armendariz Rings

Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q of R exists, then R is weak α-skew Armendariz if and only if Q is weak α-skew Armendariz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly ci...

Grothendieck Rings of Basic Classical Lie Superalgebras

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids, which we call Weyl groupoids.

Cluster algebras and classical invariant rings

Let V be a k-dimensional complex vector space. The Plücker ring of polynomial SL(V ) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of the Grassmannian Gr(k, n). In 2003, using combinatorial tools developed by A. Postnikov, J. Scott showed that the Plücker ring carries a cluster algebra structure. Over the ensuing decade, this has ...

Differential Polynomial Rings of Triangular Matrix Rings

Rings of Regular Functions on Spherical Nilpotent Orbits for Complex Classical Groups

Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a Gmodule. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. Thesis Supervisor: David Alexander Vogan Title: Professor of Mathematics