A construction of dual box

Let R be a quasi-hereditary algebra, F(∆) and F(∇) its categories of good and cogood modules correspondingly. In [6] these categories were characterized as the categories of representations of some boxes A = A∆ and A∇. These last are the box theory counterparts of Ringel duality ([8]). We present an implicit construction of the box B such that B − mo is equivalent to F(∇)

Graded commutative algebras: examples, classification, open problems

We consider \G-graded commutative algebras, where \G is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on the subject. We then give a recent classification result and formulate an open problem

On the Classification of Automorphic Lie Algebras

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2-Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.Comment: 29 pages, 1 diagram, 9 tables, standard LaTeX2e, submitted for publicatio

On Gelfand-Zetlin modules

summary:[For the entire collection see Zbl 0742.00067.]\par Let {\germ g}\sb k be the Lie algebra {\germ gl}(k,\mathcal{C}), and let U\sb k be the universal enveloping algebra for {\germ g}\sb k. Let Z\sb k be the center of U\sb k. The authors consider the chain of Lie algebras {\germ g}\sb n\supset {\germ g}\sb{n-1}\supset\dots\supset {\germ g}\sb 1. Then Z=\langle Z\sb k\mid k=1,2,\dots n\rangle is an associative algebra which is called the Gel'fand-Zetlin subalgebra of U\sb n. A {\germ g}\sb n module $V$ is called a $GZ$-module if V=\sum\sb x\oplus V(x), where the summation is over the space of characters of $Z$ and V(x)=\{v\in V\mid(a-x(a))\sp mv=0, m\in\mathcal{Z}\sb +, $a\in\mathcal{Z}\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V(x)=0$ for some $x$ and the module $V$ is simple, then $V$ is a $GZ$-module. Indecomposable $GZ$- modules are also described. The authors give three conjectures on $GZ$- modules and