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Representation fields for commutative orders

Luis Arenas-Carmona (2012)

Annales de l’institut Fourier

A representation field for a non-maximal order ℌ in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of ℌ . Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.

Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.

Friedrich Wehrung (2000)

Publicacions Matemàtiques

We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...

Representation theory of two-dimensionalbrauer graph rings

Wolfgang Rump (2000)

Colloquium Mathematicae

We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of ℤ [ q , q - 1 ] at (p,q-1) for some rational prime p . For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction)...

Representation-directed algebras form an open scheme

Stanislaw Kasjan (2002)

Colloquium Mathematicae

We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.

Representation-finite triangular algebras form an open scheme

Stanisław Kasjan (2003)

Open Mathematics

Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

Representations of a class of positively based algebras

Shiyu Lin, Shilin Yang (2023)

Czechoslovak Mathematical Journal

We investigate the representation theory of the positively based algebra A m , d , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that A m , d is of finite representative type if d ≤ 4 , of tame type if d = 5 , and of wild type if d ≥ 6 . In the case when d ≤ 4 , all indecomposable representations of A m , d are constructed. Furthermore, their right cell representations as well as left cell representations of A m , d are described.

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