Dieudonne Determinant Research Articles

RE-algebras, quasi-determinants and the full Toda system

In 1991, Gelfand and Retakh embodied the idea of a noncommutative Dieudonne determinant for a generating matrix of RTT algebra, namely, they found a representation of the quantum determinant of RTT algebra in the form of a product of principal quasi-determinants. In this note we construct an analogue of the above statement for the RE-algebra corresponding to the Drinfeld R-matrix for the order n=2,3. Namely, we have found a family of quasi-determinants that are principal with respect to the antidiagonal, commuting among themselves, whose product turns out to be the quantum determinant of this algebra. This family generalizes the construction of integrals of the full Toda system due to Deift et al. for the quantum case of RE-algebras. In our opinion, this result also clarifies the role of RE-algebras as a quantum homogeneous spaces and can be used to construct effective quantum field theories with a boundary.

Computing the Degree of Determinants via Discrete Convex Optimization on Euclidean Buildings

In this paper, we consider the computation of the degree of the Dieudonne determinant of a linear symbolic matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$, where each $A_i$ is an $n \times n$ polynom...

On subgroups in the special linear group over a division algebra that contain the subgroup of diagonal matrices

For an arbitrary division algebra T we study the arrangement of subgroups of the special linear group Γ = SL n ( T)( n ≥ 3) that contain the subgroup Δ = SD n ( T) of diagonal matrices with Dieudonne's determinant (see [1]) equal to 1. We show that the description of these subgroups is standard in the following sense: For any subgroup H, Δ ≤ H ≤ Γ there exists a unique D-net σ such that Γ( σ) ≤ H ≤ N Γ ( σ), where Γ(σ) is the D-net subgroup corresponding to the net σ and N Γ ( σ) is the normalizer of Γ(σ) in Γ.

Abelian categories in which linear algebra is possible

We determine a class of abelian categories, properly containing the class of all categories vect-k of finite dimensional right vector spaces over a division ring k, in which all the elementary results of linear algebra over division rings can be proved. The ideas we present follow the program of von Neumann's Continuous Geometry, essentially introducing category theory and the Dieudonne determinant.

Ordering Epic R-fields

We are concerned with the following problem. Given a ring R and an epic R-field K, under what conditions can K be fully ordered? Epic R-fields can be constructed in terms of matrices over R; this makes it natural to consider matrix cones over R rather than ordinary cones of elements of K. Essentially, a matrix cone over R, associated with a given ordering of K, consists of all square matrices which either become singular or have positive Dieudonne determinant over K. We give necessary and sufficient conditions in terms of matrix cones for (i) an epic R-field to be orderable, (ii) a full order on R to be extendable to a field of fractions of R and (iii) for such an extension to be unique.