Related papers: The algebra of Boolean matrices, correspondence fu…
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized…
Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by non-principal ideals yields simple…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad. The associated notion of morphism of left modules ("Linear" natural transformations) captures an important…
We establish a correspondence between modules and spans of algebras within a general monoidal 2-category $\mathfrak{C}$. Specifically, for an algebra $A$ in $\mathfrak{C}$, we construct a normalized lax 3-functor from the 2-category of…
Intuitively, the filter dimension of an algebra or a module measures how `close' standard filtrations of the algebra or the module are. In particular, for a simple algebra it also measures the growth of how `fast' one can prove that the…
Let $K$ be a field. We attach to each finite poset $\mathbb P$ a von Neumann regular $K$-algebra $Q_K(\mathbb P)$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $Q_K(\mathbb P)$-modules…
The goal of the paper is to relate complexity measures associated with the evaluation of Boolean functions (certificate complexity, decision tree complexity) and learning dimensions used to characterize exact learning (teaching dimension,…
A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The…
We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious…
This is the second paper in a series of papers aimed at providing a geometric construction of modular functors and topological quantum field theories from conformal field theory building on the constructions in [TUY] and [KNTY]. We give a…
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof…
Let g be a complex simple Lie algebra and let V be a finite dimensional U(g) module. A relative Yangian is defined with respect to this pair. According to recent work of Khoroshkin and Nazarov the finite dimensional simple modules of the…
We provide a unified framework of Mahler measure, dessins d'enfants, and gauge theory. With certain physically motivated Newton polynomials from reflexive polygons, the Mahler measure and the dessin are in one-to-one correspondence. From…
Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the…
We study finiteness properties, especially the noetherian property, the Krull dimension and a variation of finite presentation, in categories of polynomial functors from a small symmetric monoidal category whose unit is an initial object to…
We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study…
We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we…
We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up…