Related papers: Linear Algebra Over a Ring
Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is…
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying…
We classify gradings on matrix algebras by a finite abelian group. A grading is called good if all elementary matrices are homogeneous. For cyclic groups, all gradings on a matrix algebra over an algebraically closed field are good. We can…
The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix…
A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant…
The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is…
The aim of this paper is to prove the following result. For any commutative formal group ${\frak F}(x\otimes 1,1\otimes x),$ which is considered as a formal group over $H_\mathbb{Q},$ there exists a homomorphism to a formal group of the…
A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be…
The rings of linear continuous operators on the topological spaces of $\mathfrak{G}$-zero maps were described, where $\mathfrak{G}$ is a filter on a set with an involution. This applies to modules of formal series with well ordered support…
A notion of one-dimensional formal ring is presented. It consists of a triple $(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket…
We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of…
We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix…
The representation ring of an affine algebraic group scheme can be endowed with the structure of a (special) $\lambda$-ring. We show that the same is true for the ring of symmetric representations, i.e. for the Grothendieck-Witt ring of the…
Elements of the Riordan group $\cal R$ over a field $\mathbb F$ of characteristic zero are infinite lower triangular matrices which are defined in terms of pairs of formal power series. We wish to bring to the forefront, as a tool in the…
Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for…
We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space,…
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
The notion of formal Siegel modular forms for an arithmetic subgroup $\Gamma$ of the symplectic group of genus $n$ is a generalization of symmetric formal Fourier-Jacobi series. Assuming an upper bound on the affine covering number of the…
We describe a method for solving linear systems over the localization of a commutative ring $R$ at a multiplicatively closed subset $S$ that works under the following hypotheses: the ring $R$ is coherent, i.e., we can compute finite…
We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…