Hiromasa Nakayama
We give an algorithm to compute the local $b$ function. In this algorithm, we use the Mora division algorithm in the ring of differential operators and an approximate division algorithm in the ring of differential operators with power series coefficient.
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Let T be a complete discrete valuation ring and $\hat{X}$ a smooth projective curve over $S=\spec(T)$ with closed fibre $X$. Denote by $F$ the function field of $\hat{X}$ and by $\hat{F}$ the completion of $F$ with respect to the discrete valuation defined by $X$, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over $F$. This is done by defining an index preserving group homomorphism $s:\br(\hat{F})'\to\br(F)'$, and using it to lift indecomposable and noncrossed product division algebras over $\hat{F}$.
We construct indecomposable and noncrossed product division algebras over function fields of smooth curves X over Z_p. This is done by defining an index preserving morphism s:Br(\hat K(X))' -> Br(K(X))' which splits res:Br(K(X)) -> Br(\hat K(X)), where \hat K(X) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over \hat K(X).
Let $(K, v)$ be a Henselian valued field satisfying the following conditions, for a given prime number $p$: (i) central division $K$-algebras of (finite) $p$-primary dimensions have Schur indices equal to their exponents; (ii) the value group $v(K)$ properly includes its subgroup $pv(K)$. The paper shows that if $\hat K$ is the residue field of $(K, v)$ and $\hat R$ is an intermediate field of the maximal $p$-extension $\hat K (p)/\hat K$, then the natural homomorphism Br$(\hat K) \to $ Br$(\hat R)$ of Brauer groups maps surjectively the $p$-component Br$(\hat K)_{p}$ on Br$(\hat R)_{p}$. It p…
Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the $\mathbb{F}$-vector space $M_{n,p}(\mathbb{D})$ in which every matrix has rank less than or equal to $r$. Using a new method, we prove that $\dim_{\mathbb{F}} \mathcal{V} \leq \max(n,p)\,rd$ and we characterize the spaces for which equality holds. This extends a famous theorem of Flanders which was known only for fields.
The concept of an $I$-matrix in the full $2\times 2$ matrix ring $M_2(R/I)$, where $R$ is an arbitrary UFD and $I$ is a nonzero ideal in $R$, was introduced in \cite{mar}. Moreover a concrete description of the centralizer of an $I$-matrix $\hat B$ in $M_2(R/I)$ as the sum of two subrings $\mathcal S_1$ and $\mathcal S_2$ of $M_2(R/I)$ was also given, where $\mathcal S_1$ is the image (under the natural epimorphism from $M_2(R)$ to $M_2(R/I)$) of the centralizer in $M_2(R)$ of a pre-image of $\hat B$, and where the entries in $\mathcal S_2$ are intersections of certain annihilators of elements…
We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^σ$ where $\hat{\mathcal{H}}^σ$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^σ-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat…
Given a ring $R$, the notion of Sylvester rank function was conceived within the context of Cohn's classification theory of epic division $R$-rings. In this paper we study and describe the space of Sylvester rank functions on certain families of rings, including Dedekind domains, simple left noetherian rings and skew Laurent polynomial rings $\mathcal{D}[t^{\pm 1};τ]$ for any division ring $\mathcal{D}$ and any automorphism $τ$ of $\mathcal{D}$.
Our aim in this thesis is to use the language of deformation-quantization to understand certain quantized algebras by looking at properties of the corresponding commutative ones, and conversely to obtain results about the commutative algebras (upon which a Poisson structure is induced) using existing results for the non-commutative ones. We consider two main cases: firstly, the division ring of fractions of the quantum plane, which we view as a deformation of the commutative field of rational functions in two variables with respect to the bracket $\{x,y\} = xy$, and secondly, quantum matrices …
In this paper, we continue our study of the tensor product structure of category $\mathcal W$ of weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $χ$ and $χ(a)$ are different. Then we describe the Green ring $r(\mathcal W)$ of the tensor category $\mathcal W$. It is shown that $r(\mathcal W)$ is isomorphic to the polynomial algebra over the group ring $\math…
Partially Alternative Real Division Algebras With A Few Imaginary Units
本文扩展了对四维部分交替代数的研究，在至少三个不同虚数单位和一个反射条件下，完全分类了四维部分交替实可除代数的同构类，发现存在无穷多类，并刻画了其自同构群。
The concept of an $I$-matrix in the full $2\times 2$ matrix ring $M_2(R/I)$, where $R$ is an arbitrary UFD and $I$ is a nonzero ideal in $R$, is introduced. We obtain a concrete description of the centralizer of an $I$-matrix $\hat B$ in $M_2(R/I)$ as the sum of two subrings $\mathcal S_1$ and $\mathcal S_2$ of $M_2(R/I)$, where $\mathcal S_1$ is the image (under the natural epimorphism from $M_2(R)$ to $M_2(R/I)$) of the centralizer in $M_2(R)$ of a pre-image of $\hat B$, and where the entries in $\mathcal S_2$ are intersections of certain annihilators of elements arising from the entries of …
Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(α, β)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(α, β)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensio…
Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is generated by elements of small height once $S$ contains an explicit finite set of places of $k$. This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when $B = k$. Our height bound is an explicit function of the number field and the discriminant of a maximal order in $B$ used to define its $S$-units.
Let $C$ denote a closed convex cone $C$ in $\mathbb{R}^d$ with apex at 0. We denote by $\mathcal{E}'(C)$ the set of distributions having compact support which is contained in $C$. Then $\mathcal{E}'(C)$ is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on $\hat{f}_1,..., \hat{f}_n$ for $f_1,...,f_n \in \mathcal{E}'(C)$ to generate the ring $\mathcal{E}'(C)$. (Here $\hat{\cdot}$ denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Hörmander. En…
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