Related papers: Exponential factorizations of holomorphic maps

Let $F$ be a $p$-adic field. Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\{GL_n(F)\}_{n=0}^\infty$ taken together, with multiplication defined in the sense of parabolic…

We derive an explicit formula for the holonomy $R$-matrix of quantum $\mathfrak{sl}_2$ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization…

Let $X=X_1 \times X_2$ be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the…

We establish new strong factorization properties for the smooth vectors of representations of exponential solvable Lie groups on Fr\'{e}chet spaces. In particular, our results improve upon the Dixmier-Malliavin factorization theorem for…

We study the matrix factorization problem associated with an SO(2) spinning top by using the algebro-geometric approach. We derive the explicit expressions in terms of Riemann theta functions and discus some related problems including a…

In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when…

We study $2\times 2$ matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group $\mathrm{Sp}_2(\mathcal{A},\sigma)$. We define traces and determinants of such matrices and use them to prove a…

We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with…

Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more…

A ladder structure of operators is presented for the associated Legendre polynomials and the spherical harmonics showing that both belong to the same irreducible representation of so(3,2). As both are also bases of square-integrable…

We analyse the recursive formula found for various Ext groups for $\SL_2(k)$, $k$ a field of characteristic $p$, and derive various generating functions for these groups. We use this to show that the growth rate for the cohomology of…

We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by…

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M}…

Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of…

We (a) prove that continuous morphisms from locally compact groups to locally exponential (possibly infinite-dimensional) Lie groups factor through Lie quotients, recovering a result of Shtern's on factoring norm-continuous representations…

Let $S \subset \mathbb{Z}^{d}$ be a finitely generated subsemigroup. Let $E$ be a product system over $S$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:=\{\alpha_x\}_{x \in…

Let $(S,\mathfrak n)$ be a regular local ring and $f$ a non-zero element of $\mathfrak n^2$. A theorem due to Kn\"orrer states that there are finitely many isomorphism classes of maximal Cohen-Macaulay $R=S/(f)$-modules if and only if the…

Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear…

Let $G$ be a finite group. To every smooth $G$-action on a compact, connected and oriented Riemann surface we can associate its data of singular orbits. The set of such data becomes an Abelian group $B_G$ under the $G$-equivariant connected…

In this paper we prove that a multiplicative quadratic map between a unital ring $K$ and a field $L$ is induced by a homomorphism from $K$ into $L$ or a composition algebra over $L$. Especially we show that if $K$ is a field, then every…