scholarly journals Division algebras of slice functions

2019 ◽  
Vol 150 (4) ◽  
pp. 2055-2082 ◽  
Author(s):  
Riccardo Ghiloni ◽  
Alessandro Perotti ◽  
Caterina Stoppato
Keyword(s):  
Maximum Modulus ◽  
Division Algebras ◽  
Weierstrass Theorem ◽  
Minimum Modulus ◽  
Slice Regular Functions ◽  
Open Mapping Theorem ◽  
Zero Sets ◽  
Regular Functions ◽  
Finite Dimensional

AbstractThis work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

Bohr theorems for slice regular functions over octonions

2020 ◽  
pp. 1-16
Author(s):  
Zhenghua Xu
Keyword(s):  
Unit Ball ◽  
Division Algebras ◽  
Coefficient Estimate ◽  
Coefficient Estimates ◽  
Covering Theorem ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Carathéodory Class

In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions $\mathbb {O}$ . To this end, we establish the coefficient estimates for self-maps of the unit ball of $\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.

scholarly journals Graded-division algebras over arbitrary fields

2020 ◽  
pp. 2140009
Author(s):  
Yuri Bahturin ◽  
Alberto Elduque ◽  
Mikhail Kochetov
Keyword(s):  
Finite Groups ◽  
Closed Field ◽  
Arbitrary Field ◽  
Division Algebras ◽  
Real Numbers ◽  
Arbitrary Group ◽  
Matrix Algebras ◽  
Finite Dimensional ◽  
Group Gradings

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

scholarly journals Classification of the four-dimensional power-commutative real division algebras

2011 ◽  
Vol 141 (6) ◽  
pp. 1207-1223 ◽  
Author(s):  
Erik Darpö ◽  
Abdellatif Rochdi
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Finite Dimensional ◽  
Special Case

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterized.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

Sign in / Sign up

Export Citation Format

Share Document