Extension Of Algebra Research Articles

Symmetry and reduction for second order degenerate Lagrangians

For the chiral oscillator described by a second order and degenerate Lagrangian with special Euclidean group of symmetries, we show, by cotangent bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of oscillator algebra, the central extension of special Euclidean algebra in two dimensions.

Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L, we are interested in finding the pairs (ϕ,ψ)∈Aut(V)×Aut(L), which are inducible by an automorphism in Aut(L˜). We connect the inducibility problem to the (2,3)-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut(V)×Aut(L) to be inducible lies in the (2,3)-cohomology group H(2,3)(L,V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2,3)-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2.

Pseudo-differential extension for graded nilpotent Lie groups

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group G form a ^* -subalgebra of the bounded operators on L^2(G) . We show that its \mathrm{C}^* -closure is an extension of a noncommutative algebra of principal symbols by compact operators. As a new approach, we use the generalized fixed point algebra of an \mathbb{\R}_{>0} -action on a certain ideal in the \mathrm{C}^* -algebra of the tangent groupoid of G . The action takes the graded structure of G into account. Our construction allows to compute the \mathrm{K} -theory of the algebra of symbols.

On compatible Leibniz algebras

In this paper, we study compatible Leibniz algebras. We explore the classification of complex Leibniz algebras in dimensions 2 and 3 to provide compatible pairs. We characterize compatible Leibniz algebras in terms of Maurer–Cartan elements of a suitable differential graded Lie algebra. Moreover, we define a cohomology theory of compatible Leibniz algebras, which in particular controls one-parameter formal deformations of this algebraic structure. Motivated by a classical application of cohomology, we moreover study abelian extensions of compatible Leibniz algebras.

Nijenhuis operators on Leibniz algebras

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras.

Possible ambient kinematics

In a seminal paper, Bacry and Lévy–Leblond classified kinematical algebras, a class of Lie algebras encoding the symmetries of spacetime. Homogeneous spacetimes (infinitesimally, Klein pairs) associated with these possible kinematics can be partitioned into four families—riemannian, lorentzian, galilean, and carrollian—based on the type of invariant metric structure they admit. In this work, we classify possible ambient kinematics—defined as extensions of kinematical algebras by a scalar ideal—as well as their associated Klein pairs. Kinematical Klein pairs arising as quotient space along the extra scalar ideal are said to admit a lift into the corresponding ambient Klein pair. While all non-galilean Klein pairs admit a unique—trivial and torsionfree—higher-dimensional lift, galilean Klein pairs are constructively shown to admit lifts into two distinct families of ambient Klein pairs. The first family includes the bargmann algebra as well as its curved/torsional avatars while the second family is novel and generically allows lifts into torsional ambient spaces. We further comment on the relation between these two families and the maximally symmetric family of leibnizian Klein pairs.

On the representability of actions of Leibniz algebras and Poisson algebras

AbstractIn a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

Totally real algebraic integers in short intervals, Jacobi polynomials, and unicritical families in arithmetic dynamics

AbstractWe classify all postcritically finite unicritical polynomials defined over the maximal totally real algebraic extension of . Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion that allows one to give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than 4.

Flux vacua and modularity for $\mathbb{Z}_2$ symmetric Calabi-Yau manifolds

We find continuous families of supersymmetric flux vacua in IIB Calabi-Yau compactifications for multiparameter manifolds with an appropriate \mathbb{Z}_{2}ℤ2 symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weight-two modular forms, these manifolds being said to be weight-two modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to F-theory on elliptically fibred Calabi-Yau fourfolds. If conjectural expressions for Deligne’s periods are true, then these imply that the F-theory fibre is complex-isomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in \mathbb{Q}ℚ but also in algebraic extensions of \mathbb{Q}ℚ, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.

Characterizations of Lie n -Centralizers on Certain Trivial Extension Algebras

In this paper, we describe the structure of Lie n -centralizers of a trivial extension algebra. We then present some conditions under which a Lie n -centralizer on a trivial extension algebra is proper. As an application, we consider Lie n -centralizers on a triangular algebra.

A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF

AbstractFor any subset $Z \subseteq {\mathbb {Q}}$ , consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$ . Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ . Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $ -definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

On Amitsur cohomology of monads

We introduce and study zero and one-dimensional non-abelian Amitsur cohomology theory for monads equipped with involutive distributive law satisfying the Yang–Baxter equation. Some applications to extensions of (non-commutative) rings and Banach algebras are given.

Algebraic extensions of some results by Yang

By means of differential geometry arguments Yang showed that for every unital real division algebra A of dimension > 1 the square map x ↦ x 2 is onto and, consequently, A contains a subalgebra isomorphic to C . The first assertion is extended algebraically for an algebra with only a non-zero flexible element. It persists if the unit is replaced by a non-zero central element. Next, always by algebraic approach, we give examples of both four-dimensional and eight-dimensional real division algebras A with left-unit e containing no two-dimensional subalgebras with the additional property R e 2 = I A .

Pura Vida Neutrosophic Algebra

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors.

New perspectives on semi-primal varieties

We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.

Normal Extensions

Summary In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E. We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F[T] = {p(a 1, . . . an ) | p ∈ F[X], ai ∈ T} and F(T) = F[T] for finite algebraic T ⊆ E. We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).

ON THE ESSENTIAL TORSION FINITENESS OF ABELIAN VARIETIES OVER TORSION FIELDS

AbstractThe classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.

Pseudo finite fields with the Laurent property

We construct an algebraic extension F of Q which is pseudo finite and has the “Laurent property”. In addition, F has an extension F * which is a non-principal ultraproduct of distinct finite fields (so F * is pseudo finite), F * has the Laurent property, and F is the algebraic part of F * .

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

Existential definability and diophantine stability

Let K be a number field, let L be an algebraic (possibly infinite degree) extension of K, and let OK ⊂ OL be their rings of integers. Suppose A is an abelian variety defined over K such that A(K) is infinite and A(L)/A(K) is a torsion group. If at least one of the following conditions is satisfied:1.L is a number field,2.L is totally real,3.L is a quadratic extension of a totally real field, then OK has a diophantine definition over OL.