Minimal Ideals Research Articles

Extremal elements in Lie algebras, buildings and structurable algebras

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element x∈g such that [x,[x,g]] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g, i.e. a subspace I satisfying [I,[I,g]]≤I. We show that in characteristic different from 2,3 the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set.This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from 2,3 each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra g and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.

Two-State Quantum Systems Revisited: A Clifford Algebra Approach

We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions $\mathcal G_3$. In this description, both the quantum states and Hermitian operators are written as elements of $\mathcal G_3$. By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in GA. We then use this approach to revisit the problem of a spin-$1/2$ particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of $\mathcal G_3$.

Hom-Lie Algebras with a Set Grading

In this paper we study the regular Hom-Lie algebra ${\mathscr{L}}$ graded by an arbitrary set $\mathcal {S}$ (set grading). We show that ${\mathscr{L}}$ decomposes as ${\mathscr{L}}=\mathcal {U}\oplus {\sum }_{[\lambda ]\in ({\varLambda }_{\mathcal {S}}\setminus \{0\})/\sim }{\mathscr{L}}_{[\lambda ]}$ , where $\mathcal {U}$ is a linear complement of ${\sum }_{[\lambda ]\in ({\varLambda }_{\mathcal {S}}\setminus \{0\})/\sim }{\mathscr{L}}_{0,[\lambda ]}$ in ${\mathscr{L}}_{0}$ and any ${\mathscr{L}}_{[\lambda ]}$ a well-described graded ideals of ${\mathscr{L}}$ , satisfying $[{\mathscr{L}}_{[\lambda ]}, {\mathscr{L}}_{[\mu ]}]=0$ if [λ]≠[μ]. Under certain conditions, the simplicity of ${\mathscr{L}}$ is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals, each one being a simple set-graded regular Hom-Lie algebras.

Relative Ideals in Ordered Semigroups

In this paper, after introducing the notion of relative ideals in ordered semigroups, some important properties of these ideals are studied. Then notions of relatively prime and relatively weakly prime ideals are defined and some vital results have been proved. We also find some conditions under which relative ideals become left, right and two-sided ideals in any regular ordered semigroup. Finally, the notion of maximal and minimal relative ideals are introduced and some of their properties are studied.

Complete regularity of Ellis semigroups of -actions

AbstractIt is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

A general formulation based on algebraic spinors for the quantum computation

In this work, we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially, we present a general formulation through elements of minimal left ideals in tensor products of Clifford algebras. Posteriorly, we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages compared to standard Hilbert space formulation.

A topological model of composite preons from the minimal ideals of two Clifford algebras

This paper demonstrates a direct correspondence between a recent algebraic characterization of leptons and quarks as basis elements of the minimal one-sided ideals of the complex Clifford algebras Cℓ(6) and Cℓ(4), shown earlier to transform as a single generation of leptons and quarks under the Standard Model's unbroken SU(3)c×U(1)em and SU(2)L gauge symmetries respectively, and a topological formulation of the Harari-Shupe preon model in which leptons and quarks are represented in terms of braids.It was previously shown that mapping a Witt basis of Cℓ(6) to particular braids in the circular Artin braid group B3c makes it possible to replicate the topological structure describing electrocolor symmetries in this preon model. This paper extends this curious correspondence, which involves only the minimal ideals of Cℓ(6) under SU(3)c×U(1)em, to include the SU(2)L chiral weak symmetry. This is achieved by mapping a Witt basis of an additional Cℓ(4) algebra to braids in B3, taken to be a subgroup of B3c. The braids corresponding to the charged vector bosons are determined, and it is demonstrated that chiral weak interactions can be described via the composition of braids.

2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

AbstractLet${\mathcal{A}}$be a semisimple Banach algebra with minimal left ideals and$\text{soc}({\mathcal{A}})$be the socle of${\mathcal{A}}$. We prove that if$\text{soc}({\mathcal{A}})$is an essential ideal of${\mathcal{A}}$, then every 2-local derivation on${\mathcal{A}}$is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and${\mathcal{J}}$-subspace lattice algebras, is a derivation.

Regular Hom-algebras admitting a multiplicative basis

Abstract Let ( ℌ , μ , α ) {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽 {{\mathbb{F}}} . A basis ℬ = { e i } i ∈ I {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of ℌ {{\mathfrak{H}}} is called multiplicative if for any i , j ∈ I {i,j\in I} , we have that μ ⁢ ( e i , e j ) ∈ 𝔽 ⁢ e k {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α ⁢ ( e i ) ∈ 𝔽 ⁢ e p {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some k , p ∈ I {k,p\in I} . We show that if ℌ {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum ℌ = ⊕ r ℑ r {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of ℌ {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case ℬ {{\mathcal{B}}} , in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra mathbb {C}ell (6) and two minimal right ideals of mathbb {C}ell (4) transform as one generation of leptons and quarks under the gauge symmetry SU(3)_Ctimes U(1)_{EM} and SU(2)_L respectively. The SU(2)_L weak symmetries are naturally chiral. Combining the mathbb {C}ell (6) and mathbb {C}ell (4) ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented. The combined ideals can be written as minimal left ideals of mathbb {C}ell (6)otimes mathbb {C}ell (4)cong mathbb {C}ell (10) in a way that preserves individually the mathbb {C}ell (6) structure and mathbb {C}ell (4) structure of physical states. This resulting model captures many of the attractive features of the Georgi and Glashow SU(5) Grand Unified Theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not preserve the underlying algebraic structure.

Rings in which idempotents generate maximal or minimal ideals

We characterize rings in which every left ideal generated by an idempotent different from 0 and 1 is either a maximal left ideal or a minimal left ideal. In the commutative case, we give a characterization in terms of topological properties of the maximal spectrum with the Zariski topology. We also consider a strictly weaker variant of this property, defined almost similarly, and characterize those rings that have the property in question.

Lie superalgebras with a set grading

This paper studies Lie superalgebras graded by an arbitrary set [Formula: see text] (set grading). We show that the set-graded Lie superalgebra [Formula: see text] decomposes as the sum of well-described set-graded ideals plus a certain linear subspace. Under certain conditions, the simplicity of [Formula: see text] is characterized and it is shown that the above decomposition is exactly the direct sum of the family of its minimal set-graded ideals, each one being a simple set-graded Lie superalgebra.

Glorious pairs of roots and Abelian ideals of a Borel subalgebra

Let $${{\mathfrak {g}}}$$ be a simple Lie algebra with a Borel subalgebra $${{\mathfrak {b}}}$$ . Let $$\Delta ^+$$ be the corresponding (po)set of positive roots and $$\theta $$ the highest root. A pair $$\{\eta ,\eta '\}\subset \Delta ^+$$ is said to be glorious, if $$\eta ,\eta '$$ are incomparable and $$\eta +\eta '=\theta $$ . Using the theory of abelian ideals of $${{\mathfrak {b}}}$$ , we (1) establish a relationship of $$\eta ,\eta '$$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types $${{\mathbf {\mathsf{{{DE}}}}}}_{}$$ , we prove that if $$\{\eta ,\eta '\}$$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $$\eta \wedge \eta '$$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type $${{\mathbf {\mathsf{{{A}}}}}}_{}$$ . As an application, we describe the minimal non-abelian ideals of $${{\mathfrak {b}}}$$ .

Boundedness and absoluteness of some dynamical invariants in model theory

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

Analysis on semihypergroups: function spaces, homomorphisms and ideals

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by Dunkl (Am Math Soc 179:331–348, 1973), Jewett (Adv Math 18(1):1–101, 1975) and Spector (Apercu de la theorie des hypergroups, (French) Analyse harmonique sur les groupes de Lie (Sem. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups.

Leibniz triple systems admitting a multiplicative basis

Let T be an arbitrary Leibniz triple system over an arbitrary field of scalars . A basis of T is multiplicative if for we have for some . We show that if T admits a multiplicative basis then it decomposes as the orthogonal direct sum of well-described ideals admitting each one a multiplicative basis. Also the minimality of T is characterized in terms of the multiplicative basis and under new conditions we prove that the above direct sum is by means of its minimal ideals.

Minimal Ideals and Primitivity in Near-rings

We address and answer the question when a minimal ideal of a zero symmetric near-ring is a primitive near-ring. This implies that a minimal ideal of a zero symmetric near-ring is a simple near-ring in many natural situations.

ON LEAVITT PATH ALGEBRAS OVER COMMUTATIVE RINGS

In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.

Three fermion generations with two unbroken gauge symmetries from the complex sedenions

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)_ctimes U(1)_{em} can be described using the algebra of complexified sedenions {mathbb {C}}otimes {mathbb {S}}. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of {mathbb {C}}otimes {mathbb {S}} can be used to uniquely split the algebra into three complex octonion subalgebras {mathbb {C}}otimes {mathbb {O}}. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 {mathbb {C}}-dimensional {mathbb {C}}otimes {mathbb {O}} subalgebras on themselves generates three copies of the Clifford algebra {mathbb {C}}ell (6). It was previously shown that the minimal left ideals of {mathbb {C}}ell (6) describe a single generation of fermions with unbroken SU(3)_ctimes U(1)_{em} gauge symmetry. Extending this construction from {mathbb {C}}otimes {mathbb {O}} to {mathbb {C}}otimes {mathbb {S}} naturally leads to a description of exactly three generations.

Braided fermions from Hurwitz algebras

Some curious structural similarities between a recent braid - and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups that can be represented using the four normed division algebras are B2 and , exactly those required to represent a single generation of fermions in terms of simple three strand ribbon braids. These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra Cℓ(6), generated from the nested left actions of the complex octonions on itself. That is, the ribbon spectrum can be related to octonion algebras. Some speculative ideas relating to ongoing research that attempts to construct a unified theory based on braid groups and Hurwitz algebras are discussed.