Split Algebra Research Articles

On split regular Hom-Leibniz algebras

We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra [Formula: see text] is of the form [Formula: see text] with [Formula: see text] a subspace of the abelian subalgebra [Formula: see text] and any [Formula: see text], a well described ideal of [Formula: see text], satisfying [Formula: see text] if [Formula: see text]. Under certain conditions, in the case of [Formula: see text] being of maximal length, the simplicity of the algebra is characterized.

О ПОЧТИ (ПАРА) КОМПЛЕКСНЫХ СТРУКТУРАХ КЭЛИ НА СФЕРАХ S2,4 И S3,3

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ,V) and we describe R(Λ,V) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.

The Hyperbolic Spinor Representation of Transformations in $$\mathbb {R}_1^3$$ R 1 3 by Means of Split Quaternions

In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space $$\mathbb {R}_1^3$$ . In addition, we obtain an automorphism of the split quaternion algebra $$H'$$ corresponding to a rotation in $$\mathbb {R}_1^3$$ . Then, we give the relationship between the hyperbolic spinors and rotations in $$\mathbb {R}_1^3$$ . Finally, we associate to a split quaternion with a hyperbolic spinor by means of a transformation. In this way, we show that the rotation of a rigid body in the Minkowski 3-space $$\mathbb {R}_1^3$$ expressed the split quaternions can be written by means of the hyperbolic spinors with two hyperbolic components. So, we obtain a new and short representation (hyperbolic spinor representation) of transformation in the 3-dimensional Minkowski space $$\mathbb {R}_1^3$$ expressed by means of split quaternions.

On split regular Hom-Lie superalgebras

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra L is of the form L = U + ∑ j I j with U a linear subspace of a maximal abelian graded subalgebra H and any I j a well described (split) ideal of L satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.

The Cyclic Universes Model Based on the Split Division Algebras

The proposed cyclic universes model based on the split division algebras accounts for the inflation, the Big Bang, gravity, dark energy, dark matter, the standard model, and the masses of all elementary particles. The split algebras (complex quaternion and complex octonion) as the Furey model generate the fixed spacetime dimension number for the observable universe with the fixed 4-dimensional spacetime (4D) standard model particles and the oscillating spacetime dimension number for the oscillating universes (hidden or dark energy) with the oscillation between 11D and 11D through 10D and between 10D and 10D through 4D. 11D has the lowest rest mass, the highest speed of light, and the highest vacuum energy, while 4D has the highest rest mass, the lowest (observed) speed of light, and zero vacuum energy. In the cyclic universes model, the universes start with the positive-energy and the negative-energy 11D membrane-antimembrane dual universes from the zero-energy inter-universal void, and are followed by the transformation of the 11D membrane-antimembrane dual universes into the 10D string-antistring dual universes and the external dual gravities as in the Randall-Sundrum model, resulting in the four equal and separate universes consisting of the positive-energy 10D universe, the positive-energy external gravity, the negative-energy 10D universe, and the negative-energy external gravity. Under the fixed spacetime dimension number, the positive-energy 10D universe is transformed into 4D standard model particles through the inflation and the Big Bang. Dark matter is the right-handed neutrino, exactly five times of baryonic matter in total mass in the universe. Under the oscillating spacetime dimension number, the other three universes oscillate between 10D and 10D through 4D, resulting in the hidden universes when D > 4 and dark energy (the maximum dark energy = 3/4 = 75%) when D = 4. Eventually, all four universes return to the 10D universes.

Clifford parallelisms defined by octonions

We define (left and right) Clifford parallelisms on a seven-dimensional projective space algebraically, using an octonion division algebra. Thus, we generalize the two well-known Clifford parallelisms on a three-dimensional projective space, obtained from a quaternion division algebra. We determine (for both the octonion and quaternion case) the automorphism groups of these parallelisms. A geometric description of the parallel classes is given with the help of a hyperbolic quadric in a Baer superspace, obtained from the split octonion algebra over a quadratic extension of the ground field, again generalizing results that are known for the quaternion case. In contrast to the quaternion case, the orbits of the two Clifford parallelisms under the group of direct similitudes of the norm form of the algebra are non-trivial in the octonion case. The two spaces of parallelisms can be seen as the point sets of two point-line geometries, both isomorphic to the seven-dimensional projective space. Together with the original space, we thus have three versions of this projective space. We introduce a triality between them which is closely related to the triality of the polar space of split type $$\mathrm {D}_4$$ .

On an Intrinsically Local Gauge Symmetric $$\mathbf {SU(3)}$$ SU ( 3 ) Field Theory for Quantum Chromodynamics

The $$SU(3)_{c}$$ gauge theory for eight massless vector gauge fields is assimilated into the formalism which generated the intrinsic local gauge invariant U(1) and $$SU(2)_{L}$$ theories [30, 31], thus establishing the theoretical structure for intrinsically local gauge invariant quantum chromodynamics. Use is made of both the standard and split algebra of the octonions, the last of the Hurwitz algebras, in devising the technology. Numerous interesting results are obtained, including an explanation of the inherently non-chiral nature of the strong and electromagnetic interactions in contradistinction to the weak interaction. The formalism’s novelty and universality compels contemplation of its potential ability to assimilate the gravitational interaction as well.

Nonsupersymmetric magic theories and Ehlers truncations

We consider the nonsupersymmetric “magic” theories based on the split quaternion and the split complex division algebras. We show that these theories arise as “Ehlers” [Formula: see text] and [Formula: see text] truncations of the maximal supergravity theory, exploiting techniques related to the very-extended Kac–Moody algebras. We also generalize the procedure to other [Formula: see text] truncations, resulting in additional classes of nonsupersymmetric theories, as well as to truncations of nonmaximal theories. Finally, we discuss duality orbits of extremal black hole solutions in some of these nonsupersymmetric theories.

Orthogonal involutions and totally singular quadratic forms in characteristic two

We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.

Split 3-Leibniz algebras

We introduce and describe the class of split 3-Leibniz algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie triple systems and split 3-Lie algebras. More precisely, we show that any of such split 3-Leibniz algebras T is of the form T=U+∑jIj, with U a subspace of the 0-root space T0, and Ij an ideal of T satisfying [T,Ij,Ik]+[Ij,T,Ik]+[Ij,Ik,T]=0 for j≠k. Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.

On split regular Hom-Lie color algebras

We introduce the class of split regular Hom-Lie color algebras as the natural generalization of split Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Lie color algebra $L$ is of the form $L = U + \sum\limits_{[j] \in \Lambda/\sim}I_{[j]}$ with $U$ a subspace of the abelian graded subalgebra $H$ and any $I_{[j]}$, a well described ideal of $L$, satisfying $[I_{[j]}, I_{[k]}] = 0$ if $[j]\neq [k]$. Under certain conditions, in the case of $L$ being of maximal length, the simplicity of the algebra is characterized.

Quaternion algebras and symbol algebras over algebraic number field K, with the degree [K:Q] even

Let F be a quadratic field (F = ℚ (√d), where d ≠ 0, 1 is a square-free integer). Let K be the pure algebraic quartic field K = ℚ (4√d). In this paper we determine connections between split quaternion algebras over F and split quaternion algebras over K.

Totally decomposable symplectic and unitary involutions

We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise the symplectic result was shown in Becher (Invent Math 173(1):1–6, 2008) and the unitary result was partly shown in Black and Queguiner-Mathieu (Enseign Math 60(3–4):377–395, 2014).

Derived categories of Grassmannians over integers and modular representation theory

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category Db(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GLk. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13], which was constructed over a field of characteristic zero.We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of GLn−k. The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors.We also construct a tilting vector bundle on Gr(k,n). We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z, and we identify the objects of Db(Gr(k,n)), which correspond to the standard and costandard modules in both structures.All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change).Similar results over fields of arbitrary characteristic were obtained independently in [7], by different methods.

The structure of split regular BiHom-Lie algebras

We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U + ∑ j I j with U a linear subspace of a fixed maximal abelian subalgebra H and any I j a well described (split) ideal of L , satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.

More on regular subgroups of the affine group

This paper is a new contribution to the study of regular subgroups of the affine group AGLn(F), for any field F. In particular we associate to each partition λ≠(1n+1) of n+1 abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for n≤4. Our classification is equivalent to the classification of split local algebras of dimension n+1 over F. Our methods, based on classical results of linear algebra, are computer free.

Totally decomposable quadratic pairs

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in Becher (Invent Math 173:1–6, 2008).

The structure of split regular Hom-Poisson algebras

The structure of split regular Hom-Poisson algebras

Decomposition Matrices are Generically Trivial

We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.