Split Algebra Research Articles

Quasi-hereditary structure of twisted split category algebras revisited

Let k be a field of characteristic 0, let C be a finite split category, let α be a 2-cocycle of C with values in the multiplicative group of k, and consider the resulting twisted category algebra A:=kαC. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over k is equivalent to a module category over a condensed algebra εAε, for an idempotent ε of A. In [2] the authors proved that A is quasi-hereditary (with respect to an explicit partial order ⩽ on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order ⩽ by introducing a coarser order ⊴ leading to the same results on A, but which allows to pass the quasi-heredity result to the condensed algebra εAε describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [21]. The new partial order ⊴ has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.

The Column and Row Immanants Over A Split Quaternion Algebra

The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a split quaternion algebra are introduced, and their basic properties are investigated. The key theorem about the column and row immanants of a Hermitian matrix over a split quaternion algebra is proved. Based on this theorem an immanant (permanent, determinant) of a Hermitian matrix over a split quaternion algebra is introduced.

Squaring the magic

We construct and classify all possible Magic Squares (MS’s) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras. Besides the known Freudenthal-Rozenfeld-Tits MS, the single-split Gunaydin-Sierra-Townsend MS, and the double-split Barton-Sudbery MS, we obtain other 7 Euclidean and 10 Lorentzian novel MS’s. We elucidate the role and the meaning of the various non-compact real forms of Lie algebras, entering the MS’s as symmetries of theories of Einstein-Maxwell gravity coupled to non-linear sigma models of scalar fields, possibly endowed with local supersymmetry, in D = 3, 4 and 5 space-time dimensions. In particular, such symmetries can be recognized as the U -dualities or the stabilizers of scalar manifolds within space-time with standard Lorentzian signature or with other, more exotic signatures, also relevant to suitable compactifications of the so-called M∗and M ′theories. Symmetries pertaining to some attractor U -orbits of magic supergravities in Lorentzian space-time also arise in this framework. ar X iv :1 20 8. 61 53 v2 [ m at hph ] 2 5 Se p 20 12

Projective planes over 2-dimensional quadratic algebras

The split version of the Freudenthal–Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [5,20,21]. The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2×A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of “projective planes over 2-dimensional quadratic algebras”, in cases of the split and non-split Galois extensions, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.

Quasi-Jordan Banach Algebras

We initiate a study of quasi-Jordan normed algebras. It is demonstrated that any quasi-Jordan Banach algebra with a norm1unit can be given an equivalent norm making the algebra isometrically isomorphic to a closed right ideal of a unital split quasi-Jordan Banach algebra; the set of invertible elements may not be open; the spectrum of any element is nonempty, but it may be neither bounded nor closed and hence not compact. Some characterizations of the unbounded spectrum of an element in a split quasi-Jordan Banach algebra with certain examples are given in the end.

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution is shown to have an affirmative answer.

Diagonalization of matrices over rings

We propose a method for diagonalizing matrices with entries in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory.

Twisted split category algebras as quasi-hereditary algebras

We show that if C is a finite split category, k is a field of characteristic 0, and α is a 2-cocycle of C with values in k × , then the twisted category algebra k α C is quasi-hereditary.

On the structure of split non-commutative Poisson algebras

A non-commutative Poisson algebra is a Lie algebra endowed with an, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we part from a split Lie algebra, with respect to a maximal abelian subalgebra H, we obtain a so-called split non-commutative Poisson algebra. This article is devoted to the study of the structure of split non-commutative Poisson algebras, by showing that any of such an algebra 𝒫 is of the form with U a linear subspace of H and any a well-described ideal of 𝒫, satisfying if j ≠ k. Under certain conditions the simplicity of 𝒫 is characterized and it is shown that 𝒫 is the direct sum of the family of its simple ideals.

Some Identities in Algebras Obtained by the Cayley-Dickson Process

Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the quaternion algebra. The idea of this work is to generalize the Hall identity to algebras obtained by the Cayley-Dickson process.Starting from the above remarks, in this paper, we prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse of this statement is also true for split quaternion algebras. From Hall identity, we will find some new properties and identities in algebras obtained by the Cayley-Dickson process.

Differential operators on quantized flag manifolds at roots of unity

The quantized flag manifold, which is a q-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of D-modules on it using the framework of non-commutative algebraic geometry; however, when the parameter q is a root of unity, Lusztig’s Frobenius morphism allows us to handle D-modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold. In this paper, we will show that this sheaf of rings on the ordinary flag manifold is an Azumaya algebra over its center. We also show that its restriction to certain subsets are split Azumaya algebras. These are analogues of some results of Bezrukavnikov–Mirković–Rumynin on D-modules on flag manifolds in positive characteristics.

ON OCTONIONIC GRAVITY, EXCEPTIONAL JORDAN STRINGS AND NONASSOCIATIVE TERNARY GAUGE FIELD THEORIES

Novel nonassociative octonionic ternary gauge field theories are proposed based on a ternary bracket. This paves the way to the many physical applications of exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A real-valued analog of the Einstein–Hilbert Lagrangian [Formula: see text] involving sums of all the possible contractions of the Ricci tensors plus their octonionic-complex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also re-examined. Finally, we construct (to our knowledge) generalized novel octonionic string and p-brane actions and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of 7,8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, black-holes in string theory and quantum information theory.

Split Malcev algebras

We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras M is of the form \(M={\mathcal U} +\sum_{j}I_{j}\) with \({\mathcal U}\) a subspace of the abelian Malcev subalgebra H and any Ij a well described ideal of M satisfying [Ij,Ik] = 0 if j ≠ k. Under certain conditions, the simplicity of M is characterized and it is shown that M is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.

Not Every Splitting Heyting or Interior Algebra is Finitely Presentable

We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Godel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.

Splitting algebras and Schubert calculus

We continue previous work on Schubert calculus of Grassmann schemes via splitting and factorization algebras. The aim of the project is to describe the intersection theory of flag schemes under ver ...

Split 3-Lie algebras

In order to begin an approach to the structure of 3-Lie algebras (with restrictions neither on the dimension nor on the base field), we introduce the class of split 3-Lie algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of ternary algebras, we show that any of such split 3-Lie algebras \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is of the form \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}={\mathcal U} +\sum \limits _{j}I_{j}$\end{document}T=U+∑jIj with \documentclass[12pt]{minimal}\begin{document}${\mathcal U}$\end{document}U a subspace of the 0-root space \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}_0$\end{document}T0 and any Ij a well described ideal of \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T, satisfying \documentclass[12pt]{minimal}\begin{document}$[I_j,{\mathfrak T},I_k]=0$\end{document}[Ij,T,Ik]=0 if j ≠ k. Under certain conditions the simplicity of \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is characterized and it is shown that \documentclass[12pt]{minimal}\begin{document}${\mathfrak T}$\end{document}T is the direct sum of the family of its minimal ideals, each one being a simple split 3-Lie algebra.

Galois subfields of inertially split division algebras

Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the residue division algebra D ¯ contains a maximal subfield which is Galois over the residue field F ¯ . This theorem captures an essential argument of previously known noncrossed product proofs in the more general language of noncommutative valuations. The result is particularly useful in connection with explicit constructions.

Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3):158–169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.

Restrictive, split and unital quasi-Jordan algebras

It is well known that by means of the right and left products of an associative dialgebra we can build a new product over the same vector space with respect to which it becomes a right version of a Jordan algebra (in fact, this new product is right commutative) called quasi-Jordan algebra. Recently, Kolesnikov and Bremner independently have discovered an interesting property of this new product. As the results of this paper indicate, when the said property is incorporated as an axiom in the definition of quasi-Jordan algebra then in a natural way one can introduce and study concepts in this new structure such as derivations (in particular inner derivations), quadratic representations, and the structure groups of a quasi-Jordan algebra.

On split Leibniz algebras

In order to begin an approach to the structure of arbitrary Leibniz algebras, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Leibniz algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that any of such algebras (L,[·,·]) is of the form L=U+∑jIj with U a subspace of the abelian subalgebra H, (in the sense [H,H]=0), and any Ij a well described ideal of L, satisfying [Ij,Ik]=0 if j≠k. In the case of L being of maximal length we characterize the simplicity of the algebra in terms of connections of roots.