Special Linear Algebra Research Articles

Gauging the Maxwell Extended GLn,R and SLn+1,R Algebras

We consider the extension of the general-linear and special-linear algebras by employing the Maxwell symmetry in D space-time dimensions. We show how various Maxwell extensions of the ordinary space-time algebras can be obtained by a suitable contraction of generalized algebras. The extended Lie algebras could be useful in the construction of generalized gravity theories and the objects that couple to them. We also consider the gravitational dynamics of these algebras in the framework of the gauge theories of gravity. By adopting the symmetry-breaking mechanism of the Stelle–West model, we present some modified gravity models that contain the generalized cosmological constant term in four dimensions.

Why Quaternions and Octonions Exist

Summary There is a simple combinatorial anomaly, making possible some special linear algebra and thereby some special geometry, that occurs only in dimensions 1, 2, 4, and 8. The consequences are wide ranging and in particular lead to the existence of the complex numbers, the quaternions and the octonions. This article explains why the anomaly exists only in these dimensions using elementary linear algebra.

On the conformal symmetry of exceptional scalar theories

The DBI and special galileon theories exhibit a conformal symmetry at unphysical values of the spacetime dimension. We find the Lagrangian form of this symmetry. The special conformal transformations are non-linearly realized on the fields, even though conformal symmetry is unbroken. Commuting the conformal transformations with the extended shift symmetries, we find new symmetries, which when taken together with the conformal and shift symmetries close into a larger algebra. For DBI this larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. For the special galileon it is a real form of the special linear algebra. We also find the Weyl transformations corresponding to the conformal symmetries, as well as the necessary improvement terms to make the theories Weyl invariant, to second order in the coupling in the DBI case and to lowest order in the coupling in the special galileon case.

On the combinatorics of the universal enveloping algebra $$\widehat{U}_h({{\mathfrak {sl}}}_2)$$ U ^ h ( sl 2 )

Using combinatorial methods we study the structural coefficients of the formal homogeneous universal enveloping algebra $$\widehat{U}_h({\mathfrak {sl}}_2) $$ of the special linear algebra $$ {\mathfrak {sl}}_2$$ over a characteristic zero field. We provide explicit formulae for the product of generic elements in $$ \widehat{U}_h({\mathfrak {sl}}_2),$$ and construct combinatorial objects giving flesh to these formulae; in particular, we provide explicit formulae and combinatorial interpretations for the structural coefficients of divided power Poincare–Birkhoff–Witt basis.

Orthogonal and symplectic Yangians and Yang–Baxter R-operators

Yang–Baxter R operators symmetric with respect to the orthogonal and symplectic algebras are considered in an uniform way. Explicit forms for the spinorial and metaplectic R operators are obtained. L operators, obeying the RLL relation with the orthogonal or symplectic fundamental R matrix, are considered in the interesting cases, where their expansion in inverse powers of the spectral parameter is truncated. Unlike the case of special linear algebra symmetry the truncation results in additional conditions on the Lie algebra generators of which the L operators is built and which can be fulfilled in distinguished representations only. Further, generalized L operators, obeying the modified RLL relation with the fundamental R matrix replaced by the spinorial or metaplectic one, are considered in the particular case of linear dependence on the spectral parameter. It is shown how by fusion with respect to the spinorial or metaplectic representation these first order spinorial L operators reproduce the ordinary L operators with second order truncation.

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.

(ANTI)COMMUTATIVITY OF THE HARMONIC CREATION AND ANNIHILATION OPERATORS

It is known that para-particles, together with fermions and bosons, of a single mode can be described as an irreducible representation of the Lie (super) algebra 𝔰𝔩2(ℂ) (2-dimensional special linear algebra over the complex number ℂ), that is, they satisfy the equation of motion of a harmonic oscillator. Under the equation of motion of a harmonic oscillator, we obtain the set of the commutation relations which is isomorphic to the irreducible representation, to find that the equation of motion, conversely, can be derived from the commutation relation only for the case of either fermion or boson. If Nature admits of the existence of such a sufficient condition for the equation of motion of a harmonic oscillator, no para-particle can be allowed.

On Generating Relations Involving Generalized Gegenbauer Polynomials

Abstract In this paper, generating relations involving generalized Gegenbauer polynomials are obtained by constructing a three-dimensional Lie algebra isomorphic to special linear algebra sl(2). Further, a number of new interesting relations involving various generalized polynomials are obtained as applications of these generating relations.

The Combinatorics of Orbital Varieties Closures of Nilpotent Order 2 in sl${}_n$

We consider two partial orders on the set of standard Young tableaux. The first one is induced to this set from the weak right order on symmetric group by Robinson-Schensted algorithm. The second one is induced to it from the dominance order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We prove that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in special linear algebra $sl_n.$

On certain maximal cyclic modules for the quantized special linear algebra at a root of unity

By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type A may be transformed into modules over Lusztig's infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.

Lie-theoretic generating relations of two variable Laguerre polynomials

Generating relations involving two variable laguerre polynomials L n ( x, y) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L n ( x, y).

Computing the Isolated Roots by Matrix Methods

Two main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The first one involves Gröbner basis computation, and applies to any zero-dimensional system. But, it is performed withexact arithmetic and, usually, large numbers appear during the computation. The other approach is based on resultant formulations and can be performed with floating point arithmetic. However, it applies only to generic situations, leading to singular problems in several systems coming from robotics and computational vision, for instance.In this paper, reinvestigating the resultant approach from the linear algebra point of view, we handle the problem of genericity and present a new algorithm for computing the isolated roots of an algebraic variety, not necessarily of dimension zero. We analyse two types of resultant formulations, transform them into eigenvector problems, and describe special linear algebra operations on the matrix pencils in order to reduce the root computation to a non-singular eigenvector problem. This new algorithm, based on pencil decompositions, has a good complexity even in the non-generic situations and can be executed with floating point arithmetic.

Generators and relations for special linear algebras and groups

Generators and relations for special linear algebras and groups

Simple supersymmetries

Two infinite families of simple graded Lie algebras (GLA’s) over the complex numbers are described: the special linear algebras SL(m‖n) [whose Bose sector is the direct sum of a one-dimensional algebra with the ordinary Lie algebra SL(m) ×SL(n)], and the orthosymplectic algebras OSp(2r‖s) [with Bose sector Sp(2r) ×O(s)]. The GLA’s of physics fit into these two families either directly or via Inönü–Wigner contraction. These algebras along with further exceptional GLA’s constitute all the classical GLA’s, i.e., all GLA’s with a nondegenerate metric (not necessarilly Killing) form. The existence of infinite families of hyperexceptional GLA’s, i.e., of GLA’s that are simple but not classical is pointed out.