Simple Algebra Research Articles

Roll Stabilisation of a Flexible Spacecraft using Quantitative Feedback Theory

Quantitative Feedback Theory (QFT) has achieved the status as a powerful design technique for the achievement of assigned performance over the specified range of plant uncertainties. This is due to the relative simple algebra and the transparency of the design methodology. A QFT solution to a Single Input Single Output (SISO) problem of roll stabilization of a flexible satellite is presented in this paper. A simple controller is designed which ensures robustness against parameter variations and disturbances. The results of the study, are compared with those obtained using an H* controller.

An alternate approach to Habiro and Lê’s universal invariant of homology three-spheres

An alternative construction of the invariant of homology three spheres valued in a completion of an integral polynomial ring associated to each quantized complex simple Lie algebra by Habiro and Lê [Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol. 20 (2016) 2687–2835, doi:10.2140/gt.2016.20.2687, arXiv:1503:03549v1] is given.

Classification of D-bialgebra structures on power series algebras

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A〚z〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A〚z〛 as well as of non-degenerate solutions of the usual CYBE.If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A〚z〛 as well as of all non-degenerate solutions of an associative version of the CYBE.

Equivalence of categories of simple Lie algebras in positive characteristic

In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent.

Local automorphisms of complex solvable Lie algebras of maximal rank

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism.

Symbol length of classes in Milnor 𝐾-groups

Given a field F F , a positive integer m m and an integer n ≥ 2 n\geq 2 , it is proved that the symbol length of classes in Milnor’s K K -groups K n F / 2 m K n F K_n F/2^m K_n F that are equivalent to single symbols under the embedding into K n F / 2 m + 1 K n F K_n F/2^{m+1} K_n F is at most 2 n − 1 2^{n-1} under the assumption that F ⊇ μ 2 m + 1 F \supseteq \mu _{2^{m+1}} . Since K 2 F / 2 m K 2 F ≅ 2 m B r ( F ) K_2 F/2^m K_2 F \cong {_{2^m}Br(F)} for n = 2 n=2 , this coincides with the upper bound of 2 2 (proved by Tignol in 1983) for the symbol length of central simple algebras of exponent 2 m 2^m that are Brauer equivalent to a single symbol algebra of degree 2 m + 1 2^{m+1} . The cases where the embedding into K n F / 2 m + 1 K n F K_n F/2^{m+1} K_n F is of symbol length 2 2 , 3 3 , and 4 4 (the last when n = 2 n=2 ) are also considered. The paper finishes with the study of the symbol length for classes in K 3 / 3 m K 3 F K_3/3^m K_3 F whose embedding into K 3 F / 3 m + 1 K 3 F K_3 F/3^{m+1} K_3 F is one symbol when F ⊇ μ 3 m + 1 F \supseteq \mu _{3^{m+1}} .

A Prolog assisted search for new simple Lie algebras

We describe some recent computer investigations with the ‘Constraint Logic Programming over Finite Domains’ library in the Prolog programming environment to search for new simple Lie algebras over the field GF ⁡ ( 2 ) \operatorname {GF}(2) of 2 2 elements. Motivated by a paper of Grishkov et al., we specifically look for those with a thin decomposition, and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension 31 31 . We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known.

Wild Local Structures of Automorphic Lie Algebras

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

Simple weight modules for Yangian Y (𝔰𝔩2)

Let [Formula: see text] be a finite-dimensional simple Lie algebra over [Formula: see text]. A [Formula: see text]-module is said to be weight if it is a weight [Formula: see text]-module. We give a complete classification of simple weight modules for [Formula: see text] which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical [Formula: see text]-representation theory, we show that there exists a class of [Formula: see text] irreducible modules which have uniformly two-dimensional weight spaces.

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

AbstractLet G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$ . We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$ . It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$ . Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$ . As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G-orbits in place of ${\mathcal O}_{\textsf{min}}$ .

Chevalley involutions for Lie tori and extended affine Lie algebras

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of “Chevalley involutions” for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove by a type-dependent argument the existence of Chevalley involutions for centerless Lie tori.

Finite splittings of differential matrix algebras

It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. In [4] a counterpart of this result was studied in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of differential matrix algebras divide their degrees.

The Blow-Up Analysis on B2(1) Affine Toda System: Local Mass and Affine Weyl Group

Abstract It has been established that the local mass of blow-up solutions to Toda systems associated with the simple Lie algebras $\textbf{A}_{n}$, $\textbf{B}_{n}$, $\textbf{C}_{n}$, and $\textbf{G}_{2}$ can be represented by a finite Weyl group. In particular, at each blow-up point, after a sequence of bubbling steps (via scaling) is performed, the transformation of the local mass at each step corresponds to the action of an element in the Weyl group. In this article, we present the results in the same spirit for the affine $\textbf{B}_{2}^{(1)}$ Toda system with singularities. Compared with the Toda system with simple Lie algebras, the computation of local masses is more challenging due to the infinite number of elements of the affine Weyl group of type $\textbf{B}_{2}^{(1)}$. In order to give an explicit expression for the local mass formula, we introduce two free integers and write down all the possibilities into $8$ types. This shows a striking difference to previous results on Toda systems with simple Lie algebras. The main result of this article seems to provide the first major advance in understanding the relation between the blow-up analysis of affine Toda system and the affine Weyl group of the associated Lie algebras.

Two-fold refinement of non simply laced Chern-Simons theories

Inspired by the two-parameter Macdonald's deformation of the formulae for non simply laced simple Lie algebras, we propose a two-fold refinement of the partition function of the corresponding Chern-Simons theory on S3. It is based on a two-fold refinement of the Kac-Peterson formula for the volume of the fundamental domain of the coroot lattice of non simply laced Lie algebras. We further derive explicit integral representations of the two-fold refined Chern-Simons partition functions. We also present the corresponding generalized universal-like expressions for them. With these formulae in hand, one can try to investigate a possible duality of the corresponding Chern-Simons theories with hypothetical two-fold refined topological string theories.

Integral preserving discretization of 2D Toda lattices

There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin’s discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction.

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored.

Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let [Formula: see text] be a finite Lie conformal algebra. We investigate the conformal derivation algebra [Formula: see text], conformal triple derivation algebra [Formula: see text] and generalized conformal triple derivation algebra [Formula: see text], focusing mainly on the connections among these derivation algebras. We also give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, [Formula: see text], where [Formula: see text] is a finite simple Lie conformal algebra. But for [Formula: see text], we obtain a conclusion that is closely related to [Formula: see text]. Finally, we introduce the definition of a triple homomorphism of Lie conformal algebras. Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

Relationship of the Hennings and Witten–Reshetikhan–Turaev invariants for higher rank quantum groups

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen et al. [On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163; Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999] which relates the Hennings and WRT invariants for [Formula: see text] and [Formula: see text] for arbitrary rational homology three-spheres.

Deformations of Symmetric Simple Modular Lie (Super)Algebras

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.