Algebras Of Rank Research Articles

О порождающем множестве подалгебры инвариантов свободной ограниченной алгебры Ли

About Generating Set of the Invariant Subalgebra of Free Restricted Lie Algebra V. M. Petrogradsky, I. A. Subbotin Department of Mathematics, University of Brasilia, 70910-900 Brasilia DF, Brazil, petrogradsky@rambler.ru Ulyanovsk State University, Russia, 432970, Ulyanovsk, ul. L’va Tolstogo, 42, shelby888@yandex.ru Suppose thatL = L(X) is the free Lie p–algebra of finite rank k with free generating setX = {x1, . . . , xk} on a field of positive characteristic. LetG is nontrivial finite group of homogeneous automorphismsL(X). Our main purpose to prove thatL subalgebra of invariants is is infinitely generated. We have more strongly result. Let Y = ∪∞n=1Yn be homogeneous free generating set for the algebra of invariants L, elements Yn are of degree n relatively X , n ≥ 1. Consider the corresponding generating function H (Y, t) = ∞

Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case

We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases.

STABLE RANK OF LEAVITT PATH ALGEBRAS OF ARBITRARY GRAPHS

AbstractThe stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

A global steering method for nonholonomic systems

In this paper, we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in Jean et al. (2005) [21] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in Murray and Sastry (1993) [29] for chained-form systems.

Control analysis of an underactuated spacecraft under disturbance

Based on the nonlinear controllability theory, this paper analyzes the attitude controllability for an underactuated spacecraft using two or one thrusters in sequence. In order to provide a preconditional guide on designing the control law for the underactuated attitude control system in the actual case, the underactuated spacecraft with respect to the orbit frame is investigated in the presence of periodical oscillation disturbance. First, attitude dynamic model was established for an underactuated spacecraft actuated by three thrusters, one or two of which has failed separately, and the special orthogonal group (SO(3)) is deployed to describe the attitude motions. Second, Liouville theorem and Poincaré’s recurrence theorem were used to confirm that the drift field is Weakly Positively Poisson Stable (WPPS). Furthermore, the sufficient and necessary condition of controllability was obtained on the basis of Lie Algebra Rank Condition (LARC). Finally, according to the case of two available thrusters, angular velocity stabilization and three-axis attitude stabilization control laws are designed, and proved by Lyapunov stability theory and LaSalle invariant theorem. The analysis and simulation results illustrate the feasibility of the proposed control law.

The local structure of groups of triangular automorphisms of relatively free algebras

Let K be an arbitrary field and C n a relatively free algebra of rank n. In particular, as C n we may treat a polynomial algebra P n , a free associative algebra A n , or an absolutely free algebra F n . For the algebras C n = P n , A n , F n , it is proved that every finitely generated subgroup G of a group TC n of triangular automorphisms admits a faithful matrix representation over a field K; hence it is residually finite by Mal’tsev’s theorem. For any algebra C n , the triangular automorphism group TC n is locally soluble, while the unitriangular automorphism group UC n is locally nilpotent. Consequently, UC n is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup G of UC n can be arbitrarily large with increasing n or transcendence degree of a field K over the prime subfield.

Localizing the Elliott conjecture at strongly self-absorbing C* -algebras

Abstract. We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C* -algebra 𝒟 ${\mathcal {D}}$ ; we also explain how the known classification theorems for nuclear C* -algebras fit into this concept. As a new result in this direction, we employ recent results of H. Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C* -algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang–Su algebra 𝒵 $\mathcal {Z}$ . Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C* -algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang–Su algebra itself.

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.

Inner and Outer Automorphisms of Free Metabelian Nilpotent Lie Algebras

Let L m, c be the free metabelian nilpotent of class c Lie algebra of rank m over a field K of characteristic 0. We describe the groups of inner and outer automorphisms of L m, c . To obtain this result, we first describe the groups of inner and continuous outer automorphisms of the completion with respect to the formal power series topology of the free metabelian Lie algebra F m of rank m.

Irreducible Harish-Chandra modules over extended Witt algebras

Let d be a positive integer, \(A={\mathbb{C}} [t_{1}^{\pm1},\ldots ,t_{d}^{\pm1}]\) be the Laurent polynomial algebra, and \(W=\operatorname{Der} (A)\) be the derivation Lie algebra of A. Then we have the semidirect product Lie algebra W⋉A which we call the extended Witt algebra of rank d. In this paper, we classify all irreducible Harish-Chandra modules over W⋉A with nontrivial action of A.

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.

Almost primitive elements of free nonassociative (anty)commutative algebras of small rank

Criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed for free nonassociative commutative and anticommutative algebras of rank 1 and 2.

Almost primitive elements of free nonassociative algebras of small ranks

Let K be a field, X = {x 1 ,…, x n }, and let F(X) be the free nonassociative algebra over the field K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free. A subset M of nonzero elements of the algebra F(X) is said to be primitive if there is a set Y of free generators of F(X), F(X) = F(Y ), such that M ⊆ Y (in this case we have |Y| = |X| = n). A nonzero element u of the free algebra F(X) is said to be almost primitive if u is not a primitive element of the algebra F(X), but u is a primitive element of any proper subalgebra of F(X) that contains it. In this article, for free nonassociative algebras of rank 1 and 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed. New examples of almost primitive elements of free nonassociative algebras of rank 3 are constructed.

Additive mappings on C ∗ -algebras sub-preserving absolute values of products

Let A be a C * -algebra of real rank zero and B be a C * -algebra with unit I.I t is shown that if φ : A -→ B is an additive mapping which satisfies |φ(A)φ(B) |≤ φ(|AB| )f or every A, B ∈ A+ and φ(A )= I for some A ∈ As withA �≤ 1, then the restriction of mapping φ to As is a Jordan homomorphism, where As denotes the set of all self-adjoint elements. We will also show that if φ is surjective preserving the product and an absolute value, then φ is a C-linear or C-antilinear *-homomorphism on A. MSC: Primary 47B49; Secondary 46L05; 47L30

Skew-symmetric cluster algebras of finite mutation type

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems E_6 , E_7 , E_8 , \widetilde E_6 , \widetilde E_7 , \widetilde E_8 , E_6^{(1,1)} , E_7^{(1,1)} , E_8^{(1,1)} ; two remaining were found by Derksen and Owen in [DO]. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.

Controllability of an underactuated spacecraft with one thruster under disturbance

For an underactuated spacecraft using only one thruster, the attitude controllability with respect to the orbit frame is studied in the presence of periodical oscillation disturbance, which provides a preconditional guide on designing control law for underactuated attitude control system. Firstly, attitude dynamic model was established for an underactuated spacecraft, and attitude motion was described using the special orthogonal group (SO (3)). Secondly, Liouville theorem was used to confirm that the flow generated by the drift vector of the underactuated attitude control system is volume-preserving. Furthermore, according to Poincare’s recurrence theorem, we draw conclusions that this drift field is weakly positively poisson stable (WPPS). Thirdly, the sufficient and necessary condition of controllability was obtained on the basis of lie algebra rank condition (LARC). Finally, the controllable conditions were analyzed and simulated in different cases of inertia matrix with the installed position of thruster.

Super Duality and Homology of Unitarizable Modules of Lie Algebras

The \mathfrak u -homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types \mathfrak gl (n) , \mathfrak sp (2n) and \mathfrak so (2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU(p,q),SU(p)\times SU(q)) , (Sp(2n),U(n)) and (SO^\ast(2n),U(n)) .

Krull–Gabriel dimension and Cantor–Bendixson rank of 1-domestic string algebras

Krull–Gabriel dimension and Cantor–Bendixson rank of 1-domestic string algebras

Graphs from Generalized Kac--Moody Algebras

In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac--Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen.

Feedback control of a redundant wheeled snake mechanism using transverse functions SO(4)

The Transverse Function (TF) approach is applied to the control of a nonholonomic three-segments/snake-like wheeled mechanism, similar to the planar low-dimensional version of Hirose's Active Cord Mechanism (ACM) previously studied by the authors with the same control approach, but with two additional internal degrees of freedom (d.o.f.) whose actuation yields more flexible and efficient control solutions. From a theoretical point of view, these complementary d.o.f. modify the Control Lie Algebra of the system so that only first-order Lie brackets of the control vector fields are needed to satisfy the Lie Algebra Rank Condition (LARC). The fact that four independent (angular velocity) control inputs are used also implies for this system the existence of Transverse Functions (TF) defined on the six-dimensional special orthogonal group(4). Several examples of mechanisms whose control involve TF defined on(3) have been pointed out in the past. Beyond the specific control problem addressed here, a motivation for the present study is to illustrate for the first time how functions defined on the larger set(4) can be determined and used for the control of a physical system. This study is complemented with recalls concerning the parametrization of(4) by pairs of isoclinic quaternions and with the derivation of complementary differential calculus relations associated with this parametrization.