Pre-Lie algebra characterization of SISO feedback invariants

2014 ◽  
Author(s):  
W. Steven Gray ◽  
Makhin Thitsa ◽  
Luis A. Duffaut Espinosa
Keyword(s):  
Lie Algebra ◽  
Algebra Characterization

Some studies on central derivation of nilpotent Lie superalgebra

2018 ◽  
Vol 13 (04) ◽  
pp. 2050068
Author(s):  
Rudra Narayan Padhan ◽  
K. C. Pati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Nilpotent Lie Algebra ◽  
New Concepts

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150; On [Formula: see text]-derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced.

Representations and cohomologies of regular Hom-pre-Lie algebras

2019 ◽  
Vol 19 (08) ◽  
pp. 2050149
Author(s):  
Shanshan Liu ◽  
Lina Song ◽  
Rong Tang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Theory ◽  
Trivial Representation ◽  
Linear Deformation ◽  
Dual Representations ◽  
Regular Representations ◽  
Equivalent Characterization ◽  
Tensor Representations

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an [Formula: see text]-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

THE BANACH-LIE *-ALGEBRA OF MULTIPLICATION OPERATORS ON A W*-ALGEBRA

2011 ◽  
Vol 04 (02) ◽  
pp. 235-261
Author(s):  
Maysaa Alqurashi ◽  
Najla A. Altwaijry ◽  
C. Martin Edwards ◽  
Christopher S. Hoskin
Keyword(s):  
State Space ◽  
Lie Algebra ◽  
The State ◽  
Multiplication Operators ◽  
Dual Cone ◽  
Positive Real ◽  
Linear Isometry ◽  
Real Linear ◽  
Special Case

The hermitian part [Formula: see text] of the Banach-Lie *-algebra [Formula: see text] of multiplication operators on the W *-algebra A is a unital GM-space, the base of the dual cone in the dual GL-space [Formula: see text] of which is affine isomorphic and weak*-homeomorphic to the state space of [Formula: see text]. It is shown that there exists a Lie *-isomorphism ϕ from the quotient (A ⊕∞ Aop)/K of an enveloping W *-algebra A ⊕∞ Aop of A by a weak*-closed Lie *-ideal K onto [Formula: see text], the restriction to the hermitian part ((A ⊕∞ Aop)/K)h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of [Formula: see text]. In the special case in which A is a W *-factor this leads to a further identification of the state space of [Formula: see text] in terms of the state space of A. For any W *-algebra A, the Banach-Lie *-algebra [Formula: see text] coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of [Formula: see text] in terms of a centre-valued 'norm' on A, which is similar to that previously obtained by non-order-theoretic methods.

scholarly journals A characterization of the Kostrikin radical of a Lie algebra

2011 ◽  
Vol 346 (1) ◽  
pp. 266-283 ◽  
Author(s):  
Esther García ◽  
Miguel Gómez Lozano
Keyword(s):  
Lie Algebra

Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras

2018 ◽  
Vol 30 (4) ◽  
pp. 1049-1060
Author(s):  
Mohammad Hossein Jafari ◽  
Ali Reza Madadi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Engel Elements ◽  
Central Automorphisms

Abstract In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.

THE GENERAL STRUCTURE OF G-GRADED CONTRACTIONS OF LIE ALGEBRAS, II: THE CONTRACTED LIE ALGEBRA

2006 ◽  
Vol 18 (06) ◽  
pp. 655-711 ◽  
Author(s):  
EVELYN WEIMAR-WOODS
Keyword(s):  
Abelian Group ◽  
Detailed Analysis ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Abelian Group ◽  
General Structure ◽  
Complete Characterization ◽  
Proper Subset

We continue our study of G-graded contractions γ of Lie algebras where G is an arbitrary finite Abelian group. We compare them with contractions, especially with respect to their usefulness in physics. (Note that the unfortunate terminology "graded contraction" is confusing since they are, by definition, not contractions.)We give a complete characterization of continuous G-graded contractions and note that they are equivalent to a proper subset of contractions. We study how the structure of the contracted Lie algebra Lγdepends on γ, and show that, for discrete graded contractions, applications in physics seem unlikely.Finally, with respect to applications to representations and invariants of Lie algebras, a comparison of graded contractions with contractions reveals the insurmountable defects of the graded contraction approach. In summary, our detailed analysis shows that graded contractions are clearly not useful in physics.

scholarly journals Diophantine properties of nilpotent Lie groups

2015 ◽  
Vol 151 (6) ◽  
pp. 1157-1188 ◽  
Author(s):  
Menny Aka ◽  
Emmanuel Breuillard ◽  
Lior Rosenzweig ◽  
Nicolas de Saxcé
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Finitely Generated ◽  
Derived Length ◽  
Simple Lie Groups ◽  
The Ideal

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

scholarly journals A Characterization of the Block Lie Algebra and Itsq-Forms in Characteristic 0

1998 ◽  
Vol 207 (2) ◽  
pp. 367-408 ◽  
Author(s):  
J.Marshall Osborn ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebra

scholarly journals Para-Kähler hom-Lie algebras

2019 ◽  
Vol 18 (03) ◽  
pp. 1950044 ◽  
Author(s):  
E. Peyghan ◽  
L. Nourmohammadifar
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Product ◽  
Kähler Structures

In this paper, we introduce the notions of pseudo-Riemannian, para-Hermitian and para-Kähler structures on hom-Lie algebras. In addition, we present the characterization of these structures. Also, we provide an example including these structures. We then introduce the phase space of a hom-Lie algebra and using the hom-left symmetric product, we show that a para-Kähler hom-Lie algebra gives a phase space and conversely, we can construct a para-Kähler hom-Lie algebra using a phase space.

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