ON HAMILTONIAN GROUP OF MULTISYMPLECTIC MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 929-935 ◽  
Author(s):  
M. SHAFIEE
Keyword(s):  
Lie Algebra ◽  
Equivalence Classes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hamiltonian Paths ◽  
Hamiltonian Group ◽  
Identity Component ◽  
Necessary And Sufficient ◽  
Hamiltonian Forms

In this paper the Hamiltonian group Ham (M, Ω) is defined for a compact k-plectic manifold (M, Ω) and it is shown that its Lie algebra is the space of equivalence classes of Hamiltonian forms, modulo closed forms. Also if ψ be a multisymplectomorphism in the identity component Msymp 0(M, Ω) of the group of multisymplectomorphisms Msymp (M, Ω), we obtain a necessary and sufficient condition under which ψ belongs to Ham (M, Ω). As two consequences, we show that Hamiltonian paths are generated by Hamiltonian forms and if Hk (M, ℝ) = 0, then Ham (M, Ω) is equal to Msymp 0(M, Ω).

scholarly journals Bracing Zonohedra With Special Faces

2015 ◽  
Vol 3 (1-2) ◽  
pp. 88-95 ◽  
Author(s):  
Gyula Nagy
Keyword(s):  
Convex Polyhedron ◽  
Three Dimensional ◽  
Equivalence Classes ◽  
Frame Structure ◽  
Sufficient Condition ◽  
Preliminary Design ◽  
Necessary And Sufficient Condition ◽  
Special Assumption ◽  
Centrally Symmetric ◽  
Necessary And Sufficient

Abstract The analysis of simpler preliminary design gives useful input for more complicated three-dimensional building frame structure. A zonohedron, as a preliminary structure of design, is a convex polyhedron for which each face possesses central symmetry. We considered zonohedron as a special framework with the special assumption that the polygonal faces can be deformed in such a way that faces remain planar and centrally symmetric, moreover the length of all edges remains unchanged. Introducing some diagonal braces we got a new mechanism. This paper deals with the flexibility of this kind of mechanisms, and investigates the rigidity of the braced framework. The flexibility of the framework can be characterized by some vectors, which represent equivalence classes of the edges. A necessary and sufficient condition for the rigidity of the braced rhombic face zonohedra is posed. A real mechanical construction, based on two simple elements, provides a CAD prototype of these new mechanisms.

scholarly journals CUBIC DIRAC OPERATORS AND THE STRANGE FREUDENTHAL–DE VRIES FORMULA FOR COLOUR LIE ALGEBRAS

2021 ◽  
Author(s):  
PHILIPPE MEYER
Keyword(s):  
Dirac Operator ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Dirac Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Representation ◽  
Basic Colour ◽  
De Vries ◽  
Necessary And Sufficient

AbstractThe aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an ϵ-orthogonal representation of an ϵ-quadratic colour Lie algebra. This is used to prove a strange Freudenthal–de Vries formula for basic colour Lie algebras as well as a Parthasarathy formula for cubic Dirac operators of colour Lie algebras. We calculate the cohomology induced by this Dirac operator, analogously to the algebraic Vogan conjecture proved by Huang and Pandžić.

scholarly journals Quotient groups of IA-automorphisms of a free group of rank 3

2018 ◽  
Vol 28 (01) ◽  
pp. 115-131 ◽  
Author(s):  
V. Metaftsis ◽  
A. I. Papistas ◽  
I. Sevaslidou
Keyword(s):  
Lie Algebra ◽  
Free Group ◽  
Quotient Group ◽  
Lower Central Series ◽  
Central Series ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Graded Lie Algebra ◽  
Necessary And Sufficient ◽  
Quotient Groups

We prove that, for any positive integer [Formula: see text], the quotient group [Formula: see text] of the lower central series of the McCool group [Formula: see text] is isomorphic to two copies of the quotient group [Formula: see text] of the lower central series of a free group [Formula: see text] of rank [Formula: see text] as [Formula: see text]-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra [Formula: see text] of [Formula: see text] is naturally embedded into the Johnson Lie algebra [Formula: see text] of the IA-automorphisms of [Formula: see text].

On Engel Fuzzy Subpolygroups

2017 ◽  
Vol 13 (02) ◽  
pp. 195-206 ◽  
Author(s):  
R. A. Borzooei ◽  
E. Mohammadzadeh ◽  
Violeta Fotea
Keyword(s):  
Equivalence Classes ◽  
Sufficient Condition ◽  
Fundamental Relation ◽  
Necessary And Sufficient Condition ◽  
Engel Group ◽  
Fuzzy Subgroup ◽  
Necessary And Sufficient

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup [Formula: see text] of [Formula: see text] and [Formula: see text], the fundamental relation on a given polygroup [Formula: see text], we construct an Engel fuzzy subgroup [Formula: see text]. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group [Formula: see text]/[Formula: see text], the group of equivalence classes derived from a fuzzy subpolygroup of [Formula: see text]. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

scholarly journals On the Lie structure of locally matrix algebras

2020 ◽  
Vol 12 (2) ◽  
pp. 311-316
Author(s):  
O. Bezushchak
Keyword(s):  
Lie Algebra ◽  
Matrix Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Matrix Algebras ◽  
Necessary And Sufficient ◽  
Algebra Over A Field

Let $A$ be a unital locally matrix algebra over a field $\mathbb{F}$ of characteristic different from $2.$ We find a necessary and sufficient condition for the Lie algebra $A\diagup\mathbb{F}\cdot 1$ to be simple and for the Lie algebra of derivations $\text{Der}(A)$ to be topologically simple. The condition depends on the Steinitz number of $A$ only.

scholarly journals Extensions and automorphisms of Lie algebras

2016 ◽  
Vol 16 (09) ◽  
pp. 1750162 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Mahender Singh
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Short Exact Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nilpotent Lie Algebra ◽  
Lie Algebra Cohomology ◽  
Cohomology Of Lie Algebras ◽  
Necessary And Sufficient

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

scholarly journals Rotation-Equivalence Classes of Binary Vectors

2016 ◽  
Vol 67 (1) ◽  
pp. 93-98
Author(s):  
Otokar Grošek ◽  
Viliam Hromada
Keyword(s):  
Algebraic Approach ◽  
Equivalence Classes ◽  
Linear Feedback ◽  
Sufficient Condition ◽  
Hamming Weight ◽  
Necessary And Sufficient Condition ◽  
Binary Vectors ◽  
Linear Feedback Shift Registers ◽  
Feedback Shift Registers ◽  
Necessary And Sufficient

Abstract In this paper we study equivalence classes of binary vectors with regards to their rotation by using an algebraic approach based on the theory of linear feedback shift registers. We state the necessary and sufficient condition for existence of an equivalence class with given cardinality and provide two formulas. The first represents the sharp distribution of cardinalities for given length and Hamming weight of binary vectors and the second enables us to determine the number of different classes with the same cardinality.

SOME PROPERTIES ON THE SCHUR MULTIPLIER OF A PAIR OF LIE ALGEBRAS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250011 ◽  
Author(s):  
MOHAMMAD REZA RISMANCHIAN ◽  
MEHDI ARASKHAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Sufficient Condition ◽  
Schur Multiplier ◽  
Necessary And Sufficient Condition ◽  
Pair Of Lie Algebras ◽  
Necessary And Sufficient

The aim of this paper is to introduce the concept of the Schur multiplier [Formula: see text] of a pair of Lie algebras and to obtain some inequalities for the dimension of [Formula: see text]. Also, we consider some of the features of central extension of an arbitrary Lie algebra. Moreover, we present a necessary and sufficient condition in which the Schur multiplier of a pair of Lie algebras can be embedded into the Schur multiplier of their factor Lie algebras.

scholarly journals On the automorphism group of a connected locally compact topological group

1992 ◽  
Vol 35 (2) ◽  
pp. 285-294
Author(s):  
Ta-Sun Wu
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Identity Component ◽  
Necessary And Sufficient ◽  
Locally Compact Topological Group

Let G be a locally compact connected topological group. Let Aut0G be the identity component of the group of all bi-continuous automorphisms of G topologized by Birkhoff topology. We give a necessary and sufficient condition for Aut0G to be locally compact.

The Steinberg Lie Algebra st2(S)

2016 ◽  
Vol 23 (01) ◽  
pp. 129-136
Author(s):  
Yongjie Wang ◽  
Yiqian Shi ◽  
Yun Gao
Keyword(s):  
Lie Algebra ◽  
Triple System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lie Triple System ◽  
Necessary And Sufficient

Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2(S) of the Steinberg Lie algebra st2(S) and give a necessary and sufficient condition for I2(S)=0.

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