The path functor and faithful representability of banach lie algebras

1973 ◽  
Vol 16 (1) ◽  
pp. 54-69 ◽  
Author(s):  
W. T. van Est ◽  
S. Świerczkowski
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Uniform Norm

In this note “vector space” will mean “Banach space” unless otherwise specified. Accordingly “Lie algebra” will stand for “Banach Lie algebra”. Morphisms between Lie algebras will be assumed continuous. A Banach algebra B will be always assumed associative, and it will be also viewed as a Lie algebra with product [X, YXY− YX. In particular, the Lie algebra gl(V) of endomorphisms of a vector space V will be equipped with the uniform norm. A morphism of Lie algebras L → gl(V) will b called a representation of L in gl(V). Also, if B is a Banach algebra, a morphism of Lie algebras L → B will be called a representation of L in B. From such one evidently obtains a representation of L in gl(B). A representation will be called faithful if it is injective.

scholarly journals Forms with O-Orthogonal Lie Algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 125-126
Author(s):  
Frank Servedio
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Homogeneous Polynomial ◽  
Algebra Structure ◽  
Orthogonal Lie Algebra ◽  
Coordinate Functions

A form P of degree r is a homogeneous polynomial in k[Yi, …, Yn] on kn, k a field; Yi are the coordinate functions on kn. Let V(n, r) denote the k-vector space of forms of degree r. Mn(k) = Endk(kn) has canonical Lie algebra structure with [A, B] = AB-BA and it acts as a k-Lie Algebra of kderivations of degree 0 on k[Yi, …, Yn] defined by setting D(A)Y= Yo(-A) for A∈Endk(kn), Y∈V(n,l) = Homk(kn, k) and extending as a k-derivation. Define the orthogonal Lie Algebra, LO(P), of P by LO(P) =

q-Deformations of 3-Lie Algebras

2017 ◽  
Vol 24 (03) ◽  
pp. 519-540 ◽  
Author(s):  
Ruipu Bai ◽  
Lixin Lin ◽  
Yan Zhang ◽  
Chuangchuang Kang
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Jacobi Identity ◽  
Group Algebras ◽  
Type I ◽  
Associative Algebras

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra [Formula: see text], where [Formula: see text] and [Formula: see text], is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]q and [Formula: see text] from [Formula: see text] to A, and a map [Formula: see text] satisfying the q-Jacobi identity [Formula: see text] for all [Formula: see text]. If the multiplications satisfy that [Formula: see text] and [Formula: see text] is skew-symmetry, then [Formula: see text] is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras [Formula: see text], and [Formula: see text] are obtained.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

Automorphism Group of a Class of Gradation Shifting Toroidal Lie Algebras

2010 ◽  
Vol 17 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Zhangsheng Xia ◽  
Shaobin Tan ◽  
Haifeng Lian
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Groups ◽  
Type B ◽  
Laurent Polynomials ◽  
Lie Bracket ◽  
Toroidal Algebra ◽  
Toroidal Lie Algebra

Let [Formula: see text] be the ring of Laurent polynomials in commuting variables. As a generalization of the toroidal Lie algebra, the gradation shifting toroidal Lie algebra [Formula: see text] is isomorphic to the corresponding (centerless) toroidal Lie algebra so(n, ℂ) ⨂ A of type B or D as a vector space, with the Lie bracket twisted by n fixed elements E1,…,En from A. In this paper, we study the automorphisms of the gradation shifting toroidal algebra [Formula: see text], which is proved to be closely related to a class of subgroups of GL(n,ℤ), called the linear groups over semilattices. We use the linear group over a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra [Formula: see text], which extends our earlier work.

scholarly journals On Nichols (braided) Lie algebras

2015 ◽  
Vol 26 (10) ◽  
pp. 1550082
Author(s):  
Weicai Wu ◽  
Shouchuan Zhang ◽  
Yao-Zhong Zhang
Keyword(s):  
Vector Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Nichols Algebra ◽  
Finite Dimensional ◽  
Braided Lie Algebras

We prove (i) Nichols algebra 𝔅(V) of vector space V is finite dimensional if and only if Nichols braided Lie algebra 𝔏(V) is finite dimensional; (ii) if the rank of connected V is 2 and 𝔅(V) is an arithmetic root system, then 𝔅(V) = F ⊕ 𝔏(V); and (iii) if Δ(𝔅(V)) is an arithmetic root system and there does not exist any m-infinity element with puu ≠ 1 for any u ∈ D(V), then dim (𝔅(V)) = ∞ if and only if there exists V′, which is twisting equivalent to V, such that dim (𝔏-(V′)) = ∞. Furthermore, we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.

I.—On the Geometry of Metrisable Lie Algebras

1958 ◽  
Vol 65 (1) ◽  
pp. 1-12
Author(s):  
H. S. Ruse
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Projective Spaces ◽  
Real Numbers ◽  
Dimensional Algebra ◽  
Doctoral Thesis ◽  
Abelian Algebra ◽  
The Relationship

SynopsisMetrisable Lie algebras have been defined by Tsou and Walker (1957). Their definition is adopted below in § I.The object of the present paper is to display something of the geometrical background of such algebras, particularly for those taken over the field of real numbers.§ I is introductory. In § 2 appears a statement, made as brief as possible because it is wholly classical, of the relationship between the vector-space of the Lie algebra L and the associated affine and projective spaces A and P. Some properties of metrisable Lie algebras are then examined in terms of the geometry of P, which provides an (n –I)-dimensional map of the n-dimensional algebra L. It is assumed throughout the paper that the Lie algebras under discussion are non-abelian, since the projective map of an abelian algebra presents nothing of interest.As the present work is intended as no more than a preliminary, it is confined, so far as its applications are concerned, to a discussion of metrisable algebras of dimensions 3 and 4 and to one example of an algebra of dimension 6.I have been privileged in preparing the paper to have access to the typescript of the paper by Tsou and Walker referred to above, and also to the doctoral thesis (1955) of the former. I am greatly indebted to them both, and also to Dr Paul Cohn and to a referee for suggestions regarding certain details of presentation.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

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