scholarly journals A New Proof of the Existence of Free Lie Algebras and an Application

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Andrea Bonfiglioli ◽  
Roberta Fulci
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Free Lie Algebra ◽  
Enveloping Algebra ◽  
Free Lie Algebras

The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that (given a set and a field of characteristic zero) the Lie algebra of the Lie polynomials in the letters of (over the field ) is a free Lie algebra generated by , all available proofs use the embedding of a Lie algebra into its enveloping algebra . The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier (1956), we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

scholarly journals UNDECIDABILITY OF THE FIRST ORDER THEORIES OF FREE NONCOMMUTATIVE LIE ALGEBRAS

2018 ◽  
Vol 83 (3) ◽  
pp. 1204-1216 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALEXEI MYASNIKOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Model Theory ◽  
Characteristic Zero ◽  
Elementary Theory ◽  
Independence Property ◽  
First Order ◽  
Free Lie Algebras ◽  
Image Position ◽  
Do So

AbstractLet R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

THE EQUATION [x,u] + [y,v] = 0 IN FREE LIE ALGEBRAS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1165-1187 ◽  
Author(s):  
VLADIMIR REMESLENNIKOV ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Solution Space ◽  
Free Generator ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Free Lie Algebras ◽  
Free Generators ◽  
Bilinear Equation

We investigate equations of the form [x,u] + [y,v] = 0 over a free Lie algebra L. In the case where u and v are free generators of L, we exhibit two series of solutions, we work out the dimensions of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the results on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1,x2] + [x3,x4] = 0.

Uniqueness theorems for left-symmetric enveloping algebras

1996 ◽  
Vol 120 (2) ◽  
pp. 193-206
Author(s):  
J. R. Bolgar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Uniqueness Theorems ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Uniqueness Results ◽  
Symmetric Algebras ◽  
Algebra Over A Field

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

scholarly journals ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

2013 ◽  
Vol 23 (01) ◽  
pp. 205-213 ◽  
Author(s):  
NIL MANSUROǦLU ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Homogeneous Component ◽  
Free Lie Algebra ◽  
Dimension Function ◽  
Free Lie Algebras ◽  
Explicit Formulae ◽  
Characteristic 2

Let L be a free Lie algebra of finite rank over a field K and let Ln denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, Lk] = [[Lm, Ln], Lk]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L2, L2, L1] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [Lm, Ln, Lk]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [Lm, Ln, Lk] in terms of Witt's dimension function.

scholarly journals Wronskian Envelope of a Lie Algebra

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Laurent Poinsot
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Free Lie Algebra ◽  
Sufficient Condition ◽  
Enveloping Algebra ◽  
Lie Bracket ◽  
Differential Algebras

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

2009 ◽  
Vol 19 (05) ◽  
pp. 699-703 ◽  
Author(s):  
RALPH STÖHR ◽  
MICHAEL VAUGHAN-LEE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Free Lie Algebra ◽  
Free Lie Algebras ◽  
Free Generators

Let L be a free Lie algebra of finite rank. If n ≥ 1 then we let Ln denote the homogeneous subspace of L spanned by Lie products of weight n in the free generators of L. We obtain formulae for the dimensions of the subspaces [Lm,Ln] for all m and n.

TEST RANK OF F/R′ LIE ALGEBRAS

2006 ◽  
Vol 16 (04) ◽  
pp. 817-825 ◽  
Author(s):  
ZERRIN ESMERLIGIL ◽  
DILEK KAHYALAR ◽  
NAIME EKICI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Integral Domain ◽  
Universal Enveloping Algebra ◽  
Free Lie Algebra ◽  
Enveloping Algebra ◽  
Test Rank ◽  
Ore Condition

Let F be a free Lie algebra of finite rank n and R be an ideal of F such that the universal enveloping algebra U(F/R) for F/R is an integral domain satisfying the Ore condition. We show that test rank for the Lie algebras of the form F/R′ is equal to n - 1 or n.

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

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