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.

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.

scholarly journals A NOTE ON DERIVATIONS OF LIE ALGEBRAS

2011 ◽  
Vol 84 (3) ◽  
pp. 444-446 ◽  
Author(s):  
M. SHAHRYARI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Abelian Algebra ◽  
Finite Dimensional ◽  
Image Position

AbstractIn this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D≤Der(L), with the property for some n>1, is necessarily solvable. As a result, we show that if L has a derivation d:L→L such that Ln⊆d(L), for some n>1, then L is solvable.

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.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Euclidean Lie Algebras

1969 ◽  
Vol 21 ◽  
pp. 1432-1454 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Cartan Matrix ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Generalized Cartan Matrix

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

scholarly journals FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

2009 ◽  
Vol 20 (11) ◽  
pp. 1347-1362 ◽  
Author(s):  
LEANDRO CAGLIERO ◽  
NADINA ROJAS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Abelian Subalgebras ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Minimal Dimension

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).

scholarly journals Irreducible representations of simple Lie algebras by differential operators

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
A. Morozov ◽  
M. Reva ◽  
N. Tselousov ◽  
Y. Zenkevich
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Differential Operators ◽  
Irreducible Representations ◽  
Explicit Formulas ◽  
Systematic Method ◽  
First Order ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Lowering Operators

AbstractWe describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $${\mathfrak {g}}$$ g . The Lie algebra generators are represented as first order differential operators in $$\frac{1}{2} \left( \dim {\mathfrak {g}} - \text {rank} \, {\mathfrak {g}}\right) $$ 1 2 dim g - rank g variables. All rising generators $$\mathbf{e}$$ e are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators $$\mathbf{f}$$ f . We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.

Palindromes in the free metabelian Lie algebras

2019 ◽  
Vol 29 (05) ◽  
pp. 885-891
Author(s):  
Şehmus Fındık ◽  
Nazar Şahi̇n Öğüşlü
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Reverse Order ◽  
Generating Set ◽  
Linear Basis ◽  
Free Metabelian Lie Algebra ◽  
Definition Of ◽  
Algebra Over A Field

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let [Formula: see text] be the free metabelian Lie algebra over a field of characteristic zero generated by [Formula: see text]. We propose the following definition of palindromes in the setting of Lie algebras: An element [Formula: see text] is called a palindrome if it is preserved under the change of generators; i.e. [Formula: see text]. We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.

XV.—Bounds for the Indices of Nilpotent and Solvable Lie Algebras

1954 ◽  
Vol 64 (2) ◽  
pp. 200-208
Author(s):  
E. M. Patterson
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Arbitrary Field ◽  
Nilpotent Lie Algebras ◽  
Solvable Lie Algebras

SynopsisBy examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.

Outer Derivations and Classical-Albert-Zassenhaus lie Algebras

1984 ◽  
Vol 36 (6) ◽  
pp. 961-972 ◽  
Author(s):  
David J. Winter
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Cartan Subalgebra ◽  
One Dimensional ◽  
Derivation Algebra ◽  
Image Position

This paper is concerned with the structure of the derivation algebra Der L of the Lie algebra L with split Cartan subalgebra H. The Fitting decompositionof Der L with respect to ad ad H leads to a decompositionwhereThis decomposition is studied in detail in Section 2, where the centralizer of ad L∞ in D0(H) is shown to bewhich is Hom(L/L2, Center L) when H is Abelian. When the root-spaces La (a nonzero) are one-dimensional, this leads to the decomposition of Der L aswhere T is any maximal torus of D0(H).

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