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.

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.

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.

Invariant Symmetric Bilinear Forms and Derivation Algebras of Unitary Lie Algebras

2016 ◽  
Vol 23 (03) ◽  
pp. 361-384 ◽  
Author(s):  
Yelong Zheng ◽  
Jiwen Gao ◽  
Zhihua Chang ◽  
Yun Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Bilinear Forms ◽  
Special Linear ◽  
Algebra Over A Field

Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: ℬ(L) ≅ (R+/([R,R] ∩ R+))* and Der (L) = Inn (L)+ Der (L)0 = Inn (L)+ SDer (R), which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field 𝔽.

scholarly journals The Representations of Lie Algebras of Prime Characteristic

1954 ◽  
Vol 2 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Quotient Ring ◽  
Irreducible Representations ◽  
Indecomposable Representation ◽  
Prime Characteristic ◽  
Representations Of Lie Algebras ◽  
Algebra Over A Field

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

scholarly journals Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550024
Author(s):  
Guillermo Ames ◽  
Leandro Cagliero ◽  
Mónica Cruz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Rank Conjecture ◽  
Algebra Over A Field ◽  
Toral Rank

If 𝔫 is a [Formula: see text]-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dim H*(𝔫) ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dim H*(𝔫) ≥ 2 dim (ℨ) for any finite-dimensional nilpotent Lie algebra 𝔫 with center ℨ. The TRC is more than 25 years old and remains open even for [Formula: see text]-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra 𝔫 with center ℨ: (A) If 𝔫 admits a [Formula: see text]-grading [Formula: see text], such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a ℤ+-grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? (B) If 𝔫 is r-step nilpotent admitting a grading 𝔫 = 𝔫1 ⊕ 𝔫2 ⊕ ⋯ ⊕ 𝔫k such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

Growth functions of lie algebras associated with associative algebras

2021 ◽  
pp. 2250110
Author(s):  
Adel Alahmadi ◽  
Fawziah Alharthi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Collection ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Growth Functions ◽  
Nilpotent Elements ◽  
Algebra Over A Field

Let [Formula: see text] be a finitely generated associative algebra over a field [Formula: see text] of characteristic [Formula: see text] and let [Formula: see text] be its associated Lie algebra. In this paper, we investigate relations between the growth functions of [Formula: see text] and the Lie algebra [Formula: see text]. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

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.

scholarly journals Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

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.

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.

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