scholarly journals On the module structure of free Lie algebras

1999 ◽  
Vol 352 (2) ◽  
pp. 901-934 ◽  
Author(s):  
R. M. Bryant ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Module Structure ◽  
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 Invariant Bases for Free Lie Algebras

1998 ◽  
Vol 204 (1) ◽  
pp. 337-346 ◽  
Author(s):  
Sean Guilfoyle ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Free Lie 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.

scholarly journals Free Lie algebras as modules for symmetric groups

1999 ◽  
Vol 67 (2) ◽  
pp. 143-156 ◽  
Author(s):  
R. M. Bryant ◽  
L. G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Free Lie Algebra ◽  
Prime Characteristic ◽  
Specht Modules ◽  
Generating Set ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Free Lie Algebras ◽  
Direct Decompositions

AbstractLet r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

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