scholarly journals On the support of the free Lie algebra: the Schutzenberger problems

2010 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Ioannis C. Michos
Keyword(s):  
Lie Algebra ◽  
Recursion Formula ◽  
Discrete Math ◽  
Binomial Coefficient ◽  
Finite Alphabet ◽  
Free Lie Algebra ◽  
Lie Ring ◽  
Linear Polynomial ◽  
International Audience ◽  
Natural Way

Combinatorics International audience M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).

scholarly journals On bijections between monotone rooted trees and the comb basis

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Fu Liu
Keyword(s):  
Lie Algebra ◽  
Rooted Trees ◽  
Close Connection ◽  
Free Lie Algebra ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Lie Brackets ◽  
Element Set

International audience Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$. Soit $A$ un ensemble à $n$ éléments. Soit $\mathscr{L} ie_2(A)$ la partie multilinéaire de l'algèbre de Lie libre sur $A$ avec une paire de crochets de Lie compatibles et $\mathscr{L} ie_2(A, i)$ le sous-espace de$\mathscr{L} ie_2(A)$ généré par tous les monômes en $\mathscr{L} ie_2(A)$ avec $i$ supports d'un même type. L'auteur et Dotsenko-Khoroshkin montrent que la dimension de $\mathscr{L} ie_2(A, i)$ est la taille de la $R_{A,i}$, l'ensemble des arbres enracinés sur $A$ avec $i$ arêtes décroissantes. Il y a trois familles de bases connues pour $\mathscr{L} ie_2(A, i)$ : la base de peigne, la base Lyndon, et la base Liu-Lyndon. Récemment, Gonzalez, D' Léon et Wachs, dans leur étude de (co)-homologie de la poset des partitions pondérés, ont demandé si il y a des bijections jolies entre$R_{A,i}$, et la base de peigne ou la base Lyndon. Nous donnons une définition naturelle de "bijection jolie " , et un conjecture qu'il y a une seule bijection jolie entre $R_{A,i}$, et la base de peigne. Nous montrons que la conjecture est vraie pour les cas extrêmes: $i = 0$, et $n − 1$.

scholarly journals Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Lorenz A. Gilch
Keyword(s):  
Random Walks ◽  
Finite Groups ◽  
Transition Probabilities ◽  
Finite Alphabet ◽  
Free Products ◽  
Rate Of Escape ◽  
International Audience ◽  
Special Case ◽  
Current Word ◽  
Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

scholarly journals Weighted partitions

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Rafael González S. D'León ◽  
Michelle L. Wachs
Keyword(s):  
Lie Algebra ◽  
Characteristic Polynomial ◽  
Rooted Trees ◽  
Free Lie Algebra ◽  
International Audience

International audience In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted <mathfrak>S</mathfrak>_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n.

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 Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

TEST RANK OF THE LIE ALGEBRA F/[R′, F]

2014 ◽  
Vol 13 (04) ◽  
pp. 1350122 ◽  
Author(s):  
NAZAR ŞAHİN ÖĞÜŞLÜ
Keyword(s):  
Lie Algebra ◽  
Free Lie Algebra ◽  
Invariant Ideal ◽  
Test Rank

Let F be a free Lie algebra of rank n ≥ 2 and R be a fully invariant ideal of F. We show that the test rank of the Lie algebra F/[R′, F] is equal to 1 when n is even and less than or equal to 2 when n is odd.

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