7
votes
Accepted
How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly?
The natural map is rather from $TV \to U(FreeLie(V))$: consider the forgetful functors $Assoc \to Lie \to Vec$ and compose their left adjoints to get the left adjoint $T$ of the composite. Then, as ...
- 27.8k
6
votes
Accepted
Free groups and free restricted Lie algebras
This is Theorem 6.5 p.130 from Lazard, Michel.
Sur les groupes nilpotent et les anneaux de Lie.
Ann. Sci. Ecole Norm. Sup (3) 71 (1954) 101-190.
- 1,089
4
votes
super Lyndon words
An article you probably want to look at is E. S. Chibrikov, "The Right-Normed Basis for a Free Lie Superalgebra and Lyndon–Shirshov Words", Algebra Logika 45 (2006), issue 4, pp. 458--483. This ...
3
votes
list of Hall basis
much of the computational work in rough path theory relies on working with Hall bases. You can find an interactive web page that generates Hall bases for you at
https://coropa.sourceforge.io/
Half ...
- 31
3
votes
Accepted
Lyndon basis of free Lie superalgebras
Theorem 2.2 in Leonid A. Bokut, Seok-Jin Kang, Kyu-Hwan Lee, Peter Malcolmson, Gröbner–Shirshov Bases for Lie Superalgebras and Their Universal Enveloping Algebras, Journal of Algebra 217, Issue 2, 15 ...
- 33.8k
3
votes
Accepted
Free Lie algebra and nilpotent groups in Rothschild and Stein's paper
First, the notation is a little confusing. I think you are supposed to understand that $n$ is always the number of vector fields in the fixed set $\{X_1, \dots, X_n\}$. But the symbol $n$ in the ...
- 29.7k
2
votes
Criterion to construct a $\mathbb{Z}$-basis of a free $\mathbb{Z}$-Lie algebra
It turns out my intuition was wrong and the answer is "no".
Take for instance $n=3$, and attribute the degree $\alpha_i$ to $x_i$, so that $L(\mathbb{Q},3)$ becomes graded by the free $\mathbb{Z}$-...
- 71
1
vote
Ideal of the free Lie algebra L(x,y) generated by x
Now I know the answer on my question. First of all we note that the ideal $(x)$ is freely generated by the elements
$$x,[x,y],[x,y,y],\dots$$
as a Lie algebra. Indeed, it is easy to check that they ...
- 1,484
1
vote
Lyndon words and Hall basis
There is no simple connection. Note that there are many different Hall sets H. The Lyndon basis is not such a Hall basis.
However, one can easily relax the ...
- 380
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