# Free Lie algebra
A **free Lie algebra** is a [Lie algebra](https://www.edgepedia.org/lie-algebra) \( L = L(X) \) over a ring \( R \) with a distinguished free generating set \( X \), characterized by the universal property that any mapping of \( X \) into an arbitrary Lie algebra \( G \) over \( R \) extends to a homomorphism from \( L \) into \( G \)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). It is the Lie-algebra analog of a free group or a free associative algebra: it imposes no relations on its generators beyond the defining Lie identities of anti-commutativity and the Jacobi identity[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) • [2](https://encyclopediaofmath.org/wiki/Lie_algebra).
| Key fact | Detail |
|---|---|
| Universal property | A map from the free generating set \( X \) into any Lie algebra \( G \) over \( R \) extends to a homomorphism \( L(X) \to G \)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) |
| Rank | The cardinality of \( X \) completely determines \( L \) and is called its rank[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) |
| Module structure | A free Lie algebra is a free \( R \)-module[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) |
| Subalgebras | Over a field, every subalgebra of a free Lie algebra is itself free (Shirshov's theorem)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) |
| Enveloping algebra | \( L(X) \) is the Lie subalgebra of the free associative algebra \( A(X) \) generated by \( X \), and \( A(X) \) is the universal enveloping algebra of \( L(X) \)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) |
| Standard bases | Hall bases (from Philip Hall's 1930s work on commutators), the Lyndon basis, and the Shirshov basis, which coincides with the Lyndon basis up to symmetries[3](https://www.researchgate.net/publication/2846238_Bases_for_Lie_algebras_and_a_continuous_CBH_formula) • [4](https://encyclopediaofmath.org/index.php?title=Shirshov_basis) |
## Definition and universal property
A Lie algebra over a commutative ring \( k \) is a unitary \( k \)-module \( L \) with a bilinear bracket \( (x,y) \mapsto [x,y] \) satisfying \( [x,x] = 0 \), equivalently \( [x,y] = -[y,x] \), and the Jacobi identity \( [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \)[2](https://encyclopediaofmath.org/wiki/Lie_algebra). The most important case is that in which \( k \) is a field, especially \( \mathbb{R} \) or \( \mathbb{C} \)[2](https://encyclopediaofmath.org/wiki/Lie_algebra).
The free Lie algebra \( L(X) \) on a set \( X \) satisfies the universal mapping property stated above: generators can be sent anywhere, and the rest of the homomorphism is forced[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). The cardinality of \( X \), called the rank, completely determines the algebra up to isomorphism[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). Concretely, the algebra can be built as the free magmatic algebra on \( X \), the vector space of all formal bracketings of the generators, quotiented by the ideal generated by \( xx \), \( xy + yx \), and \( x(yz) + y(zx) + z(xy) \), which encode the two Lie identities[5](https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/free_lie_algebra.html).
## Relation to free associative algebras and free groups
Let \( A(X) \) be the free associative algebra on \( X \), constructible as the tensor algebra over a [free module](https://www.edgepedia.org/free-module) with basis \( X \)[6](https://planetmath.org/freeliealgebra). Then \( L(X) \) is the Lie subalgebra of \( A(X) \), with bracket the commutator, generated by \( X \), and \( A(X) \) is the universal enveloping algebra of \( L(X) \)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free) • [6](https://planetmath.org/freeliealgebra). W. Magnus established canonical connections between free Lie algebras and both free groups and free associative algebras, connections that underlie the commutator calculus in free groups[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free).
## Bases
Because \( L(X) \) is a free \( R \)-module, it has bases, and several explicit combinatorial bases are known[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). The first bases build on [Philip Hall](https://www.edgepedia.org/philip-hall)'s work in the 1930s on commutators; a Hall basis indexes basis elements by bracketed words built inductively from the generators[3](https://www.researchgate.net/publication/2846238_Bases_for_Lie_algebras_and_a_continuous_CBH_formula). One natural way to prove that Hall words form a basis is a direct construction: start with a linear space spanned by Hall words, define the Lie product of Hall words, and check that the product yields the Lie identities[7](https://arxiv.org/abs/0804.0914).
**Lyndon and Shirshov bases.** A word \( P \) over an ordered alphabet is Lyndon if it is smaller in the dictionary ordering than any of its nonempty suffixes[8](https://link.springer.com/article/10.1007/s00220-023-04748-z). The Lie monomials \( \ell[P] \), defined recursively by \( \ell[P] = [\ell[u], \ell[v]] \) for the standard factorization \( P = uv \) and \( \ell[a] = a \) for a single letter, form a basis of the free Lie algebra as \( P \) ranges over Lyndon words[8](https://link.springer.com/article/10.1007/s00220-023-04748-z). The Shirshov basis, introduced by A. I. Shirshov in 1962, is identical to the Lyndon basis up to symmetries; Shirshov's lemma shows that any word is a non-decreasing product of words strictly greater than all their circular shifts, and a left-normed bracketing \( \pi(w) = [\pi(f_1), [\pi(f_2), \dots [\pi(f_n), a]]] \) of such words gives the basis elements[4](https://encyclopediaofmath.org/index.php?title=Shirshov_basis). The Lyndon basis does not respect the grading by word length, while the Hall basis does[9](https://mathoverflow.net/questions/97703/list-of-hall-basis).
## Structure constants and computation
The structure constants of \( L(X) \), the coefficients expressing a bracket of two basis elements in a basis, have been studied for their growth. A priori bounds valid for all Hall bases follow from the classical recursive decomposition algorithm, and lower bounds show that the constants grow at least geometrically in every Hall basis[10](https://arxiv.org/abs/2111.14398). For the historical length-compatible Hall bases and the Lyndon basis, sharp upper bounds are geometric in the length of the brackets[10](https://arxiv.org/abs/2111.14398). In the two-indeterminate case, one can construct a Hall basis whose structure constants grow as slowly as possible, matching the general lower bound with constants linked to the [Fibonacci sequence](https://www.edgepedia.org/fibonacci-sequence), and another whose constants grow super-geometrically[10](https://arxiv.org/abs/2111.14398).
These bases are implemented in computer algebra systems. SageMath realizes the free Lie algebra in both the Hall basis and the Lyndon basis and converts between them; for three generators over \( \mathbb{Q} \), the degree-2 graded piece of either basis consists of the three brackets \( [x_0, x_1] \), \( [x_0, x_2] \), and \( [x_1, x_2] \)[5](https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/free_lie_algebra.html). The free Lie algebra also carries a duality with the shuffle product: Ree's theorem characterizes Lie polynomials as the elements orthogonal to all proper shuffles, and the Lyndon-word basis of the algebra is dual to the Lyndon-word basis of its dual space[8](https://link.springer.com/article/10.1007/s00220-023-04748-z).
## Subalgebras
Shirshov's theorem states that a subalgebra \( M \) of a free Lie algebra over a field is itself a free Lie algebra[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). Over the integers the statement needs a hypothesis: it holds only when the quotient \( L/M \) is a free [Abelian group](https://www.edgepedia.org/abelian-group)[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free). When a free Lie algebra is defined over a field, its finitely generated subalgebras constitute a sublattice within the lattice formed by all of its subalgebras.[1](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free)
## References
1. [Lie algebra, free, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lie_algebra%2C_free)
2. [Lie algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lie_algebra)
3. [Bases for Lie algebras and a continuous CBH formula, ResearchGate](https://www.researchgate.net/publication/2846238_Bases_for_Lie_algebras_and_a_continuous_CBH_formula)
4. [Shirshov basis, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Shirshov_basis)
5. [Free Lie Algebras, SageMath Reference Manual: Algebras](https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/free_lie_algebra.html)
6. [Free Lie algebra, PlanetMath](https://planetmath.org/freeliealgebra)
7. [Anti-commutative Groebner-Shirshov basis of a free Lie algebra, arXiv:0804.0914](https://arxiv.org/abs/0804.0914)
8. [A Lie Bracket for the Momentum Kernel, Communications in Mathematical Physics (2023)](https://link.springer.com/article/10.1007/s00220-023-04748-z)
9. [List of Hall basis, MathOverflow](https://mathoverflow.net/questions/97703/list-of-hall-basis)
10. [Growth of structure constants of free Lie algebras relative to Hall bases, arXiv:2111.14398](https://arxiv.org/abs/2111.14398)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Computational and symbolic algebra › Algebraic combinatorics and graph theory › Combinatorics on words and free structures*
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