Lie algebra, free
over a ring
A Lie algebra over
in which one can distinguish a free generating set
, a mapping from which into an arbitrary algebra
over
can be be extended to a homomorphism from
into
. The cardinality of
completely determines
and is called its rank. A free Lie algebra is a free
-module (for bases of it see Basic commutator). A subalgebra
of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [1]). If
, then this is true only under the condition that
is a free Abelian group [2]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [3]. W. Magnus [4] established canonical connections between free Lie algebras and both free groups and free associative algebras.
References
[1] | A.I. Shirshov, "Subalgebras of free Lie algebras" Mat. Sb. , 33 : 2 (1953) pp. 441–452 (In Russian) |
[2] | E. Witt, "Die Unterringe der freien Lieschen Ringe" Math. Z. , 64 (1956) pp. 195–216 |
[3] | G.P. Kukin, "Intersection of subalgebras of a free Lie algebra" Algebra and Logic , 16 (1977) pp. 387–394 Algebra i Logika , 16 (1977) pp. 577–587 |
[4] | W. Magnus, "Ueber Beziehungen zwischen höheren Kommutatoren" J. Reine Angew. Math. , 177 (1937) pp. 105–115 |
[5] | Yu.A. Bakhturin, "Identical relations in Lie algebras" , VNU , Utrecht (1987) (Translated from Russian) |
Comments
To construct one can start from the free associative algebra
generated by
, which is made into a Lie algebra
by taking as Lie product
. Then
is the Lie subalgebra of
generated by
, and
is the universal enveloping algebra of
.
In case is a field of characteristic 0, more precise results on which elements of
belong to
are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element
of degree
belongs to
if and only if
, where
is the linear mapping defined by
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for . Friedrichs' theorem says for the case of finite
that
belongs to
if and only if
, where
is the homomorphism
defined by
for
.
Free Lie algebras are the best context for the formulation of the Campbell–Baker–Hausdorff formula in its most general form.
References
[a1] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |
[a2] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
Lie algebra, free. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_free&oldid=36855