Poincaré–Birkhoff–Witt theorem
Encyclopedia
In the theory of Lie algebra
s, the Poincaré–Birkhoff–Witt theorem (Poincaré
(1900), G. D. Birkhoff
(1937), Witt
(1937); frequently contracted to PBW theorem) is a result giving an explicit description of the universal enveloping algebra
of a Lie algebra. The term 'PBW type theorem' or even 'PBW theorem' may also refer to various analogues of the original theorem, comparing a filtered algebra
to its associated graded algebra, in particular, in the area of quantum groups.
V over a field
has a basis
; this is a set S such that any element of V is a unique (finite) linear combination
of elements of S. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are totally ordered
by some relation which we denote ≤.
If L is a Lie algebra over a field K, there is a canonical K-linear map
h from L into the universal enveloping algebra
U(L).
Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite sequence (x1, x2 ..., xn) of elements of X which is non-decreasing in the order ≤, that is, x1 ≤x2 ≤ ... ≤ xn. Extend h to all canonical monomials as follows: If (x1, x2, ..., xn) is a canonical monomial, let
Then h is injective on the set of canonical monomials and its range is a basis of the K-vector space U(L).
Stated somewhat differently, consider Y = h(X). Y is totally ordered by the induced ordering from X. The set of monomials
where y12 < ... < yn are elements of Y, and the exponents are non-negative, together with the multiplicative unit 1, form a basis for U(L). Note that the unit element 1 corresponds to the empty canonical monomial.
The multiplicative structure of U(L) is determined by the structure constants in the basis X, that is, the coefficients cu,v,x such that
This relation allows one to reduce any product of ys to a linear combination of canonical monomials: The structure constants determine yiyj – yjyi, i.e. what to do in order to change the order of two elements of Y in a product. This fact, modulo an inductive argument on the degree of (non-canonical) monomials, shows one can always achieve products where the factors are ordered in a non-decreasing fashion.
The Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the end result of this reduction is unique and does not depend on the order in which one swaps adjacent elements.
Corollary. If L is a Lie algebra over a field, the canonical map L → U(L) is injective. In particular, any Lie algebra over a field is isomorphic to a Lie subalgebra of an associative algebra.
To extend to the case when L is no longer a free K-module, one needs to make a reformulation that does not use bases. This involves replacing the space of monomials in some basis with the Symmetric algebra
, S(L), on L.
In the case that K contains the field of rational numbers, one can consider the natural map from S(L) to U(L), sending a monomial
. for 
, to the element 
. Then, one has the theorem that this map is an isomorphism of K-modules.
Still more generally and naturally, one can consider U(L) as a filtered algebra
, equipped with the filtration given by specifying that
lies in filtered degree 
. The map 
of K-modules canonically extends to a map 
of algebras, where 
is the tensor algebra
on L (for example, by the universal property of tensor algebras), and this is a filtered map equipping
with the filtration putting L in degree one (actually, 
is graded). Then, passing to the associated graded, one gets a canonical morphism 
, which kills the elements 
for 
, and hence descends to a canonical morphism 
. Then, the (graded) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism.
This is not true for all K and L (see, for example, the last section of Cohn's 1961 paper), but is true in many cases. These include the aforementioned ones, where either L is a free K-module, or K contains the field of rational numbers. More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free
as an abelian group
, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain
. See, for example, the 1969 paper by Higgins for these statements.
Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonical morphism
lifts to a K-module isomorphism 
, without taking associated graded. This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rational numbers, using the construction outlined here (in fact, the result is a coalgebra
isomorphism, and not merely a K-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra structures such that
for 
). This stronger statement, however, might not extend to all of the cases in the previous paragraph.
Birkhoff and Witt do not mention Poincaré's work in their 1937 papers. Cartan
and Eilenberg
in their 1956 book call the theorem Poincaré-Witt Theorem and attribute the complete proof to Witt. Bourbaki were the first to use all three names in their 1960 book. Knapp
presents a clear illustration of the shifting tradition. In his 1986 book he calls it Birkhoff-Witt Theorem while in his later 1996 book he switches to Poincaré-Birkhoff-Witt Theorem.
It is not clear whether Poincaré's result was complete. Ton-That and Tran conclude that Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff. On the other hand, they point out that Poincaré makes several statements without bothering to prove them. Their own proofs of all the steps are rather long according to their admission.
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s, the Poincaré–Birkhoff–Witt theorem (Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
(1900), G. D. Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
(1937), Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
(1937); frequently contracted to PBW theorem) is a result giving an explicit description of the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of a Lie algebra. The term 'PBW type theorem' or even 'PBW theorem' may also refer to various analogues of the original theorem, comparing a filtered algebra
Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
to its associated graded algebra, in particular, in the area of quantum groups.
Statement of the theorem
Recall that any vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
has a basis
Basis
Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...
; this is a set S such that any element of V is a unique (finite) linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of elements of S. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
by some relation which we denote ≤.
If L is a Lie algebra over a field K, there is a canonical K-linear map
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
h from L into the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
U(L).
Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite sequence (x1, x2 ..., xn) of elements of X which is non-decreasing in the order ≤, that is, x1 ≤x2 ≤ ... ≤ xn. Extend h to all canonical monomials as follows: If (x1, x2, ..., xn) is a canonical monomial, let
Then h is injective on the set of canonical monomials and its range is a basis of the K-vector space U(L).
Stated somewhat differently, consider Y = h(X). Y is totally ordered by the induced ordering from X. The set of monomials
where y1
The multiplicative structure of U(L) is determined by the structure constants in the basis X, that is, the coefficients cu,v,x such that
This relation allows one to reduce any product of ys to a linear combination of canonical monomials: The structure constants determine yiyj – yjyi, i.e. what to do in order to change the order of two elements of Y in a product. This fact, modulo an inductive argument on the degree of (non-canonical) monomials, shows one can always achieve products where the factors are ordered in a non-decreasing fashion.
The Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the end result of this reduction is unique and does not depend on the order in which one swaps adjacent elements.
Corollary. If L is a Lie algebra over a field, the canonical map L → U(L) is injective. In particular, any Lie algebra over a field is isomorphic to a Lie subalgebra of an associative algebra.
More general contexts
Already at its earliest stages, it was known that K could be replaced by any commutative ring, provided that L is a free K-module, i.e., has a basis as above.To extend to the case when L is no longer a free K-module, one needs to make a reformulation that does not use bases. This involves replacing the space of monomials in some basis with the Symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
, S(L), on L.
In the case that K contains the field of rational numbers, one can consider the natural map from S(L) to U(L), sending a monomial
. for , to the element . Then, one has the theorem that this map is an isomorphism of K-modules.Still more generally and naturally, one can consider U(L) as a filtered algebra
Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
, equipped with the filtration given by specifying that
lies in filtered degree . The map of K-modules canonically extends to a map of algebras, where is the tensor algebraTensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
on L (for example, by the universal property of tensor algebras), and this is a filtered map equipping
with the filtration putting L in degree one (actually, is graded). Then, passing to the associated graded, one gets a canonical morphism , which kills the elements for , and hence descends to a canonical morphism . Then, the (graded) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism.This is not true for all K and L (see, for example, the last section of Cohn's 1961 paper), but is true in many cases. These include the aforementioned ones, where either L is a free K-module, or K contains the field of rational numbers. More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free
Torsion-free
In mathematics, the term torsion-free may refer to several unrelated notions:* In abstract algebra, a group is torsion-free if the only element of finite order is the identity....
as an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
. See, for example, the 1969 paper by Higgins for these statements.
Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonical morphism
lifts to a K-module isomorphism , without taking associated graded. This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rational numbers, using the construction outlined here (in fact, the result is a coalgebraCoalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
isomorphism, and not merely a K-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra structures such that
for ). This stronger statement, however, might not extend to all of the cases in the previous paragraph.History of the theorem
Ton-That and Tran have investigated the history of the theorem. They have found out that the majority of the sources before Bourbaki's 1960 book call it Birkhoff-Witt theorem. Following this old tradition, Fofanova in her encyclopaedic entry says that H. Poincaré obtained the first variant of the theorem. She further says that the theorem was subsequently completely demonstrated by E. Witt and G.D. Birkhoff. It appears that pre-Bourbaki sources were not familiar with Poincaré's paper.Birkhoff and Witt do not mention Poincaré's work in their 1937 papers. Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
and Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
in their 1956 book call the theorem Poincaré-Witt Theorem and attribute the complete proof to Witt. Bourbaki were the first to use all three names in their 1960 book. Knapp
Anthony Knapp
Anthony W. Knapp is a mathematician at the US State University of New York, Stony Brook working on representation theory who classified the tempered representations of a semisimple Lie group.He won the Leroy P...
presents a clear illustration of the shifting tradition. In his 1986 book he calls it Birkhoff-Witt Theorem while in his later 1996 book he switches to Poincaré-Birkhoff-Witt Theorem.
It is not clear whether Poincaré's result was complete. Ton-That and Tran conclude that Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff. On the other hand, they point out that Poincaré makes several statements without bothering to prove them. Their own proofs of all the steps are rather long according to their admission.