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In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms ''PBW type theorem'' and ''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. == Statement of the theorem == Recall that any vector space ''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, let ''h'' denote the canonical K-linear map 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 (''x''1, ''x''2 ..., ''x''''n'') of elements of ''X'' which is non-decreasing in the order ≤, that is, ''x''1 ≤''x''2 ≤ ... ≤ ''x''''n''. Extend ''h'' to all canonical monomials as follows: If (''x''1, ''x''2, ..., ''x''''n'') 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 ''y''1 <''y''2 < ... < ''y''''n'' 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 ''c''''u,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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poincaré–Birkhoff–Witt theorem」の詳細全文を読む スポンサード リンク
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