an:01538694
Zbl 0958.01012
Ton-That, Tuong; Tran, Thai-Duong
Poincar??'s proof of the so-called Birkhoff-Witt theorem
EN
Rev. Hist. Math. 5, No. 2, 249-284 (1999).
00065527
1999
j
01A60 17-03 01A55 17B35
Henri Poincar??; Lie algebra; universal enveloping algebra; Garrett Birkhoff; Ernst Witt; symmetric algebra
Who developed the universal enveloping algebra of a Lie algebra (real or complex)? Who introduced the canonical map of the symmetric algebra onto the universal enveloping algebra? \textit{Garrett Birkhoff}'s article is dated 1937 [Ann. Math. (2) 38, 526-532 (1937; Zbl 0016.24402)] as is \textit{Ernst Witt}'s [J. Reine Angew. Math. 177, 152-160 (1937; Zbl 0016.24401)]. Poincar??'s proof appeared in 1900 [see C. R. 128, 1065-1069 (1899; JFM 30.0334.01)]. There are three pages of historical introduction; four pages of Poincar?? on Lie groups; 25 pages on concepts, algebras, polynomial bases, and proof; and a conclusion and bibliography of 40 items, and E. T. Bell's book of fairy tales. Much of the argument is long formulae, but the paper is clear, easy to read, and bites where it should.
James J.Cross (Parkville)
Zbl 0016.24401; Zbl 0016.24402; JFM 30.0334.01