Skip to Main Content
Algebraic Operads
book

Algebraic Operads

by Murray R. Bremner, Vladimir Dotsenko
April 2016
Intermediate to advanced content levelIntermediate to advanced
383 pages
11h 46m
English
Chapman and Hall/CRC
Content preview from Algebraic Operads
Noncommutative Associative Algebras 47
2.5.3 Universal enveloping algebras of Lie algebras
Theorem 2.5.3.1 (Poincaré–Birkhoff–Witt (PBW) theorem). If L is a fi-
nite dimensional Lie algebra over a field F with a totally ordered basis
X = {x
1
, . . . , x
n
}, then a basis of its universal associative enveloping alge-
bra U (L) consists of the monomials x
e
1
1
···x
e
n
n
with e
1
, . . . , e
n
0. Therefore:
(i) U(L) is infinite dimensional.
(ii) The canonical map α: L U (L) is injective.
(iii) L is isomorphic to a subalgebra of the Lie algebra U(L)
.
Proof. By Definition 2.1.1.2, the algebra U (L) is a quotient of T (L)
=
T (X).
We equip T (X) with the glex order with x
1
··· x
n
. Since both the
Lie bracket and the product in T (L)
=
T (X) are bilinear, it is sufficien ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Reinventing the Organization for GenAI and LLMs

Reinventing the Organization for GenAI and LLMs

Ethan Mollick
Algebraic and Stochastic Coding Theory

Algebraic and Stochastic Coding Theory

Dave K. Kythe, Prem K. Kythe

Publisher Resources

ISBN: 9781482248579