
52 Algebraic Operads: An Algorithmic Companion
We will recall his original definition, and explain how to view it in the language
of Gröbner bases.
Definition 2.5.4.1 (PBW basis). Let A = T (x
1
, . . . , x
n
)/(R) be a quadratic
algebra. Suppose that for each i ≥ 0, cosets of monomials belonging to the set
B := B
i
⊂ X
i
is a basis for the homogeneous component A
i
. This means, as
usual, that the structure constants of A with respect to B are defined:
(b
1
+ I)(b
2
+ I) =
X
b∈B
c
b
b
1
,b
2
(b + I). (2.14)
Equip T (x
1
, . . . , x
n
) with the glex order. The basis B is called a Poincaré–
Birkhoff–Witt (PBW) basis of A if the following two conditions hold:
1. for each two basis