Last Updated：2026/01/01
Sentence
It
is
easily
seen
that
the
bracket
[x,y]:=xy-yx
of
two
primitive
elements
is
again
a
primitive
element.
It
follows
that
primitive
elements
form
a
Lie
algebra.
For
H=U(g)
any
element
of
g
is
primitive
and
in
fact
using
the
Poincaré-Birkhoff-Win
theorem,
one
can
show
that
the
set
of
primitive
elements
of
U(g)
coincides
with
the
Lie
algebra
g.
Quizzes for review
A primitive element of a Hopf algebra is an element h∈H such that 𝛥h=1⊗h+h⊗1. It is easily seen that the bracket [x,y]:=xy-yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For H=U(g) any element of g is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.
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