In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."
Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map
X
↦
X
[
p
]
satisfying
a
d
(
X
[
p
]
)
=
a
d
(
X
)
p
for all
X
∈
L
,
(
t
X
)
[
p
]
=
t
p
X
[
p
]
for all
t
∈
k
,
X
∈
L
,
(
X
+
Y
)
[
p
]
=
X
[
p
]
+
Y
[
p
]
+
∑
i
=
1
p
−
1
s
i
(
X
,
Y
)
i
, for all
X
,
Y
∈
L
, where
s
i
(
X
,
Y
)
is the coefficient of
t
i
−
1
in the formal expression
a
d
(
t
X
+
Y
)
p
−
1
(
X
)
.
If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map.
For any associative algebra A defined over a field of characteristic p, the bracket operation
[
X
,
Y
]
:=
X
Y
−
Y
X
and p operation
X
[
p
]
:=
X
p
make A into a restricted Lie algebra
L
i
e
(
A
)
.
Let G be an algebraic group over a field k of characteristic p, and
L
i
e
(
G
)
be the Zariski tangent space at the identity element of G. Each element of
L
i
e
(
G
)
uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on
L
i
e
(
G
)
just as in the Lie group case. If p>0, the Frobenius map
x
↦
x
p
defines a p operation on
L
i
e
(
G
)
.
The functor
A
↦
L
i
e
(
A
)
has a left adjoint
L
↦
U
[
p
]
(
L
)
called the restricted universal enveloping algebra. To construct this, let
U
(
L
)
be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form
x
p
−
x
[
p
]
, we set
U
[
p
]
(
L
)
=
U
(
L
)
/
I
. It satisfies a form of the PBW theorem.
