In physics and mathematics, the κ-Poincaré algebra, named after Henri Poincaré, is a deformation of the Poincaré algebra into an Hopf algebra. In the bicrossproduct basis, introduced by Majid-Ruegg its commutation rules reads:
[
P
μ
,
P
ν
]
=
0
[
R
j
,
P
0
]
=
0
,
[
R
j
,
P
k
]
=
i
ε
j
k
l
P
l
,
[
R
j
,
N
k
]
=
i
ε
j
k
l
N
l
,
[
R
j
,
R
k
]
=
i
ε
j
k
l
R
l
[
N
j
,
P
0
]
=
i
P
j
,
[
N
j
,
P
k
]
=
i
δ
j
k
(
1
−
e
−
2
λ
P
0
2
λ
+
λ
2
|
P
→
|
2
)
−
i
λ
P
j
P
k
,
[
N
j
,
N
k
]
=
−
i
ε
j
k
l
R
l
Where
P
μ
are the translation generators,
R
j
the rotations and
N
j
the boosts. The coproducts are:
Δ
P
j
=
P
j
⊗
1
+
e
−
λ
P
0
⊗
P
j
,
Δ
P
0
=
P
0
⊗
1
+
1
⊗
P
0
Δ
R
j
=
R
j
⊗
1
+
1
⊗
R
j
Δ
N
k
=
N
k
⊗
1
+
e
−
λ
P
0
⊗
N
k
+
i
λ
ε
k
l
m
P
l
⊗
R
m
.
The antipodes and the counits:
S
(
P
0
)
=
−
P
0
S
(
P
j
)
=
−
e
λ
P
0
P
j
S
(
R
j
)
=
−
R
j
S
(
N
j
)
=
−
e
λ
P
0
N
j
+
i
λ
ε
j
k
l
e
λ
P
0
P
k
R
l
ε
(
P
0
)
=
0
ε
(
P
j
)
=
0
ε
(
R
j
)
=
0
ε
(
N
j
)
=
0
The κ-Poincaré algebra is the dual Hopf algebra to the κ-Poincaré group, and can be interpreted as its “infinitesimal” version.
