JHEP02(2020)009
Using the relativistic commutation relations (2.3) together with the reduction condi-
tion (2.10) for the zero element λ
2
, the commutation relations for the generators (2.11) read
[G
a
, H] = P
a
, (2.12a)
[J
ab
, P
c
] = 2δ
c[b
P
a]
, (2.12b)
[J
ab
, G
c
] = 2δ
c[b
G
a]
, (2.12c)
[J
ab
, J
cd
] = 4δ
[a[c
J
d]b]
, (2.12d)
[P
a
, H] = ΛG
a
, (2.12e)
which corresponds to the Newton-Hooke
∓
algebra (with Λ
≶
0) without extensions [19, 20],
this algebra reduces to the Galilei symmetry for Λ = 0. Therefore the use of the semi-
group S
(N)
E
will create a family of generalised Newton-Hooke algebras for all the possible
values of N.
The case with N = 2 leads to a Newton-Hooke algebra with some extensions. Unlike
the previous case, when using the semigroup S
(2)
E
, the element λ
2
is no longer the zero of
the semigroup. In this case the reduction condition (2.10) holds for a new semigroup zero
element λ
3
, and (2.11) has to be supplemented with two extra expanded generators
S
ab
= λ
2
⊗
˜
J
ab
, M = λ
2
⊗
˜
H . (2.13)
Using the semigroup product law (2.6) for N = 2, we find that the non-vanishing commuta-
tion relations for the expanded algebra are given by the Newton-Hooke commutators (2.12)
together with
[G
a
, P
b
] = δ
ab
M , (2.14a)
[G
a
, G
b
] = S
ab
, (2.14b)
[J
ab
, S
cd
] = 4δ
[a[c
S
d]b]
, (2.14c)
[P
a
, P
b
] = −ΛS
ab
. (2.14d)
This expansion produces the Bargmann central extension M and a non-central extension
S
ab
[40]. This algebra can be understood as the generalisation of the double central ex-
tension [41–43] of the Newton-Hooke algebra to D > 3. In fact, in 2+1 dimensions the
generator S
ab
can be dualised to a scalar and becomes central. In that case the alge-
bra (2.14) becomes exactly the extended Newton-Hooke algebra (Λ 6= 0) [44, 45] or the
extended Bargmann algebra (Λ = 0) [46, 47].
Extended (post-)Newtonian gravity algebra. For N = 3 we obtain the Newton-
Hooke version of the algebra found in [9] as the symmetry of post-Newtonian gravity.
Moreover, when considering N = 4, we find an extension of the post-Newtonian gravity
algebra that generalises the algebra of [11] to any dimension. Explicitly, for N = 3 the
generators of the expanded algebra are given by (2.11), (2.13) and
B
a
= λ
3
⊗
˜
G
a
,
T
a
= λ
3
⊗
˜
P
a
.
(2.15)
The zero of the semigroup S
(3)
E
is λ
4
, and the reduction condition (2.10) together with
the semigroup law and the (A)dS commutation relations (2.3) leads to the following non-
– 5 –
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