In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of
H
⊗
H
such that
R
Δ
(
x
)
R
−
1
=
(
T
∘
Δ
)
(
x
)
for all
x
∈
H
, where
Δ
is the coproduct on H, and the linear map
T
:
H
⊗
H
→
H
⊗
H
is given by
T
(
x
⊗
y
)
=
y
⊗
x
,
(
Δ
⊗
1
)
(
R
)
=
R
13
R
23
,
(
1
⊗
Δ
)
(
R
)
=
R
13
R
12
,
where
R
12
=
ϕ
12
(
R
)
,
R
13
=
ϕ
13
(
R
)
, and
R
23
=
ϕ
23
(
R
)
, where
ϕ
12
:
H
⊗
H
→
H
⊗
H
⊗
H
,
ϕ
13
:
H
⊗
H
→
H
⊗
H
⊗
H
, and
ϕ
23
:
H
⊗
H
→
H
⊗
H
⊗
H
, are algebra morphisms determined by
ϕ
12
(
a
⊗
b
)
=
a
⊗
b
⊗
1
,
ϕ
13
(
a
⊗
b
)
=
a
⊗
1
⊗
b
,
ϕ
23
(
a
⊗
b
)
=
1
⊗
a
⊗
b
.
R is called the R-matrix.
As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang-Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity,
(
ϵ
⊗
1
)
R
=
(
1
⊗
ϵ
)
R
=
1
∈
H
; moreover
R
−
1
=
(
S
⊗
1
)
(
R
)
,
R
=
(
1
⊗
S
)
(
R
−
1
)
, and
(
S
⊗
S
)
(
R
)
=
R
. One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element:
S
2
(
x
)
=
u
x
u
−
1
where
u
:=
m
(
S
⊗
1
)
R
21
(cf. Ribbon Hopf algebras).
It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element
F
=
∑
i
f
i
⊗
f
i
∈
A
⊗
A
such that
(
ε
⊗
i
d
)
F
=
(
i
d
⊗
ε
)
F
=
1
and satisfying the cocycle condition
(
F
⊗
1
)
∘
(
Δ
⊗
i
d
)
F
=
(
1
⊗
F
)
∘
(
i
d
⊗
Δ
)
F
Furthermore,
u
=
∑
i
f
i
S
(
f
i
)
is invertible and the twisted antipode is given by
S
′
(
a
)
=
u
S
(
a
)
u
−
1
, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.
