- Source: Quasitriangular Hopf algebra
In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of
H
⊗
H
{\displaystyle H\otimes H}
such that
R
Δ
(
x
)
R
−
1
=
(
T
∘
Δ
)
(
x
)
{\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)}
for all
x
∈
H
{\displaystyle x\in H}
, where
Δ
{\displaystyle \Delta }
is the coproduct on H, and the linear map
T
:
H
⊗
H
→
H
⊗
H
{\displaystyle T:H\otimes H\to H\otimes H}
is given by
T
(
x
⊗
y
)
=
y
⊗
x
{\displaystyle T(x\otimes y)=y\otimes x}
,
(
Δ
⊗
1
)
(
R
)
=
R
13
R
23
{\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}}
,
(
1
⊗
Δ
)
(
R
)
=
R
13
R
12
{\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}}
,
where
R
12
=
ϕ
12
(
R
)
{\displaystyle R_{12}=\phi _{12}(R)}
,
R
13
=
ϕ
13
(
R
)
{\displaystyle R_{13}=\phi _{13}(R)}
, and
R
23
=
ϕ
23
(
R
)
{\displaystyle R_{23}=\phi _{23}(R)}
, where
ϕ
12
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H}
,
ϕ
13
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H}
, and
ϕ
23
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H}
, are algebra morphisms determined by
ϕ
12
(
a
⊗
b
)
=
a
⊗
b
⊗
1
,
{\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}
ϕ
13
(
a
⊗
b
)
=
a
⊗
1
⊗
b
,
{\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}
ϕ
23
(
a
⊗
b
)
=
1
⊗
a
⊗
b
.
{\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes 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
{\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H}
; moreover
R
−
1
=
(
S
⊗
1
)
(
R
)
{\displaystyle R^{-1}=(S\otimes 1)(R)}
,
R
=
(
1
⊗
S
)
(
R
−
1
)
{\displaystyle R=(1\otimes S)(R^{-1})}
, and
(
S
⊗
S
)
(
R
)
=
R
{\displaystyle (S\otimes 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
{\displaystyle S^{2}(x)=uxu^{-1}}
where
u
:=
m
(
S
⊗
1
)
R
21
{\displaystyle u:=m(S\otimes 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.
If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding
c
U
,
V
(
u
⊗
v
)
=
T
(
R
⋅
(
u
⊗
v
)
)
=
T
(
R
1
u
⊗
R
2
v
)
{\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)}
.
Twisting
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
{\displaystyle F=\sum _{i}f^{i}\otimes f_{i}\in {\mathcal {A\otimes A}}}
such that
(
ε
⊗
i
d
)
F
=
(
i
d
⊗
ε
)
F
=
1
{\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}
and satisfying the cocycle condition
(
F
⊗
1
)
⋅
(
Δ
⊗
i
d
)
(
F
)
=
(
1
⊗
F
)
⋅
(
i
d
⊗
Δ
)
(
F
)
{\displaystyle (F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}
Furthermore,
u
=
∑
i
f
i
S
(
f
i
)
{\displaystyle u=\sum _{i}f^{i}S(f_{i})}
is invertible and the twisted antipode is given by
S
′
(
a
)
=
u
S
(
a
)
u
−
1
{\displaystyle S'(a)=uS(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.
See also
Quasi-triangular quasi-Hopf algebra
Ribbon Hopf algebra
Notes
References
Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.