- Source: Quasi-triangular quasi-Hopf algebra
A quasi" target="_blank">quasi-triangular quasi" target="_blank">quasi-Hopf algebra is a specialized form of a quasi" target="_blank">quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi" target="_blank">quasi-triangular Hopf algebra.
A quasi" target="_blank">quasi-triangular quasi" target="_blank">quasi-Hopf algebra is a set
H
A
=
(
A
,
R
,
Δ
,
ε
,
Φ
)
{\displaystyle {\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )}
where
B
A
=
(
A
,
Δ
,
ε
,
Φ
)
{\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}
is a quasi" target="_blank">quasi-Hopf algebra and
R
∈
A
⊗
A
{\displaystyle R\in {\mathcal {A\otimes A}}}
known as the R-matrix, is an invertible element such that
R
Δ
(
a
)
=
σ
∘
Δ
(
a
)
R
{\displaystyle R\Delta (a)=\sigma \circ \Delta (a)R}
for all
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
, where
σ
:
A
⊗
A
→
A
⊗
A
{\displaystyle \sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}}
is the switch map given by
x
⊗
y
→
y
⊗
x
{\displaystyle x\otimes y\rightarrow y\otimes x}
, and
(
Δ
⊗
id
)
R
=
Φ
231
R
13
Φ
132
−
1
R
23
Φ
123
{\displaystyle (\Delta \otimes \operatorname {id} )R=\Phi _{231}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}}
(
id
⊗
Δ
)
R
=
Φ
312
−
1
R
13
Φ
213
R
12
Φ
123
−
1
{\displaystyle (\operatorname {id} \otimes \Delta )R=\Phi _{312}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}}
where
Φ
a
b
c
=
x
a
⊗
x
b
⊗
x
c
{\displaystyle \Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}}
and
Φ
123
=
Φ
=
x
1
⊗
x
2
⊗
x
3
∈
A
⊗
A
⊗
A
{\displaystyle \Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}}
.
The quasi" target="_blank">quasi-Hopf algebra becomes triangular if in addition,
R
21
R
12
=
1
{\displaystyle R_{21}R_{12}=1}
.
The twisting of
H
A
{\displaystyle {\mathcal {H_{A}}}}
by
F
∈
A
⊗
A
{\displaystyle F\in {\mathcal {A\otimes A}}}
is the same as for a quasi" target="_blank">quasi-Hopf algebra, with the additional definition of the twisted R-matrix
A quasi" target="_blank">quasi-triangular (resp. triangular) quasi" target="_blank">quasi-Hopf algebra with
Φ
=
1
{\displaystyle \Phi =1}
is a quasi" target="_blank">quasi-triangular (resp. triangular) Hopf algebra as the latter two conditions in the definition reduce the conditions of quasi" target="_blank">quasi-triangularity of a Hopf algebra.
Similarly to the twisting properties of the quasi" target="_blank">quasi-Hopf algebra, the property of being quasi" target="_blank">quasi-triangular or triangular quasi" target="_blank">quasi-Hopf algebra is preserved by twisting.
See also
Ribbon Hopf algebra
References
Vladimir Drinfeld, "quasi" target="_blank">Quasi-Hopf algebras", Leningrad mathematical journal (1989), 1419–1457
J. M. Maillet and J. Sanchez de Santos, "Drinfeld Twists and Algebraic Bethe Ansatz", American Mathematical Society Translations: Series 2 Vol. 201, 2000