- Source: Quasi-Hopf algebra
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989.
A quasi-Hopf algebra is a quasi-bialgebra
B
A
=
(
A
,
Δ
,
ε
,
Φ
)
{\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}
for which there exist
α
,
β
∈
A
{\displaystyle \alpha ,\beta \in {\mathcal {A}}}
and a bijective antihomomorphism S (antipode) of
A
{\displaystyle {\mathcal {A}}}
such that
∑
i
S
(
b
i
)
α
c
i
=
ε
(
a
)
α
{\displaystyle \sum _{i}S(b_{i})\alpha c_{i}=\varepsilon (a)\alpha }
∑
i
b
i
β
S
(
c
i
)
=
ε
(
a
)
β
{\displaystyle \sum _{i}b_{i}\beta S(c_{i})=\varepsilon (a)\beta }
for all
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
and where
Δ
(
a
)
=
∑
i
b
i
⊗
c
i
{\displaystyle \Delta (a)=\sum _{i}b_{i}\otimes c_{i}}
and
∑
i
X
i
β
S
(
Y
i
)
α
Z
i
=
I
,
{\displaystyle \sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb {I} ,}
∑
j
S
(
P
j
)
α
Q
j
β
S
(
R
j
)
=
I
.
{\displaystyle \sum _{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})=\mathbb {I} .}
where the expansions for the quantities
Φ
{\displaystyle \Phi }
and
Φ
−
1
{\displaystyle \Phi ^{-1}}
are given by
Φ
=
∑
i
X
i
⊗
Y
i
⊗
Z
i
{\displaystyle \Phi =\sum _{i}X_{i}\otimes Y_{i}\otimes Z_{i}}
and
Φ
−
1
=
∑
j
P
j
⊗
Q
j
⊗
R
j
.
{\displaystyle \Phi ^{-1}=\sum _{j}P_{j}\otimes Q_{j}\otimes R_{j}.}
As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting.
Usage
Quasi-Hopf algebras form the basis of the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in Statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the Heisenberg XXZ model in the framework of the algebraic Bethe ansatz. It provides a framework for solving two-dimensional integrable models by using the quantum inverse scattering method.
See also
Quasitriangular Hopf algebra
Quasi-triangular quasi-Hopf algebra
Ribbon Hopf algebra
References
Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad Math J. 1 (1989), 1419-1457
J. M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000