- Source: Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element
Φ
{\displaystyle \Phi }
which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.
Definition
A quasi-bialgebra
B
A
=
(
A
,
Δ
,
ε
,
Φ
,
l
,
r
)
{\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi ,l,r)}
is an algebra
A
{\displaystyle {\mathcal {A}}}
over a field
F
{\displaystyle \mathbb {F} }
equipped with morphisms of algebras
Δ
:
A
→
A
⊗
A
{\displaystyle \Delta :{\mathcal {A}}\rightarrow {\mathcal {A\otimes A}}}
ε
:
A
→
F
{\displaystyle \varepsilon :{\mathcal {A}}\rightarrow \mathbb {F} }
along with invertible elements
Φ
∈
A
⊗
A
⊗
A
{\displaystyle \Phi \in {\mathcal {A\otimes A\otimes A}}}
, and
r
,
l
∈
A
{\displaystyle r,l\in A}
such that the following identities hold:
(
i
d
⊗
Δ
)
∘
Δ
(
a
)
=
Φ
[
(
Δ
⊗
i
d
)
∘
Δ
(
a
)
]
Φ
−
1
,
∀
a
∈
A
{\displaystyle (id\otimes \Delta )\circ \Delta (a)=\Phi \lbrack (\Delta \otimes id)\circ \Delta (a)\rbrack \Phi ^{-1},\quad \forall a\in {\mathcal {A}}}
[
(
i
d
⊗
i
d
⊗
Δ
)
(
Φ
)
]
[
(
Δ
⊗
i
d
⊗
i
d
)
(
Φ
)
]
=
(
1
⊗
Φ
)
[
(
i
d
⊗
Δ
⊗
i
d
)
(
Φ
)
]
(
Φ
⊗
1
)
{\displaystyle \lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)}
(
ε
⊗
i
d
)
(
Δ
a
)
=
l
−
1
a
l
,
(
i
d
⊗
ε
)
∘
Δ
=
r
−
1
a
r
,
∀
a
∈
A
{\displaystyle (\varepsilon \otimes id)(\Delta a)=l^{-1}al,\qquad (id\otimes \varepsilon )\circ \Delta =r^{-1}ar,\quad \forall a\in {\mathcal {A}}}
(
i
d
⊗
ε
⊗
i
d
)
(
Φ
)
=
r
⊗
l
−
1
.
{\displaystyle (id\otimes \varepsilon \otimes id)(\Phi )=r\otimes l^{-1}.}
Where
Δ
{\displaystyle \Delta }
and
ϵ
{\displaystyle \epsilon }
are called the comultiplication and counit,
r
{\displaystyle r}
and
l
{\displaystyle l}
are called the right and left unit constraints (resp.), and
Φ
{\displaystyle \Phi }
is sometimes called the Drinfeld associator.: 369–376 This definition is constructed so that the category
A
−
M
o
d
{\displaystyle {\mathcal {A}}-Mod}
is a tensor category under the usual vector space tensor product, and in fact this can be taken as the definition instead of the list of above identities.: 368 Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie.
l
=
r
=
1
{\displaystyle l=r=1}
the definition may sometimes be given with this assumed.: 370 Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints:
l
=
r
=
1
{\displaystyle l=r=1}
and
Φ
=
1
⊗
1
⊗
1
{\displaystyle \Phi =1\otimes 1\otimes 1}
.
Braided quasi-bialgebras
A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category
A
−
M
o
d
{\displaystyle {\mathcal {A}}-Mod}
is braided. Equivalently, by analogy with braided bialgebras, we can construct a notion of a universal R-matrix which controls the non-cocommutativity of a quasi-bialgebra. The definition is the same as in the braided bialgebra case except for additional complications in the formulas caused by adding in the associator.
Proposition: A quasi-bialgebra
(
A
,
Δ
,
ϵ
,
Φ
,
l
,
r
)
{\displaystyle ({\mathcal {A}},\Delta ,\epsilon ,\Phi ,l,r)}
is braided if it has a universal R-matrix, ie an invertible element
R
∈
A
⊗
A
{\displaystyle R\in {\mathcal {A\otimes A}}}
such that the following 3 identities hold:
(
Δ
o
p
)
(
a
)
=
R
Δ
(
a
)
R
−
1
{\displaystyle (\Delta ^{op})(a)=R\Delta (a)R^{-1}}
(
i
d
⊗
Δ
)
(
R
)
=
(
Φ
231
)
−
1
R
13
Φ
213
R
12
(
Φ
213
)
−
1
{\displaystyle (id\otimes \Delta )(R)=(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12}(\Phi _{213})^{-1}}
(
Δ
⊗
i
d
)
(
R
)
=
(
Φ
321
)
R
13
(
Φ
213
)
−
1
R
23
Φ
123
{\displaystyle (\Delta \otimes id)(R)=(\Phi _{321})R_{13}(\Phi _{213})^{-1}R_{23}\Phi _{123}}
Where, for every
a
1
⊗
.
.
.
⊗
a
k
∈
A
⊗
k
{\displaystyle a_{1}\otimes ...\otimes a_{k}\in {\mathcal {A}}^{\otimes k}}
,
a
i
1
i
2
.
.
.
i
n
{\displaystyle a_{i_{1}i_{2}...i_{n}}}
is the monomial with
a
j
{\displaystyle a_{j}}
in the
i
j
{\displaystyle i_{j}}
th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of
A
⊗
k
{\displaystyle {\mathcal {A}}^{\otimes k}}
.: 371
Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang–Baxter equation:
R
12
Φ
321
R
13
(
Φ
132
)
−
1
R
23
Φ
123
=
Φ
321
R
23
(
Φ
231
)
−
1
R
13
Φ
213
R
12
{\displaystyle R_{12}\Phi _{321}R_{13}(\Phi _{132})^{-1}R_{23}\Phi _{123}=\Phi _{321}R_{23}(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12}}
: 372
Twisting
Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume
r
=
l
=
1
{\displaystyle r=l=1}
) .
If
B
A
{\displaystyle {\mathcal {B_{A}}}}
is a quasi-bialgebra and
F
∈
A
⊗
A
{\displaystyle F\in {\mathcal {A\otimes A}}}
is an invertible element such that
(
ε
⊗
i
d
)
F
=
(
i
d
⊗
ε
)
F
=
1
{\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}
, set
Δ
′
(
a
)
=
F
Δ
(
a
)
F
−
1
,
∀
a
∈
A
{\displaystyle \Delta '(a)=F\Delta (a)F^{-1},\quad \forall a\in {\mathcal {A}}}
Φ
′
=
(
1
⊗
F
)
(
(
i
d
⊗
Δ
)
F
)
Φ
(
(
Δ
⊗
i
d
)
F
−
1
)
(
F
−
1
⊗
1
)
.
{\displaystyle \Phi '=(1\otimes F)\ ((id\otimes \Delta )F)\ \Phi \ ((\Delta \otimes id)F^{-1})\ (F^{-1}\otimes 1).}
Then, the set
(
A
,
Δ
′
,
ε
,
Φ
′
)
{\displaystyle ({\mathcal {A}},\Delta ',\varepsilon ,\Phi ')}
is also a quasi-bialgebra obtained by twisting
B
A
{\displaystyle {\mathcal {B_{A}}}}
by F, which is called a twist or gauge transformation.: 373 If
(
A
,
Δ
,
ε
,
Φ
)
{\displaystyle ({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}
was a braided quasi-bialgebra with universal R-matrix
R
{\displaystyle R}
, then so is
(
A
,
Δ
′
,
ε
,
Φ
′
)
{\displaystyle ({\mathcal {A}},\Delta ',\varepsilon ,\Phi ')}
with universal R-matrix
F
21
R
F
−
1
{\displaystyle F_{21}RF^{-1}}
(using the notation from the above section).: 376 However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by
F
1
{\displaystyle F_{1}}
and then
F
2
{\displaystyle F_{2}}
is equivalent to twisting by
F
2
F
1
{\displaystyle F_{2}F_{1}}
, and twisting by
F
{\displaystyle F}
then
F
−
1
{\displaystyle F^{-1}}
recovers the original quasi-bialgebra.
Twistings have the important property that they induce categorical equivalences on the tensor category of modules:
Theorem: Let
B
A
{\displaystyle {\mathcal {B_{A}}}}
,
B
A
′
{\displaystyle {\mathcal {B_{A'}}}}
be quasi-bialgebras, let
B
A
′
′
{\displaystyle {\mathcal {B'_{A'}}}}
be the twisting of
B
A
′
{\displaystyle {\mathcal {B_{A'}}}}
by
F
{\displaystyle F}
, and let there exist an isomorphism:
α
:
B
A
→
B
A
′
′
{\displaystyle \alpha :{\mathcal {B_{A}}}\to {\mathcal {B'_{A'}}}}
. Then the induced tensor functor
(
α
∗
,
i
d
,
ϕ
2
F
)
{\displaystyle (\alpha ^{*},id,\phi _{2}^{F})}
is a tensor category equivalence between
A
′
−
m
o
d
{\displaystyle {\mathcal {A'}}-mod}
and
A
−
m
o
d
{\displaystyle {\mathcal {A}}-mod}
. Where
ϕ
2
F
(
v
⊗
w
)
=
F
−
1
(
v
⊗
w
)
{\displaystyle \phi _{2}^{F}(v\otimes w)=F^{-1}(v\otimes w)}
. Moreover, if
α
{\displaystyle \alpha }
is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.: 375–376
Usage
Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to 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 XXZ in the framework of the Algebraic Bethe ansatz.
See also
Bialgebra
Hopf algebra
Quasi-Hopf algebra
References
Further reading
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