nLab weak bialgebra




The notion of weak bialgebra is a generalization of that of bialgebra in which the comultiplication Δ\Delta is weak in the sense that Δ(1)11\Delta(1)\neq 1\otimes 1 in general; similarly the compatibility of counit with the multiplication map is weakened (counit might fail to be a morphism of algebras). (Still a special case of sesquialgebra.)

Correspondingly weak Hopf algebras generalize Hopf algebras accordingly. Every weak Hopf algebra defines a Hopf algebroid.

Physical motivation

This kind of structures naturally comes in CFT models relation to quantum groups a root of unity: the full symmetry algebra is not quite a quantum group at root of unity, because if it were one would have to include the nonphysical quantum dimension zero finite-dimensional quantum group representations into the (pre)Hilbert space; those are the zero norm states which do not contribute to physics (like ghosts). If one quotients by these states then the true unit of a quantum group becomes an idempotent (projector), hence one deals with weak Hopf algebras instead as a price of dealing with true, physical, Hilbert space.


A weak bialgebra is a tuple (A,μ,η,Δ,ϵ)(A,\mu,\eta,\Delta,\epsilon) such that (A,μ,η)(A,\mu,\eta) is an associative unital algebra, (A,Δ,ϵ)(A,\Delta,\epsilon) is a coassociative counital coalgebra and the following compatibilities, (i),(ii) and (iii), hold:

(i) the coproduct Δ\Delta is multiplicative Δ(x)Δ(y)=Δ(xy)\Delta(x)\Delta(y)= \Delta(x y). If only (i) is satisfied, following Böhm, Caenapeel and Janssen 2011, we may speak of a prebialgebra.

(ii) the counit ϵ\epsilon satisfies weak multiplicativity

ϵ(xyz)=ϵ(xy (1))ϵ(y (2)z), \epsilon(x y z) = \epsilon(x y_{(1)})\epsilon(y_{(2)} z),
ϵ(xyz)=ϵ(xy (2))ϵ(y (1)z). \epsilon(x y z) = \epsilon(x y_{(2)})\epsilon(y_{(1)} z).

A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) monoidal.

(iii) Weak comultiplicativity of the unit:

Δ (2)(1)=(Δ(1)1)(1Δ(1)) \Delta^{(2)} (1) = (\Delta(1) \otimes 1)(1\otimes \Delta(1))
Δ (2)(1)=(1Δ(1))(Δ(1)1) \Delta^{(2)} (1) = (1 \otimes\Delta(1))(\Delta(1) \otimes 1)

A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) comonoidal.

As usually in the context of coassociative coalgebras, we denoted Δ (2):=(idΔ)Δ=(Δid)Δ\Delta^{(2)} := (id\otimes\Delta)\Delta = (\Delta\otimes id)\Delta.

A weak kk-bialgebra AA is a weak Hopf algebra if it has a kk-linear map S:AAS:A\to A (which is then called an antipode) such that for all xAx\in A

x (1)S(x (2))=ϵ(1 (1)x)1 (2), x_{(1)} S(x_{(2)}) = \epsilon(1_{(1)} x)1_{(2)},
S(x (1))x (2)=1 (1)ϵ(x1 (2)), S(x_{(1)})x_{(2)} = 1_{(1)} \epsilon(x 1_{(2)}),
S(x (1))x (2)S(x (3))=S(x) S(x_{(1)})x_{(2)} S(x_{(3)}) = S(x)

It follows that the antipode is antimultiplicative, S(xy)=S(y)S(x)S(x y)=S(y)S(x), and anticomultiplicative, Δ(S(x))=S(x) (1)S(x) (2)=S(x (2))S(x (1))\Delta(S(x)) = S(x)_{(1)}\otimes S(x)_{(2)} = S(x_{(2)})\otimes S(x_{(1)}).


Idempotents (“projections”)

For every weak bialgebra there are kk-linear maps Π L,Π R:AA\Pi^L,\Pi^R:A\to A defined by

Π L(x):=ϵ(1 (1)x)1 (2),Π R(x):=1 (1)ϵ(x1 (2)). \Pi^L(x) := \epsilon(1_{(1)} x) 1_{(2)},\,\,\,\, \Pi^R(x) := 1_{(1)}\epsilon(x 1_{(2)}).

Expressions for Π L(x),Π R(x)\Pi^L(x),\Pi^R(x) are already met above as the right hand sides in two of the axioms for the antipode. Maps Π L,Π R\Pi^L,\Pi^R are idempotents, Π RΠ R=Π R\Pi^R\Pi^R = \Pi^R and Π LΠ L=Π L\Pi^L\Pi^L = \Pi^L:

Π L(Π L(x)) = ϵ(1 (1)ϵ(1 (1)x)1 (2))1 (2)=ϵ(1 (1)x)ϵ(1 (1)1 (2))1 (2) = ϵ(1 (1)x)ϵ(1 (2))1 (3)=ϵ(1 (1)x)1 (2)=Π L(x).\array{ \Pi^L(\Pi^L(x)) &=& \epsilon\left(1_{(1')}\epsilon(1_{(1)}x) 1_{(2)}\right)1_{(2')} = \epsilon(1_{(1)}x)\epsilon(1_{(1')}1_{(2)}) 1_{(2')} \\ &=&\epsilon(1_{(1)}x)\epsilon(1_{(2)}) 1_{(3)} = \epsilon(1_{(1)}x)1_{(2)} = \Pi^L(x). }

Notice ϵ(xz)=ϵ(x1z)=ϵ(x1 (2))ϵ(1 (1)z))=ϵ(xϵ(1 (1)z))1 (2)=ϵ(xΠ L(z))=ϵ(Π R(x)z)\epsilon(x z) = \epsilon(x 1 z) = \epsilon(x 1_{(2)})\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z). The images of the idempotents A R=Π R(A)A^R = \Pi^R(A) and A L=Π L(R)A^L = \Pi^L(R) are dual as kk-linear spaces: there is a canonical nondegenerate pairing A LA RkA^L\otimes A^R\to k given by (x,y)ϵ(yx)(x,y) \mapsto \epsilon(y x).

Also Π L(xΠ L(y))=Π L(xy)\Pi^L(x\Pi^L(y)) = \Pi^L(x y) and Π R(Π R(x)y)=Π R(xy)\Pi^R(\Pi^R(x)y) = \Pi^R(x y), dually Δ(A L)AA L\Delta(A^L)\subset A\otimes A^L and Δ(A R)A RA\Delta(A^R)\subset A^R\otimes A, and in particular Δ(1)A RA L\Delta(1)\in A^R\otimes A^L.

Sometimes it is also useful to consider the idempotents Π¯ L,Π¯ R:AA\bar\Pi^L,\bar\Pi^R:A\to A defined by

Π¯ L(x):=ϵ(1 (2)x)1 (1),Π¯ R(x):=1 (2)ϵ(x1 (1)). \bar\Pi^L(x) := \epsilon(1_{(2)} x) 1_{(1)},\,\,\,\, \bar\Pi^R(x) := 1_{(2)}\epsilon(x 1_{(1)}).
Π¯ L(Π¯ L(x)) = ϵ(1 (2)ϵ(1 (2)x)1 (1))1 (1)=ϵ(1 (2)x)ϵ(1 (2)1 (1))1 (1) = ϵ(1 (3)x)ϵ(1 (2))1 (1)=ϵ(1 (2)x)1 (1)=Π¯ L(x).\array{ \bar\Pi^L(\bar\Pi^L(x))&=&\epsilon(1_{(2')}\epsilon(1_{(2)}x)1_{(1)})1_{(1')} = \epsilon(1_{(2)}x)\epsilon(1_{(2')}1_{(1)})1_{(1')} \\ &=& \epsilon(1_{(3)}x)\epsilon(1_{(2)})1_{(1)}= \epsilon(1_{(2)}x)1_{(1)} = \bar\Pi^L(x). }

Relation to fusion categories

Under Tannaka duality, every fusion category CC arises as the representation category of a weak Hopf algebra (Ostrik). However, this does not mean that every fusion category admits a fiber functor to the category of vector spaces Vect=kMod\text{Vect}= k-Mod.

Given any multi-fusion category CC, one can always construct a fiber functor F:CRModF:C\to RMod for RR the algebra spanned by a basis of orthogonal idempotents {v i} iI\{v_i\}_{i\in I} for II the equivalence classes of simple objects of CC. This functor is referred to in some sources as a generalized fiber functor. The endomorphisms of this functor then give a weak Hopf algebra that represents CC. In Hayashi 1999 (see there for the relevant definitions), this is computed as a coend, where one has that CRep(A)C\cong Rep(A) for A=coend(F *F:C op×CBmd(E))A= \text{coend}(F^*\otimes F: C^{op} \times C \to Bmd(E)), where E=R˙RE=\dot R\otimes R is equipped with a coalgebra structure

Δ(λ˙μ)= νIλ˙νν˙μ \Delta(\dot\lambda \mu) = \sum_{\nu\in I} \dot\lambda \nu\otimes \dot\nu \mu
ϵ(λ˙μ)=δ λ,μ \epsilon (\dot\lambda \mu) = \delta_{\lambda,\mu}

It is important to note that, generally speaking, CC may admit other fiber functor to different module categories RModRMod, as is the case for fusion categories of the form Rep(H)Rep(H) for HH a Hopf algebra, which admits both the fiber functor described above, as well as a fiber functor to Vect\text{Vect}.

Even further, this statement generalizes to tensor C-star-categories and C-star weak Hopf algebras (Vainerman & Vallin 2020).

Relation to Frobenius algebras

As explained in Hopf algebra, any finite-dimensional Hopf algebra can be given the structure of a Frobenius algebra. There is a similar result for weak Hopf algebras.


Any finite-dimensional weak Hopf algebra can be given the structure of a quasi-Frobenius algebra.

This is due to Bohm, Nill, and Szlachanyi (1999). While Vecsernyés (2003) seems to show that finite-dimensional weak Hopf algebras can be turned into Frobenius algebras, it is observed in Iovanov & Kadison (2008) that the proof only implies they are quasi-Frobenius algebras.


Weak comultiplications were introduced in

  • G. Mack, Volker Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B370(1992) 185.

where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in

  • G. Böhm, K. Szlachányi, A coassociative C *C^\ast-quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 437–456, arXiv:q-alg/9509008g/abs/q-alg/9509008); Weak C*C*-Hopf algebras: the coassociative symmetry of non-integral dimensions, in: Quantum groups and quantum spaces (Warsaw, 1995), 9-19, Banach Center Publ. 40, Polish Acad. Sci., Warszawa 1997.
  • Florian Nill, Axioms for weak bialgebras, math.QA/9805104
  • G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras. I. Integral theory and C *C^\ast-structure, J. Algebra 221 (1999), no. 2, 385-438, math.QA/9805116 #{BohmNillSzlachanyi}

A book exposition is in chapter weak (Hopf) bialgebras in

  • Gabriella Böhm, Hopf algebras and their generalizations from a category theoretical point of view, Lecture Notes in Math. 2226, Springer 2018, doi

Now these works are understood categorically from the point of view of weak monad theory:

The relation to fusion categories is discussed in

On the relation to Frobenius algebras

On the Tannaka duality of C-star weak Hopf algebras:

  • Leonid Vainerman, Jean-Michel Vallin. Classifying (weak) coideal subalgebras of weak Hopf C-star-algebras. Journal of Algebra 550 (2020): 333-357. (doi).
category: algebra

Last revised on March 5, 2024 at 16:30:58. See the history of this page for a list of all contributions to it.