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 hold:

(i) the coproduct Δ\Delta is multiplicative Δ(x)Δ(y)=Δ(xy)\Delta(x)\Delta(y)= \Delta(x y)

(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).

(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)

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)


Idempotents (“projections”)

For every weak bialgebra there are kk-linear maps Π L,Π R:AA\Pi^L,\Pi^R:A\to A with properties Π RΠ R=Π R\Pi^R\Pi^R = \Pi^R and Π LΠ L=Π L\Pi^L\Pi^L = \Pi^L and 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)}).

These expressions are already met above in two of the axioms for the antipode. Then 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), and 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 Δ(1)A RA L\Delta(1)\in A^R\otimes A^L.

Relation to fusion categories

Under Tannaka duality (semisimple) weak Hopf algebras correspond to (multi-)fusion categories (Ostrik).


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}

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

The relation to fusion categories is discussed in

Revised on November 16, 2015 15:19:30 by Zoran Škoda (