nLab weak bialgebra

Idea

The notion of weak bialgebra is a generalization of that of bialgebra in which the comultiplication $\Delta$ is weak in the sense that $\Delta(1)\neq 1\otimes 1$ 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.

Definitions

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

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

(ii) the counit $\epsilon$ satisfies weak multiplicativity

$$\epsilon(x y z) = \epsilon(x y_{(1)})\epsilon(y_{(2)} z),$$

$$\epsilon(x y z) = \epsilon(x y_{(2)})\epsilon(y_{(1)} z).$$

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

(iii) Weak comultiplicativity of the unit:

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

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

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 $\Delta^{(2)} := (id\otimes\Delta)\Delta = (\Delta\otimes id)\Delta$.

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

$$x_{(1)} S(x_{(2)}) = \epsilon(1_{(1)} x)1_{(2)},$$

$$S(x_{(1)})x_{(2)} = 1_{(1)} \epsilon(x 1_{(2)}),$$

$$S(x_{(1)})x_{(2)} S(x_{(3)}) = S(x)$$

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

Properties

Idempotents (“projections”)

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

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

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

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

Notice $\epsilon(x z) = \epsilon(x 1 z) = \epsilon(x 1_{(2)})\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z)$. The images of the idempotents $A^R = \Pi^R(A)$ and $A^L = \Pi^L(R)$ are dual as $k$-linear spaces: there is a canonical nondegenerate pairing $A^L\otimes A^R\to k$ given by $(x,y) \mapsto \epsilon(y x)$.

Also $\Pi^L(x\Pi^L(y)) = \Pi^L(x y)$ and $\Pi^R(\Pi^R(x)y) = \Pi^R(x y)$, dually $\Delta(A^L)\subset A\otimes A^L$ and $\Delta(A^R)\subset A^R\otimes A$, and in particular $\Delta(1)\in A^R\otimes A^L$.

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

$$\bar\Pi^L(x) := \epsilon(1_{(2)} x) 1_{(1)},\,\,\,\, \bar\Pi^R(x) := 1_{(2)}\epsilon(x 1_{(1)}).$$

$$\array{ \bar\Pi^L(\bar\Pi^L(x))&=&\epsilon(1_{(2')}\epsilon(1_{(2)}x)1_{(1)})1_{(1')} = \epsilon(1_{(2)}x)\epsilon(1_{(2')}1_{(1)})1_{(1')} \\ &=& \epsilon(1_{(3)}x)\epsilon(1_{(2)})1_{(1)}= \epsilon(1_{(2)}x)1_{(1)} = \bar\Pi^L(x). }$$

Relation to fusion categories

Under Tannaka duality, every fusion category $C$ 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 $\text{Vect}= k-Mod$.

Given any multi-fusion category $C$, one can always construct a fiber functor $F:C\to RMod$ for $R$ the algebra spanned by a basis of orthogonal idempotents $\{v_i\}_{i\in I}$ for $I$ the equivalence classes of simple objects of $C$. 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 $C$. In Hayashi 1999 (see there for the relevant definitions), this is computed as a coend, where one has that $C\cong Rep(A)$ for $A= \text{coend}(F^*\otimes F: C^{op} \times C \to Bmd(E))$, where $E=\dot R\otimes R$ is equipped with a coalgebra structure

$$\Delta(\dot\lambda \mu) = \sum_{\nu\in I} \dot\lambda \nu\otimes \dot\nu \mu$$

$$\epsilon (\dot\lambda \mu) = \delta_{\lambda,\mu}$$

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

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.

Proposition

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.

Literature

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 $C^\ast$-quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 437–456, arXiv:q-alg/9509008g/abs/q-alg/9509008); Weak $C*$-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 $C^\ast$-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:

• Gabriella Böhm, Stefaan Caenepeel, Kris Janssen, Weak bialgebras and monoidal categories, Comm. Algebra 39 (2011), no. 12 (special volume dedicated to Mia Cohen), 4584-4607. arXiv:1103.226
• Gabriella Böhm, Stephen Lack, Ross Street, Weak bimonads and weak Hopf monads, J. Algebra 328 (2011), 1-30, arXiv:1002.4493
• Gabriella Böhm, José Gómez-Torrecillas, On the double crossed product of weak Hopf algebras, arXiv:1205.2163

The relation to fusion categories is discussed in

• Takahiro Hayashi, A canonical Tannaka duality for finite semisimple tensor categories (arXiv:math/9904073)

• Victor Ostrik, Module categories, weak Hopf algebras and modular invariants (arXiv:math/0111139)

On the relation to Frobenius algebras

• Gabriella Bohm, Florian Nill?, Kornel Szlachanyi. Weak Hopf Algebras I: Integral Theory and $C^*$-structure. (1999). (arXiv:math/9805116)

• Peter Vecsernyés?. Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. Journal of Algebra. Volume 270, Issue 2, 15 December 2003, Pages 471-520. (doi)

• Miodrag Iovanov, Lars Kadison?. When weak Hopf algebras are Frobenius. (2008). (arXiv:0810.4777)

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