# weak Hopf algebra

• (1)

The comultiplication is not necessarily unit–preserving.

• (2)
• (3)

The axioms for the antipode map $S:A{\longrightarrow}A$ with respect to the counit are as follows. For all $h\in H$,

 $\displaystyle m({\rm id}\otimes S)\Delta(h)$ $\displaystyle=(\varepsilon\otimes{\rm id})(\Delta(1)(h\otimes 1))$ (0.1) $\displaystyle m(S\otimes{\rm id})\Delta(h)$ $\displaystyle=({\rm id}\otimes\varepsilon)((1\otimes h)\Delta(1))$ $\displaystyle S(h)$ $\displaystyle=S(h_{(1)})h_{(2)}S(h_{(3)})~{}.$
 ${\begin{CD}A\otimes A@>{S\otimes{\rm id}}>{}>A\otimes A\\ @A{\Delta}A{}A@V{}V{m}V\\ A@ >u\circ\varepsilon>>A\end{CD}}\qquad{\begin{CD}A\otimes A@>{{\rm id}\otimes S% }>{}>A\otimes A\\ @A{\Delta}A{}A@V{}V{m}V\\ A@ >u\circ\varepsilon>>A\end{CD}}$ (0.2)

along with the counit axiom:

 $\xymatrix@C=3pc@R=3pc{A\otimes A\ar[d]_{\varepsilon\otimes 1}&A\ar[l]_{\Delta}% \ar[dl]_{{\rm id}_{A}}\ar[d]^{\Delta}\\ A&A\otimes A\ar[l]^{1\otimes\varepsilon}}$ (0.3)

## 1 Definitions of Related Concepts

### 1.1 Definition of a Von Neumann Algebra.

Let $\mathcal{H}$ denote a complex (separable) Hilbert space  . A von Neumann algebra $\mathcal{A}$ acting on $\mathcal{H}$ is a subset of the algebra of all bounded operators   $\mathcal{L}(\mathcal{H})$ such that:

If one calls a commutant of a set $\mathcal{A}$ the special set of bounded operators on $\mathcal{L}(\mathcal{H})$ which commute with all elements in $\mathcal{A}$, then this second condition implies that the commutant of the commutant of $\mathcal{A}$ is again the set $\mathcal{A}$.

On the other hand, a von Neumann algebra $\mathcal{A}$ inherits a unital subalgebra from $\mathcal{L}(\mathcal{H})$, and according to the first condition in its definition $\mathcal{A}$ does indeed inherit a *-subalgebra structure, as further explained in the next section         on C*-algebras. Furthermore, we have notable Bicommutant Theorem which states that $\mathcal{A}$ is a von Neumann algebra if and only if $\mathcal{A}$ is a *-subalgebra of $\mathcal{L}(\mathcal{H})$, closed for the smallest topology  defined by continuous maps $(\xi,\eta)\longmapsto(A\xi,\eta)$ for all $$ where $<.,.>$ denotes the inner product  defined on $\mathcal{H}$ . For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

### 1.2 Definition of a Hopf algebra

 $\displaystyle m$ $\displaystyle:A\otimes A{\longrightarrow}A~{},~{}(multiplication)$ (1.2) $\displaystyle\eta$ $\displaystyle:\mathbb{C}{\longrightarrow}A~{},~{}(unity)$

satisfying the conditions

 $\displaystyle m(m\otimes\mathbf{1})$ $\displaystyle=m(\mathbf{1}\otimes m)$ (1.3) $\displaystyle m(\mathbf{1}\otimes\eta)$ $\displaystyle=m(\eta\otimes\mathbf{1})={\rm id}~{}.$

This first condition can be seen in terms of a commuting diagram :

 $\begin{CD}A\otimes A\otimes A@>{m\otimes{\rm id}}>{}>A\otimes A\\ @V{{\rm id}\otimes m}V{}V@V{}V{m}V\\ A\otimes A@ >m>>A\end{CD}$ (1.4)

Next suppose we consider ‘reversing the arrows’, and take an algebra $A$ equipped with a linear homorphisms $\Delta:A{\longrightarrow}A\otimes A$, satisfying, for $a,b\in A$ :

 $\displaystyle\Delta(ab)$ $\displaystyle=\Delta(a)\Delta(b)$ (1.5) $\displaystyle(\Delta\otimes{\rm id})\Delta$ $\displaystyle=({\rm id}\otimes\Delta)\Delta~{}.$

We call $\Delta$ a comultiplication, which is said to be coasociative in so far that the following diagram commutes

 $\begin{CD}A\otimes A\otimes A@<{\Delta\otimes{\rm id}}<{} (1.6)

There is also a counterpart to $\eta$, the counity map $\varepsilon:A{\longrightarrow}\mathbb{C}$ satisfying

 $({\rm id}\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes{\rm id})\circ% \Delta={\rm id}~{}.$ (1.7)

A bialgebra   $(A,m,\Delta,\eta,\varepsilon)$ is a linear space $A$ with maps $m,\Delta,\eta,\varepsilon$ satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $S:A{\longrightarrow}A$, satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$ . This map is defined implicitly via the property :

 $m(S\otimes{\rm id})\circ\Delta=m({\rm id}\otimes S)\circ\Delta=\eta\circ% \varepsilon~{}~{}.$ (1.8)

We call $S$ the antipode map. A Hopf algebra is then a bialgebra $(A,m,\eta,\Delta,\varepsilon)$ equipped with an antipode map $S$ .

Commutative   and noncommutative Hopf algebras form the backbone of quantum ‘groups’ and are essential to the generalizations  of symmetry. Indeed, in most respects a quantum ‘group’ is identifiable with a Hopf algebra. When such algebras are actually associated with proper groups of matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

## References

 Title weak Hopf algebra Canonical name WeakHopfAlgebra Date of creation 2013-03-22 18:12:43 Last modified on 2013-03-22 18:12:43 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 35 Author bci1 (20947) Entry type Definition Classification msc 08C99 Classification msc 81R15 Classification msc 57T05 Classification msc 81R50 Classification msc 16W30 Synonym quantum groupoids v.1 Related topic HopfAlgebra Related topic WeakHopfCAlgebra2 Related topic WeakHopfCAlgebra2 Related topic CommutativeDiagram Related topic GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries Related topic GrassmanHopfAlgebrasAndTheirDualCoAlgebras Related topic WeakHopfCAlgebra2 Defines weak bialgebra Defines commutant of a set Defines counit axiom Defines antipode map Defines counity Defines twisted tensor product Defines quantum double Defines QOA