# PlanetPhysics/Weak Hopf Algebra

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Definition 0.1 :


In order to define a weak Hopf algebra , one weakens' or relaxes certain axioms of a Hopf algebra as follows~:

• [(1)] The comultiplication is not necessarily unit--preserving.
• [(2)] The counit $\displaystyle \vep$ is not necessarily a homomorphism of algebras.
• [(3)] The axioms for the antipode map $\displaystyle S : A \lra A$ with respect to the counit are as follows. For all ${\displaystyle h\in H}$, $\displaystyle m(\ID \otimes S) \Delta (h) &= (\vep \otimes \ID)(\Delta (1) (h \otimes 1)) \\ m(S \otimes \ID) \Delta (h) &= (\ID \otimes \vep)((1 \otimes h) \Delta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~.$

These axioms may be appended by the following commutative diagrams $CD}"): {\displaystyle {\begin{CD} A \otimes A @> S\otimes \ID >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}} \qquad {\begin{CD} A \otimes A @> \ID\otimes S >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}}$ along with the counit axiom: $\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\vep \otimes 1} & A \ar[l]_{\Delta} \ar[dl]_{\ID_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \vep}}$ Some authors substitute the term quantum 'groupoid' for a weak Hopf algebra.

### Examples of weak Hopf algebras.

• [(1)] We refer here to Bais et al. (2002). Let ${\displaystyle G}$ be a non-Abelian group and ${\displaystyle H\subset G}$ a discrete subgroup. Let ${\displaystyle F(H)}$ denote the space of functions on ${\displaystyle H}$ and $\displaystyle \bC H$ the group algebra (which consists of the linear span of group elements with the group structure). The quantum double ${\displaystyle D(H)}$ (Drinfeld, 1987) is defined by $\displaystyle D(H) = F(H)~ \wti{\otimes}~ \bC H~,$ where, for ${\displaystyle x\in H}$, the twisted tensor product' is specified by $\displaystyle \wti{\otimes} \mapsto ~(f_1 \otimes h_1) (f_2 \otimes h_2)(x) = f_1(x) f_2(h_1 x h_1^{-1}) \otimes h_1 h_2 ~.$ The physical interpretation is often to take ${\displaystyle H}$ as the electric gauge group' and ${\displaystyle F(H)}$ as the magnetic symmetry' generated by ${\displaystyle \{f\otimes e\}}$~. In terms of the counit $\displaystyle \vep$ , the double ${\displaystyle D(H)}$ has a trivial representation given by $\displaystyle \vep(f \otimes h) = f(e)$ ~. We next look at certain features of this construction. For the purpose of braiding relations there is an ${\displaystyle R}$ matrix, $\displaystyle R \in D(H) \otimes D(H)$, leading to the [[../QuantumSpinNetworkFunctor2/|operator]] [itex] \mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be}) (R)~,$ in terms of the Clebsch--Gordan series $\displaystyle \Pi^A_{\a} \otimes \Pi^B_{\be} \cong N^{AB \gamma}_{\a \be C}~ \Pi^C_{\gamma}$ , and where ${\displaystyle \sigma }$ denotes a flip operator. The operator ${\displaystyle {\mathcal {R}}^{2}}$ is sometimes called the monodromy or Aharanov--Bohm phase factor. In the case of a condensate in a state ${\displaystyle \vert v\rangle }$ in the carrier space of some representation $\displaystyle \Pi^A_{\a}$ ~. One considers the maximal Hopf subalgebra ${\displaystyle T}$ of a Hopf algebra ${\displaystyle A}$ for which ${\displaystyle \vert v\rangle }$ is ${\displaystyle T}$--invariant; specifically ~: $\displaystyle \Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~ \forall P \in T~.$ \item[(2)] For the second example, consider ${\displaystyle A=F(H)}$~. The algebra of functions on ${\displaystyle H}$ can be broken to the algebra of functions on ${\displaystyle H/K}$, that is, to ${\displaystyle F(H/K)}$, where ${\displaystyle K}$ is normal in ${\displaystyle H}$, that is, ${\displaystyle HKH^{-1}=K}$~. Next, consider ${\displaystyle A=D(H)}$~. On breaking a purely electric condensate ${\displaystyle \vert v\rangle }$, the magnetic symmetry remains unbroken, but the electric symmetry $\displaystyle \bC H$ is broken to $\displaystyle \bC N_v$ , with ${\displaystyle N_{v}\subset H}$, the stabilizer of ${\displaystyle \vert v\rangle [itex]~.Fromthisweobtain}$T = F(H) \wti{\otimes} \bC N_v$~. \item[(3)] In Nikshych and Vainerman (2000) quantum groupoids (as weak C*--Hopf algebras, see below) were studied in relationship to the noncommutative symmetries of depth 2 von Neumann subfactors. If ${\displaystyle A\subset B\subset B_{1}\subset B_{2}\subset \ldots }$ is the Jones extension induced by a finite index depth ${\displaystyle 2}$ inclusion ${\displaystyle A\subset B}$ of ${\displaystyle II_{1}}$ factors, then ${\displaystyle Q=A'\cap B_{2}}$ admits a quantum groupoid structure and acts on ${\displaystyle B_{1}}$, so that ${\displaystyle B=B_{1}^{Q}$and}$B_2 = B_1 \rtimes Q$~. Similarly, in Rehren (1997) paragroups' (derived from weak C*--Hopf algebras) comprise (quantum) groupoids of equivalence classes such as associated with 6j--symmetry groups (relative to a fusion rules algebra). They correspond to type ${\displaystyle II}$ von Neumann algebras in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of observables within fields) have depth 2 in the Jones extension. Related is how a von Neumann algebra ${\displaystyle N}$, such as of finite index depth 2, sits inside a weak Hopf algebra formed as the crossed product ${\displaystyle N\rtimes A}$ (B\"ohm et al. 1999). \item[(4)] In Mack and Schomerus (1992) using a more general notion of the Drinfeld construction, develop the notion of a \emph{quasi triangular quasi--Hopf algebra} (QTQHA) is developed with the aim of studying a range of essential symmetries with special properties, such the quantum group algebra $\displaystyle \U_q (\rm{sl}_2)$ with ${\displaystyle \vert q\vert =1}$~. If ${\displaystyle q^{p}=1}$, then it is shown that a QTQHA is canonically associated with $\displaystyle \U_q (\rm{sl}_2)$ . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

## Definitions of Related Concepts

Let us recall two basic concepts of quantum operator algebra that are essential to algebraic quantum theories. \\

\subsection {Definition of a Von Neumann Algebra.}

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

  \item[(1)] $\displaystyle \A$
is closed under the adjoint operation (with the adjoint of an element ${\displaystyle T}$  denoted by ${\displaystyle T^{*}}$ ).  \item[(2)] $\displaystyle \A$
equals its bicommutant, namely:  $\displaystyle \A= \{A \in \cL(\mathbb{H}) : \forall B \in \cL(\mathbb{H}), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~.$



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

On the other hand, a von Neumann algebra $\displaystyle \A$ inherits a unital subalgebra from $\displaystyle \cL(\mathbb{H})$ , and according to the first condition in its definition $\displaystyle \A$ 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 $\displaystyle \A$ \emph{is a von Neumann algebra if and only if $\displaystyle \A$ is a *-subalgebra of $\displaystyle \cL(\mathbb{H})$ , closed for the smallest topology defined by continuous maps ${\displaystyle (\xi ,\eta )\longmapsto (A\xi ,\eta )}$  for all ${\displaystyle }$  where ${\displaystyle <.,.>}$  denotes the inner product defined on ${\displaystyle \mathbb {H} }$ }~. For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

### Definition of a Hopf algebra

Firstly, a unital associative algebra consists of a linear space ${\displaystyle A}$  together with two linear maps

$\displaystyle m &: A \otimes A \lra A~,~(multiplication) \eta &: \bC \lra A~,~ (unity)$ satisfying the conditions $\displaystyle m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1 \otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~.$ This first condition can be seen in terms of a commuting diagram~: $CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @> m \otimes \ID>> A \otimes A \\ @V \ID \otimes mVV @VV m V \\ A \otimes A @ > m >> A \end{CD}$ Next suppose we consider reversing the arrows', and take an algebra ${\displaystyle A}$  equipped with a linear homorphisms $\displaystyle \Delta : A \lra A \otimes A$, satisfying, for$ a,b \in A$ :

$\displaystyle \Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~.$ We call ${\displaystyle \Delta }$  a comultiplication , which is said to be coasociative in so far that the following diagram commutes $CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @< \Delta\otimes \ID<< A \otimes A \\ @A \ID \otimes \Delta AA @AA \Delta A \\ A \otimes A @ < \Delta << A \end{CD}$ There is also a counterpart to ${\displaystyle \eta }$ , the counity map $\displaystyle \vep : A \lra \bC$ satisfying $\displaystyle (\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta = \ID~.$ A bialgebra $\displaystyle (A, m, \Delta, \eta, \vep)$is a linear space$ A${\displaystyle withmaps}$ m, \Delta, \eta, \vep$ satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $\displaystyle S : A \lra A$ , satisfying ${\displaystyle S(ab)=S(b)S(a)}$ , for ${\displaystyle a,b\in A}$ ~. This map is defined implicitly via the property~: $\displaystyle m(S \otimes \ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ \vep~~.$ We call ${\displaystyle S}$  the antipode map . A Hopf algebra is then a bialgebra $\displaystyle (A,m, \eta, \Delta, \vep)$ equipped with an antipode map ${\displaystyle S}$ ~.

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.

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