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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \vep} is not necessarily a homomorphism of algebras.
  • [(3)] The axioms for the antipode map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle S : A \lra A} with respect to the counit are as follows. For all , Failed to parse (unknown function "\ID"): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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. edit

  • [(1)] We refer here to Bais et al. (2002). Let be a non-Abelian group and a discrete subgroup. Let denote the space of functions on and Failed to parse (unknown function "\bC"): {\displaystyle \bC H} the group algebra (which consists of the linear span of group elements with the group structure). The quantum double (Drinfeld, 1987) is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle D(H) = F(H)~ \wti{\otimes}~ \bC H~, } where, for , the `twisted tensor product' is specified by Failed to parse (unknown function "\wti"): {\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 as the `electric gauge group' and as the `magnetic symmetry' generated by ~. In terms of the counit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \vep} , the double has a trivial representation given by Failed to parse (unknown function "\vep"): {\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 matrix, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle R \in D(H) \otimes D(H)<math>, leading to the [[../QuantumSpinNetworkFunctor2/|operator]] <math> \mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be}) (R)~, } in terms of the Clebsch--Gordan series Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a} \otimes \Pi^B_{\be} \cong N^{AB \gamma}_{\a \be C}~ \Pi^C_{\gamma}} , and where denotes a flip operator. The operator is sometimes called the monodromy or Aharanov--Bohm phase factor. In the case of a condensate in a state in the carrier space of some representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \Pi^A_{\a}} ~. One considers the maximal Hopf subalgebra of a Hopf algebra for which is --invariant; specifically ~: Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~ \forall P \in T~. } \item[(2)] For the second example, consider ~. The algebra of functions on can be broken to the algebra of functions on , that is, to , where is normal in , that is, ~. Next, consider ~. On breaking a purely electric condensate , the magnetic symmetry remains unbroken, but the electric symmetry Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \bC H} is broken to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \bC N_v} , with , the stabilizer of T = F(H) \wti{\otimes} \bC N_v</math>~. \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 is the Jones extension induced by a finite index depth inclusion of factors, then admits a quantum groupoid structure and acts on , so that B_2 = B_1 \rtimes Q</math>~. 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 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 , such as of finite index depth 2, sits inside a weak Hopf algebra formed as the crossed product (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 Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} with ~. If , then it is shown that a QTQHA is canonically associated with Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

Definitions of Related Concepts edit

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   denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} acting on   is a subset of the algebra of all bounded operators Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} such that:

  \item[(1)] Failed to parse (unknown function "\A"): {\displaystyle \A}
 is closed under the adjoint operation (with the adjoint of an element   denoted by  ).  \item[(2)] Failed to parse (unknown function "\A"): {\displaystyle \A}
 equals its bicommutant, namely:  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} the special set of bounded operators on Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} which commute with all elements in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} , then this second condition implies that the commutant of the commutant of Failed to parse (unknown function "\A"): {\displaystyle \A} is again the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} .

On the other hand, a von Neumann algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} inherits a unital subalgebra from Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , and according to the first condition in its definition Failed to parse (unknown function "\A"): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A} \emph{is a von Neumann algebra if and only if Failed to parse (unknown function "\A"): {\displaystyle \A} is a *-subalgebra of Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , closed for the smallest topology defined by continuous maps   for all   where   denotes the inner product defined on  }~. For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

Definition of a Hopf algebra edit

Firstly, a unital associative algebra consists of a linear space   together with two linear maps

Failed to parse (syntax error): {\displaystyle m &: A \otimes A \lra A~,~(multiplication) \eta &: \bC \lra A~,~ (unity) } satisfying the conditions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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~: Failed to parse (unknown function "\begin{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   equipped with a linear homorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \Delta : A \lra A \otimes A<math>, satisfying, for } a,b \in A</math> :

Failed to parse (syntax error): {\displaystyle \Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~. } We call   a comultiplication , which is said to be coasociative in so far that the following diagram commutes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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  , the counity map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \vep : A \lra \bC} satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle (\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta = \ID~. } A bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (A, m, \Delta, \eta, \vep)<math> is a linear space } A m, \Delta, \eta, \vep</math> satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} , satisfying  , for  ~. This map is defined implicitly via the property~: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle m(S \otimes \ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ \vep~~. } We call   the antipode map . A Hopf algebra is then a bialgebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle (A,m, \eta, \Delta, \vep)} equipped with an antipode map  ~.

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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