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单词 CompactQuantumGroupoidsRelatedToCalgebras
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compact quantum groupoids related to C*-algebras


1 Compact quantum groupoids (CGQs) and C*-algebras

1.1 Introduction: von Neumann and C*-algebras. Quantum operator algebras in quantum theories

C*-algebraMathworldPlanetmathPlanetmathPlanetmath has evolved as a key concept in quantum operator algebraPlanetmathPlanetmathPlanetmath (QOA) after the introduction of thevon Neumann algebraMathworldPlanetmathPlanetmathPlanetmath for the mathematical foundationPlanetmathPlanetmath of quantum mechanics. The von Neumann algebra classification is simpler and studied in greater depth than that of general C*-algebra classification theory. The importance ofC*-algebras for understanding the geometry of quantum state spacesPlanetmathPlanetmath (viz. Alfsen and Schultz, 2003 [1]) cannot be overestimated. Moreover, the introduction of non-commutative C*-algebras in noncommutative geometryPlanetmathPlanetmath has already played important roles in expanding the Hilbert spaceMathworldPlanetmath perspective of quantum mechanics developed by von Neumann. Furthermore, extended quantum symmetries are currently being approached in terms of groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath C*- convolution algebra and their representationsPlanetmathPlanetmath; the latter also enter into the construction of compactPlanetmathPlanetmath quantum groupoids as developed in the Bibliography cited, and also briefly outlined here in the third sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The fundamental connectionsMathworldPlanetmathPlanetmath that exist between categoriesMathworldPlanetmath of C*-algebras and those of von Neumann and other quantum operator algebras, such as JB- or JBL- algebras are yet to be completed and are the subjectof in depth studies [1].

1.2 Basic definitions

Let us recall first the basic definitions of C*-algebra and involutionPlanetmathPlanetmath on a complex algebra.Further details can be found in a separate entry focused on C*-algebras (http://planetmath.org/CAlgebra).

A C*-algebra is simultaneously a *–algebra and a Banach spaceMathworldPlanetmath -with additional conditions- as defined next.

Let us consider first the definition of an involution on a complex algebra 𝔄.

Definition 1.1.

An involution on a complex algebra 𝔄 is a real–linear map TT*such that for all S,T𝔄 and λ, we have T**=T,(ST)*=T*S*,(λT)*=λ¯T*.

A *-algebra is said to be a complex associative algebra together with an involution * .

Definition 1.2.

A C*-algebra is simultaneously a *-algebra and a Banach space 𝔄,satisfying for all S,T𝔄  the following conditions:

STST,T*T2=T2.

One can easily verify that A*=A .

By the above axioms a C*–algebra is a special case of a Banach algebraMathworldPlanetmath where the latter requires the above C*-norm property, but not the involution (*) property.

Given Banach spaces E,F the space (E,F) of (boundedPlanetmathPlanetmathPlanetmathPlanetmath) linear operators from E to F forms a Banach space, where for E=F, the space (E)=(E,E) is a Banach algebra with respect to the norm

T:=sup{Tu:uE,u=1}.

In quantum field theory one may start with a Hilbert space H, and consider the Banachalgebra of bounded linear operators (H) which given to be closed underPlanetmathPlanetmath the usualalgebraic operations and taking adjointsPlanetmathPlanetmathPlanetmath, forms a *–algebra of bounded operatorsMathworldPlanetmathPlanetmath, where theadjoint operation functions as the involution, and for T(H) we have :

T:=sup{(Tu,Tu):uH,(u,u)=1}, and Tu2=(Tu,Tu)=(u,T*Tu)T*Tu2.

By a morphismMathworldPlanetmath between C*-algebras 𝔄,𝔅 we mean a linear map ϕ:𝔄𝔅, such that for all S,T𝔄, the following hold :

ϕ(ST)=ϕ(S)ϕ(T),ϕ(T*)=ϕ(T)*,

where a bijectiveMathworldPlanetmathPlanetmath morphism is said to be an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (in which case it is then anisometry). A fundamental relationMathworldPlanetmathPlanetmathPlanetmath is that any norm-closed *-algebra 𝒜 in(H) is a C*-algebra (http://planetmath.org/CAlgebra3), and conversely, any C*-algebra (http://planetmath.org/CAlgebra3) is isomorphic to a norm–closed *-algebra in (H) for some Hilbert space H .One can thus also define the category C* of C*-algebras and morphisms between C*-algebras.

For a C*-algebra (http://planetmath.org/CAlgebra3) 𝔄, we say that T𝔄 is self–adjoint if T=T* . Accordingly, the self–adjoint part 𝔄sa of 𝔄 is a realvector space since we can decompose T𝔄sa as  :

T=T+T′′:=12(T+T*)+ι(-ι2)(T-T*).

A commutativePlanetmathPlanetmath C* -algebra is one for which the associative multiplication iscommutative. Given a commutative C* -algebra 𝔄, we have 𝔄C(Y),the algebra of continuous functionsPlanetmathPlanetmath on a compact Hausdorff space Y.

The classification of C* -algebras is far more complex than that of von Neumann algebras that providethe fundamental algebraic content of quantum state and operator spaces in quantum theoriesPlanetmathPlanetmath.

1.3 Quantum groupoids and the groupoid C*-algebra

Quantum groupoid (or their dual, weak Hopf coalgebras) and algebroid symmetriesPlanetmathPlanetmathPlanetmath figure prominently both in the theory of dynamical deformationsMathworldPlanetmath of quantum groupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (or their dual Hopf algebrasMathworldPlanetmathPlanetmath) and the quantum Yang–Baxter equations (Etingof et al., 1999, 2001; [12, E2k]). On the other hand, one can also consider the naturalextensionPlanetmathPlanetmathPlanetmath of locally compact (quantum) groups to locally compact(proper) groupoids equipped with a Haar measure and a corresponding groupoid representationPlanetmathPlanetmathPlanetmathPlanetmath theory(Buneci, 2003,[MB2k3]) as a major, potentially interesting source for locally compact (butgenerally non-AbelianMathworldPlanetmathPlanetmath) quantum groupoids. The corresponding quantum groupoid representations on bundles ofHilbert spaces extend quantum symmetries well beyond those of quantum groups and their dual Hopf algebras, and also beyond the simpler operator algebra representations, and are also consistentPlanetmathPlanetmath with the locally compact quantum group representations. The latter quantum groups are neither Hopf algebras, nor are they equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to Hopf algebras or their dual coalgebras. As pointed out in the previous section, quantum groupoid representations are, however, the next important step towards unifying quantum field theories with General Relativity in a locally covariant and quantized form. Such representations need not however be restricted to weak Hopf algebra representations, as the latter have no known connection to any type of GR theory and also appear to be inconsistent with GR.

Quantum groupoids were recently considered as weak C* -Hopf algebras, and were studied in relationship to the non- commutative symmetries of depth 2 von Neumann subfactors. If

ABB1B2(1.1)

is the Jones extension induced by a finite index depth 2inclusion AB of II1 factors, then Q=AB2admits a quantum groupoid structureMathworldPlanetmath and acts on B1, so that B=B1Q and B2=B1Q . Similarly, ‘paragroups’ derived from weak C* -Hopf algebras comprise (quantum) groupoids of equivalence classesMathworldPlanetmath such as those associated with 6j-symmetry groups (relative to a fusion rules algebra). They correspond to type II von Neumann algebras in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of quantum observables within fields) have depth 2 in theJones extension. A related question is how a von Neumann algebra W*, such asof finite index depth 2, sits inside a weak Hopf algebra formed as the crossed productW*A.

1.4 Quantum compact groupoids

Compact quantum groupoids were introduced in Landsman (1998; ref. [L98]) as asimultaneous generalizationPlanetmathPlanetmath of a compact groupoid and a quantum group. Since this construction is relevant to the definition of locally compact quantum groupoids and their representations investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let 𝔄 and 𝔅 denote C*–algebras equipped with a *–homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath ηs:𝔅𝔄, and a *–antihomomorphism ηt:𝔅𝔄 whose images in 𝔄commute. A non–commutative Haar measure is defined as a completelypositive map P:𝔄𝔅 which satisfiesP(Aηs(B))=P(A)B . Alternatively, the composition =ηsP:𝔄ηs(B)𝔄 is a faithful conditional expectation.

Next consider 𝖦 to be a (topological) groupoid, and let us denote by Cc(𝖦) the space of smooth complex–valued functions with compact support on 𝖦 . In particular, for all f,gCc(𝖦), thefunction defined via convolution

(f*g)(γ)=γ1γ2=γf(γ1)g(γ2),(1.2)

is again an element of Cc(𝖦), where the convolution productPlanetmathPlanetmath defines the composition law on Cc(𝖦) . We can turn Cc(𝖦) into a * -algebra once we have defined the involution*, and this is done by specifying f*(γ)=f(γ-1)¯ .

We recall that following Landsman (1998) a representation of a groupoid 𝔾, consists of afamily (or field) of Hilbert spaces {x}xXindexed by X=Ob𝔾, along with a collectionMathworldPlanetmath of maps {U(γ)}γ𝔾, satisfying:

  • 1.

    U(γ):s(γ)r(γ),is unitaryPlanetmathPlanetmath.

  • 2.

    U(γ1γ2)=U(γ1)U(γ2), whenever(γ1,γ2)𝔾(2)  (the set of arrows).

  • 3.

    U(γ-1)=U(γ)*, for all γ𝔾 .

Suppose now 𝖦lc is a Lie groupoid. Then the isotropy groupMathworldPlanetmath𝖦x is a Lie groupMathworldPlanetmath, and for a (left or right) Haarmeasure μx on 𝖦x, we can consider the Hilbertspaces x=L2(𝖦x,μx) as exemplifying theabove sense of a representation. Putting aside some technicaldetails which can be found in Connes (1994) and Landsman (2006), theoverall idea is to define an operator of Hilbert spaces

πx(f):L2(𝖦𝗑,μx)L2(𝖦x,μx),(1.3)

given by

(πx(f)ξ)(γ)=f(γ1)ξ(γ1-1γ)𝑑μx,(1.4)

for all γ𝖦x, andξx . For each xX=Ob𝖦, πxdefines an involutive representation πx:Cc(𝖦)x . We can define a norm on Cc(𝖦) given by

f=supxXπx(f),(1.5)

whereby the completionPlanetmathPlanetmath of Cc(𝖦) in this norm, definesthe reducedPlanetmathPlanetmath C*–algebra Cr*(G) of Glc. It isperhaps the most commonly used C*–algebra for Lie groupoids(groups) in noncommutative geometry.

The next step requires a little familiarity with the theory ofHilbert modules. We define a left𝔅–action λ and a right 𝔅–actionρ on 𝔄 by λ(B)A=Aηt(B) andρ(B)A=Aηs(B) . For the sake of localization of theintended Hilbert module, we implant a 𝔅–valued innerproductMathworldPlanetmath on 𝔄 given by A,C𝔅=P(A*C)  . Let us recall that P is defined as a completely positive map.Since P is faithful, we fit a new norm on 𝔄 given by A2=P(A*A)𝔅 . The completion of 𝔄 in this newnorm is denoted by 𝔄- leading then to a Hilbertmodule over 𝔅 .

The tensor productPlanetmathPlanetmathPlanetmath 𝔄-𝔅𝔄- can be shown to be a Hilbert bimodule over 𝔅,which for i=1,2, leads to *–homorphisms φi:𝔄𝔅(𝔄-𝔄-) . Next is to define the (unital) C*–algebra 𝔄𝔅𝔄 as the C*–algebra contained in𝔅(𝔄-𝔄-) that is generated by φ1(𝔄) andφ2(𝔄) . The last stage of the recipe for defining acompact quantum groupoid entails considering a certain coproductMathworldPlanetmathoperation Δ:𝔄𝔄𝔅𝔄, together with a coinverse Q:𝔄𝔄 that it is both an algebra andbimodule antihomomorphism. Finally, the following axiomaticrelationships are observed :

(id𝔅Δ)Δ=(Δ𝔅id)Δ(1.6)
(id𝔅P)Δ=P
τ(Δ𝔅Q)Δ=ΔQ

where τ is a flip map : τ(ab)=(ba) .

There is a natural extension of the above definition of quantum compact groupoidsto locally compact quantum groupoids by taking 𝖦lc to be a locally compact groupoidPlanetmathPlanetmath (instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid 𝖦 being replaced by 𝖦lc. Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid representations (for further details, see for example the monograph by Buneci (2003).

1.4.1 Reduced C*–algebra

Consider 𝖦 to be a topological groupoid. We denote by Cc(𝖦) the space of smooth complex–valued functions with compact support on 𝖦 . In particular, for all f,gCc(𝖦), thefunction defined via convolution

(f*g)(γ)=γ1γ2=γf(γ1)g(γ2),(1.7)

is again an element of Cc(𝖦), where the convolution productdefines the composition law on Cc(𝖦) . We can turnCc(𝖦) into a *–algebra once we have defined the involution*, and this is done by specifying f*(γ)=f(γ-1)¯ .

We recall that following Landsman (1998) a representation of a groupoid 𝔾, consists of afamily (or field) of Hilbert spaces {x}xXindexed by X=Ob𝔾, along with a collection of maps {U(γ)}γ𝔾, satisfying:

  • 1.

    U(γ):s(γ)r(γ),is unitary.

  • 2.

    U(γ1γ2)=U(γ1)U(γ2), whenever(γ1,γ2)𝔾(2)  (the set of arrows).

  • 3.

    U(γ-1)=U(γ)*, for all γ𝔾 .

Suppose now 𝖦lc is a Lie groupoid. Then the isotropy group𝖦x is a Lie group, and for a (left or right) Haarmeasure μx on 𝖦x, we can consider the Hilbertspaces x=L2(𝖦x,μx) as exemplifying theabove sense of a representation. Putting aside some technicaldetails which can be found in Connes (1994) and Landsman (2006), theoverall idea is to define an operator of Hilbert spaces

πx(f):L2(𝖦𝗑,μx)L2(𝖦x,μx),(1.8)

given by

(πx(f)ξ)(γ)=f(γ1)ξ(γ1-1γ)𝑑μx,(1.9)

for all γ𝖦x, andξx . For each xX=Ob𝖦, πxdefines an involutive representation πx:Cc(𝖦)x . We can define a norm on Cc(𝖦) given by

f=supxXπx(f),(1.10)

whereby the completion of Cc(𝖦) in this norm, definesthe reduced C*–algebra Cr*(G) of Glc.

It is perhaps the most commonly used C*–algebra for Lie groupoids(groups) in noncommutative geometry.

References

  • 1 E. M. Alfsen and F. W. Schultz: Geometry of State SpacesMathworldPlanetmath of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
  • 2 I. Baianu : Categories, FunctorsMathworldPlanetmath and Automata Theory: A Novel Approach to Quantum Automata through Algebraic–Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS, (August-Sept. 1971).
  • 3 I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non–Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes 17,(3-4): 353-408(2007).
  • 4 F.A. Bais, B. J. Schroers and J. K. Slingerland: Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 No. 18 (1–4): 181–201 (2002).
  • 5 M. R. Buneci.: Groupoid Representations, Ed. Mirton: Timishoara (2003).
  • 6 M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).
  • 7 L. Crane and I.B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. TopologyMathworldPlanetmath and physics. J. Math. Phys. 35 (no. 10): 5136–5154 (1994).
  • 8 W. Drechsler and P. A. Tuckey: On quantum and parallel transport in a Hilbert bundle over spacetime., Classical and Quantum Gravity, 13:611-632 (1996). doi: 10.1088/0264–9381/13/4/004
  • 9 V. G. Drinfel’d: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798-820 (1987).
  • 10 G. J. Ellis: Higher dimensional crossed modules of algebras,J. of Pure Appl. Algebra 52 (1988), 277-282.
  • 11 P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591-640 (1998).
  • 12 P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
  • 13 P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89-129, Cambridge University Press, Cambridge, 2001.
  • 14 B. Fauser: A treatise on quantum Clifford AlgebrasPlanetmathPlanetmath. Konstanz,Habilitationsschrift. (arXiv.math.QA/0202059). (2002).
  • 15 B. Fauser: Grade Free productMathworldPlanetmath Formulae from Grassman–Hopf Gebras.Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).
  • 16 J. M. G. Fell.: The Dual SpacesMathworldPlanetmathPlanetmath of C*–Algebras., Transactions of the AmericanMathematical Society, 94: 365–403 (1960).
  • 17 F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics., Boca Raton: CRC Press, Inc (1996).
  • 18 A. Fröhlich: Non–Abelian Homological Algebra. I.Derived functorsMathworldPlanetmath and satellites, Proc. London Math. Soc., 11(3): 239–252 (1961).
  • 19 R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications.,Dover Publs., Inc.: Mineola and New York, 2005.
  • 20 P. Hahn: Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242: 1–33(1978).
  • 21 P. Hahn: The regular representationsPlanetmathPlanetmath of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).
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