q-Deforming Maps for Lie Group Covariant Heisenberg Algebras

4 Pages·0.08 MB·English

by  Gaetano Fiore

October 1997 LMU-TPW 97-27
q-Deforming Maps for Lie Group Covariant
Heisenberg Algebras1
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Gaetano Fiore
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a Sektion Physik der Ludwig-Maximilians-Universita¨t Mu¨nchen
J Theoretische Physik — Lehrstuhl Professor Wess

7 Theresienstraße 37, 80333 Mu¨nchen
Federal Republic of Germany
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e-mail: Gaetano.Fiore @ physik.uni-muenchen.de
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g Abstract
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a We briefly summarize our systematic construction procedure of q-
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q deforming maps for Lie group covariant Weyl or Clifford algebras.
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1Talk presented at the Fifth Wigner Symposium, 25-29 August 1997, Vienna, Germany.
Submittedfortheproceedings oftheConference.
1 Introduction
AnydeformationofaWeylorCliffordalgebracanberealizedthroughachange
of generators in the undeformed algebra [1, 2]. “q-Deformations” of Weyl or
Cliffordalgebrasthatwerecovariantunder the actionofa simple Lie algebrag
arecharacterizedbytheirbeingcovariantundertheactionofthequantumgroup
U g, where q = eh. Here we briefly summarize our systematic construction
h
procedure[6,7]ofallthepossiblecorrespondingchangesofgenerators,together
with the corresponding realizations of the U g-action.
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This paves the way [6] for a physical interpretation of deformed generators
as“compositeoperators”,functionsoftheundeformedones. Forinstance,ifthe
latteractascreatorsandannihilatorsonabosonicorfermionicFockspace,then
theformerwouldactascreatorsandannihilatorsofsomesortof“dressedstates”
in the same space. Since there exists [7] a basis of g-invariants that depend on
the undeformed generators in a non-polynomial way, but on the deformed ones
inapolynomialway,thesechangesofgeneratorsmightbe employedtosimplify
the dynamics of some g-covariant quantum physical systems based on some
complicated g-invariant Hamiltonian.
Let us list the essential ingredients of our construction procedure:
1. g, a simple Lie algebra.
2. The cocommutative Hopf algebra H ≡ (Ug,·,∆,ε,S) associated to Ug;
·,∆,ε,S denote the product, coproduct, counit, antipode.
3. The quantum group [4] H ≡(U g,•,∆ ,ε ,S ,R).
h h h h h
4. An algebra isomorphism[5] ϕ :U g →Ug[[h]], ϕ ◦•=·◦(ϕ ⊗ϕ ).
h h h h h
5. AcorrespondingDrinfel’dtwist[5]F ≡F(1)⊗F(2)=1⊗2+O(h)∈Ug[[h]]⊗2:
(ε⊗id)F =1=(id⊗ε)F, ∆h(a)=(ϕ−h1⊗ϕ−h1)(cid:8)F∆[ϕh(a)]F−1(cid:9).
6. γ′ :=F(2)·SF(1) and γ :=SF−1(1)·F−1(2).
7. The generators a+,ai of a ordinary Weyl or Clifford algebra A.
i
8. The action ⊲:Ug ×A→A; A is a left module algebra under ⊲.
9. The representation ρ of gto which a+,ai belong:
i
x⊲a+ =ρ(x)ja+ x⊲ai =ρ(Sx)iaj.
i i j j
10. The Jordan-Schwinger algebra homomorphism σ :Ug ∈A[[h]]:
σ(x):=ρ(x)ia+aj if x∈g σ(yz)=σ(y)σ(z)
j i
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11. The generators A˜+,A˜i of a deformed Weyl or Clifford algebra A .
i h
12. The action ⊲ :U g ×A →A ; A is a left module algebra under ⊲ .
h h h h h h
13. The representation ρ =ρ◦ϕ of U g to which A˜+,A˜i belong:
h h h i
X ⊲ A˜+ =ρ˜(X)jA˜+ X ⊲ A˜i =ρ˜(S X)iA˜j.
h i i j h h j
14. ∗-structures ∗,∗ ,⋆,⋆ in H,H ,A,A , if any.
h h h h
2 Constructing the deformed generators
Proposition 1 [6] One can realize the quantum group action ⊲ on A[[h]] by
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setting for any X ∈Uhg and β ∈A[[h]] (with X(¯1)⊗X(¯2) :=∆h(X))
X ⊲hβ :=σ[ϕh(X(¯1))]βσ[ϕh(ShX(¯2))]. (1)
Proposition 2 [6, 7] For any g-invariants u,v∈A[[h]] the elements of A[[h]]
A+ := uσ(F(1))a+σ(SF(2)γ)u−1
i i (2)
Ai := vσ(γ′SF−1(2))aiσ(F−1(1))v−1.
transform under ⊲ as A˜+,A˜i.
h i
A suitable choice of uv−1 may make A+,Aj fulfil also the QCR of A [7]. In
i h
particular we have shown the
Proposition 3 [7] If ρ is the defining representation of g, A+,Aj fulfil the
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corresponding QCR provided
uv−1 = Γ(n+1) if g =sl(N)
uv−1 = ΓqΓ2[(n12+(n1+)1+N2−l)]Γ[12(n+1+N2+l)] if g =so(N), (3)
Γq2[12(n+1+N2+l)]Γq2[21(n+1+N2−l)]
where Γ,Γq2 are Euler’s gamma-function and its q-deformation, n := a+i ai,
l :=pσ(Cso(N)), and Cso(N) is the quadratic Casimir of so(N).
If A+,Aj fulfil the QCR, then also
i
A+ :=αA+α−1 Ai,α :=αAiα−1 (4)
i,α i
will do, for any α ∈ A[[h]] of the form α = 1 + O(h). By cohomological
argumentsonecanprovethattherearenomoreelementsinA[[h]]whichdo[7].
A+ ,Ai,α transform as A˜+,A˜i under the following modified realization of ⊲ :
i,α i h
X ⊲αh β :=ασ[ϕh(X(¯1))]α−1βασ[ϕh(ShX(¯2))]α−1. (5)
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The algebra homomorphism f : A → A[[h]] such that f (A˜+) = A+ and
α h α i i,α
f (A˜i)=Ai,α is what is usually called a “q-deforming map”.
α
ForacompactsectionofUg onecanchooseaunitaryF,F∗⊗∗ =F−1. Then
the Ug-covariant ∗-structure (ai)⋆ = a+ in A is also U g-covariant in A[[h]]
i h
and has the form (Ai,α)⋆ = A+ , provided we choose u = v−1 and α⋆ = α−1.
i,α
More formally, under this assumption ⋆ ◦ f = f ◦ ⋆ , with ⋆ defined by
α α h h
(A˜i)⋆h =A˜+
i
If H is instead a triangular deformation of Ug, the previous construction
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can be equally performed andleads essentially to the same results [6], provided
we choose in the previous formulae u ≡ v ≡ 1. This follows from the triviality
of the coassociator[5], that characterizes triangular deformations H .
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Acknowledgments
ItisapleasuretothankJ.Wessforhisstimulus,supportandwarmhospitalityat
hisInstitute. ThisworkwassupportedthroughaTMRfellowshipgrantedbythe
European Commission, Dir. Gen. XII for Science, Research and Development,
under the contract ERBFMBICT960921.
References
References
[1] M. Pillin, Commun. Math. Phys. 180 (1996), 23.
[2] F. du Cloux, Asterisque (Soc. Math. France) 124-125 (1985), 129.
[3] V. G. Drinfeld, Doklady AN SSSR 273 (1983), 531.
[4] V. G. Drinfeld, Proceedings of the International Congress of Mathemati-
cians, Berkeley 1986, Vol. 1, 798.
[5] V. G. Drinfeld, Leningrad Math. J. 1 (1990), 1419.
[6] G. Fiore, Deforming Maps for Lie Group Covariant Creation & Annihila-
tion Operators, LMU-TPW 96-20,q-alg/9610005.
[7] G. Fiore, Drinfel’d Twist and q-Deforming Maps for Lie Group Covariant
Heisenberg Algebras, LMU-TPW 97-07, q-alg/9708017.
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