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Quantum homogeneous spaces as quantum quotient spacesTomasz Brzeziski Citation: Journal of Mathematical Physics 37, 2388 (1996); doi: 10.1063/1.531517 View online: http://dx.doi.org/10.1063/1.531517 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/37/5?ver=pdfcov Published by the AIP Publishing
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Quantum homogeneous spaces as quantum quotientspaces
Tomasz Brzezinskia)Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Cambridge CB3 9EW, United Kingdom
~Received 31 October 1995; accepted for publication 25 January 1996!
It is shown that certain embeddable homogeneous spaces of a quantum group thatdo not correspond to a quantum subgroup still have the structure of quantum quo-tient spaces. A construction of quantum fibre bundles on such spaces is proposed.The quantum plane and the general quantum two-spheres are discussed in detail.© 1996 American Institute of Physics.@S0022-2488~96!02705-1#
I. INTRODUCTION
A homogeneous spaceX of a Lie groupG may be always identified with the quotient spaceG/G0 , whereG0 is a Lie subgroup ofG. When the notion of a homogeneous space is generalizedto the case of quantum groups or noncommutative Hopf algebras the situation becomes muchmore complicated. A general quantum homogeneous space of a quantum groupH need not be aquotient space ofH by its quantum subgroup. By a quantum subgroup ofH we mean a HopfalgebraH0 such that there is a Hopf algebra epimorphismp:H→H0 . The quotient space is thenunderstood as a subalgebra ofH of all points that are fixed under the coaction ofH0 onH inducedby p. A quantum homogeneous spaceB of H might be such a quotient space but it is not ingeneral. There is, however, a certain class of quantum homogeneous spaces, of which the quantumtwo sphere of Podles´1 is the most prominent example, that not being quotient spaces by a quantumsubgroup ofH, may be embedded inH. One terms such homogeneous spacesembeddable.2 Thegeneral quantum two sphereSq
2~m,n! is such an embeddable homogeneous space of the quantumgroup SUq~2!, and it is a quantum quotient space in the above sense whenn50. In the latter casethe corresponding subgroup of SUq~2! may be identified with the algebra of functions onU~1!. Inthis article it is shown that certain embeddable quantum homogeneous spaces, and the generalquantum two-sphereSq
2~m,n! among them, can still be understood as quotient spaces or fixed pointsubalgebras. It is shown that there is a coalgebraC and a coalgebra epimorphismp:H→C suchthat the fixed point subspace ofH under the coaction ofC onH induced from the coproduct inHby a pushout byp is a subalgebra ofH isomorphic toB.
The interpretation of embeddable quantum homogeneous spaces as quantum quotient spacesallows one to develop the quantum group gauge theory of such spaces following the lines of Ref.3. The study of such a gauge theory becomes even more important once the appearance of thequantum homogeneous spaces in theA. Connes geometric description of the standard model wasannounced.4 For this purpose, however, one needs to generalize the notion of a quantum principalbundle of Ref. 3 so that a Hopf algebra playing the role of a quantum structure group there maybe replaced by a coalgebra. Such a generalization is proposed~see Ref. 5 for further details!. Sincethe theory of quantum principal bundles is strictly related to the theory of Hopf–Galois extensions~cf. Ref. 6!, a generalization of such extensions is proposed.
The article is organized as follows. In Sec. II the notation used in the sequel is described.Section III shows a fixed point subalgebra structure of embeddable quantum homogeneous spaces.A suitable generalization of the notion of a quantum principal bundle in Sec. IV is proposed.Sections V and VI are devoted to careful study of two examples of quantum embeddable spaces,
a!Electronic mail: [email protected]
0022-2488/96/37(5)/2388/12/$10.002388 J. Math. Phys. 37 (5), May 1996 © 1996 American Institute of Physics
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namely the quantum planeCq2 ~Ref. 7! and the quantum sphereSq
2~m,n!.1
II. PRELIMINARIES
In the sequel all the vector spaces are over the fieldk of characteristic not 2.C denotes acoalgebra with the coproductD:C→C^C and the counite:C→k which satisfy the standardaxioms, cf. Ref. 8. For the coproduct the Sweedler sigma notation is used:
Dc5c~1! ^c~2! ~D ^ id !+Dc5c~1! ^c~2! ^c~3! , etc.,
wherecPC, and the summation sign and the indices are suppressed. A vector spaceA is a leftC-comodule if there exists a mapDL :A→C^A, such that (D ^ id)+DL5( id^ DL)+DL , and(e ^ id)+DL5 id. ForDL we use the explicit notation
DLa5a~1! ^a~`! ,
whereaPA and alla(1)PC and alla(`)PA.Similarly a vector spaceA is a rightC-comodule if there exists a mapDR :A→A^C, such
that (DR^ id)+DR5( id^ D)+DR , and (id^ e)+DR5 id. ForDR the explicit notation is used:
DRa5a~0! ^a~1! ,
whereaPA and alla(1)PC and alla(0)PA.H denotes a Hopf algebra with productm:H^H→H, unit 1, coproductD:H→H^H, counit
e:H→k and antipodeS:H→H. Sweedler’s sigma notation is used as before. Similarly as for acoalgebra right and leftH-comodules are defined. For a rightH-comoduleA we denote byAcoH a vector subspace ofA of all elementsaPA such thatDRa5a^1. A right ~respectively, left!H-comoduleA is a right ~respectively, left! H-comodule algebra ifA is an algebra andDR
~respectively,DL! is an algebra map.A vector subspaceJ of H such thate(J)50 andDJ,J^H%H^J is called acoideal in H.
If J is a coideal inH thenC5H/J is a coalgebra with a coproductD given byD5~p^p!+D,wherep:H→C is a canonical surjection. The counite in C is defined by the commutative diagram
.
III. QUANTUM HOMOGENEOUS SPACES
In this section it is shown that if an embeddable quantum homogeneous space satisfies certainadditional assumption it may be identified with a quantum quotient space.
Definition 3.1:2 Let H be a Hopf algebra andB be a leftH-comodule algebra with thecoactionDL :B→H^B. B is an embeddable quantum homogeneous spaceor simply anem-beddable H-space if there exists an algebra inclusioni :B�H such thatD+ i5( id^ i )+DL , i.e., i isan intertwiner.
Proposition 3.2: ~1! A left H comodule algebra B is an embeddable H-space if and only ifthere exists an algebra characterk:B→k such that the linear map ik :B→H, i k :b°b(1)k(b(`))is injective. ~2! If B is an embeddable H-space then the linear mapxL :B^B→H^B,xL :b^b8°b(1)^b(`)b8 is injective.
Proof: ~1! If B is an embeddable quantum homogeneous space thenk5e+i is a character ofB.Sincei is an intertwiner, for anybPB, compute
2389Tomasz Brzezinski: Quantum homogeneous and quotient spaces
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i k~b!5b~1!e~ i ~b~`!!!5 i ~b!~1!e~ i ~b!~2!!5 i ~b!,
thus i k is an inclusion.Conversely assume that there is a characterk:B→k such thati k is injective. Then clearlyi k is
an algebra inclusion. Furthermore,
D~ i k~b!!5b~1! ^b~2!k~b~`!!5b~1! ^ i k~b~`!!5~ id^ i k!+DL~b!.
Thereforei k is an intertwiner as required.~2! The canonical mapcan: H^H→H^H, can: u^v°u(1)^u(2)v is a linear isomorphism.
Consider the diagram
. ~3.1!
Clearly, both the rows and the columns of diagram~3.1! are exact. Moreover for anyb,b8PB:
~ id^ i !+xL~b^b8!5b~1! ^ i ~b~`!b8!5b~1! ^ i ~b~`!!i ~b8!5 i ~b!~1! ^ i ~b!~2!i ~b8!
5can~ i ~b! ^ i ~b8!!,
and hence diagram~3.1! is also commutative. Therefore the sequence 0→B ^ B→xL
H ^ B is exact,i.e., the mapxL is injective. h
Remark 3.3:The second assertion of Proposition 3.2, i.e., the injectiveness ofxL , is a dualversion of the statement that the action of a group on its homogeneous space is transitive.
Proposition 3.4: Let B be an embeddable H space corresponding to the characterk:B→k.Define a right ideal Jk,H by Jk5$( j ( i k(bj )2k(bj ))uj ;;bjPB,;ujPH%. Then Jk is a coidealin H.
Proof: Clearly
e~ i k~b!2k~b!!5e~b~1!!k~b~`!!2k~b!5k~b!2k~b!50.
Furthermore,
D~ i k~b!2k~b!!5 i k~b!~1! ^ i k~b!~2!2k~b!1^15b~1! ^ ~ i k~b~`!!2k~b~`!!!1b~1!k~b~`!! ^1
2k~b!1^15b~1! ^ ~ i k~b~`!!2k~b~`!!!1~ i k~b!2k~b!! ^1.
Therefore for anybPB,
D~ i k~b!2k~b!!PH^Jk %Jk ^H,
so thatJk is a coideal as stated. h
Since Jk is a coideal ofH, the vector spaceC5H/Jk is a coalgebra and the canonicalsubjectionp :H→C is a coalgebra map. This in turn implies thatH is a rightC-comodule with thecoactionDR5( id^ p)+D:H→H^C. Let HcoC5$uPH;DRu5 u ^ p(1)%.
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Proposition 3.5: Let B be an embeddable H-space corresponding to the characterk:B→k,Jk be as in Proposition 3.4 and C5H/Jk . Then: (1) H
coC is a subalgebra of H; (2) B is asubalgebra of HcoC.
Proof: ~1! Since kerp5Jk is a right ideal inH there is a natural right actionr0:C^H→C ofH on C given by the commutative diagram
.
In other words for anyaPC and uPH, r0(a,u)5p(vu), where vPp21(a). For any u,vP HcoC we compute
DR~uv !5u~1!v ~1! ^ p~u~2!v ~2!!5u~1!v ~1! ^ r0~p~u~2!!,v ~2!!5uv ~1! ^ r0~p~1!,v ~2!!
5uv ~1! ^ p~v ~2!!5uv^ p~1!.
Thereforeuv P HcoC andHcoC is a subalgebra ofH as required.~2! For anybPB we compute
DR~ i k~b!!5 i k~b!~1! ^ p~ i k~b!~2!!5b~1! ^ p~ i k~b~`!!!5b~1! ^ k~b~`!!p~1!5 i k~b! ^ p~1!.
Hencei k :B�HcoC is the required algebra inclusion. h
Proposition 3.5 shows therefore that ifHcoC, i k(B) then the embeddableH-spaceB may beidentified with the quantum quotient spaceHcoC. For example, exploiting the argument of theproof of Proposition 1.2.4 of Ref. 9, one can conjecture that the above inclusion holds ifH is afaithfully flat B-module.
IV. A POSSIBLE GENERALIZATION OF QUANTUM PRINCIPAL BUNDLES
Once anH-embeddable spaceB is identified with a quotient spaceHcoC, it is natural to viewH as a total space of a principal bundle overB. Therefore one would like to apply the generaltheory of quantum principal bundles of Ref. 3 to this case too. In general, however, neitherC isa Hopf algebra nor, if it happens to be a Hopf algebra,C is a quantum subgroup ofH. Hence theinduced coaction ofC on H is not an algebra map. Therefore, to develop a gauge theory onembeddable homogeneous spaces one needs to generalize the theory of quantum principalbundles. In this section such a generalization is proposed. It is based on a simple observation thatthe structure of quantum principal bundles is mainly determined by the coalgebra structure of thequantum group. The notions studied in this section are developed to great extent in Ref. 5.
LetC be a coalgebra and letP be an algebra and a rightC-comodule. Assume that there is anactionr:P^C^P→P^C of P on P^C and an element 1PC such that
~1! For anyu,vPP,r(u^1,v)5uv (0)^v (1);~2! the following diagram:
,
wherem is a product inP, is commutative.
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Define B5PcoC5$uPP;DRu5u^1%.Lemma 4.1:B is a subalgebra of P.Proof: Take anyu,vPB. Then
DR~uv !5r~u~0! ^u~1! ,v !5r~u^1,v !5uv ~0! ^v ~1!5uv^1. h
Definition 4.2: Let P, C, r, andB be as before. It is said thatP(B,C,r) is a C-Galoisextension or a quantumr-principal bundle (with universal differential structure)if the canonicalmapx:P^ BP→P^C, x:u^ Bv°uv (0)^v (1) is a bijection.
Example 4.3: A quantum principal bundleP(B,H) as defined in Ref. 3 is ar-principalbundle with the actionr:P^H^P→P^H given byr(u^a,v)5uv (0)^av (1).
Example 4.4:Let H be a Hopf algebra,C a coalgebra, andp:H→C a coalgebra surjection.ThenH is a rightC-comodule with a coactionDR5( id^ p)+D. Denote 15p~1!PC and defineB 5 HcoC as before. Assume that kerp is a minimal right ideal inH such that $u2e(u);uPB%,kerp ~compare Sec. III!. Then a canonical right actionr0:C^H→C as in theproof of Proposition 3.5 can be defined. Furthermore,
r~u^a,v !5uv ~1! ^ r0~a,v ~2!!
for anyu, vPH, aPC. With these definitionsH(B,C,r) is a quantumr-principal bundle.Proof: First we show thatr:H^C^H→H^C is a right action and it has the properties~1!
and ~2!. Sincer0 is a right action, for anyu, v, wPH, aPC, compute
r~u^a,vw!5uv ~1!w~1! ^ r0~a,v ~2!w~2!!5uv ~1!w~1! ^ r0~r0~a,v ~2!!,w~2!!
5r~uv ~1! ^ r0~a,v ~2!!,w!5r~r~u^a,v !,w!,
and thusr is an action as required. Furthermore,
r~u^1,v !5uv ~1! ^ r0~1,v ~2!!5uv ~1! ^ p~v ~2!!5uv ~0! ^v ~1!
and
r~u~0! ^u~1! ,v !5u~1!v ~1! ^ r0~p~u~2!!,v ~2!!5u~1!v ~1! ^ p~u~2!v ~2!!5DR~uv !.
Thereforer has all the required properties.To prove that the canonical mapx is bijective we first note that, by assumption,
kerp,m+~kerpuB^H! and then use a suitably modified argument of the proof of Lemma 5.2 ofRef. 3 to deduce thatx is a bijection. It is clear thatx is a surjection since for any(kuk^akPH^C we can choose(kukSvk(1)^ Bvk(2)PH^ BH, where ;k,vkPp21(ak), andcompute
xS (kukSvk~1! ^ Bvk~2!D 5(
kuk~Svk~1!!vk~2! ^ p~vk~3!!5(
kuk^ p~vk!5(
kuk^ak .
Next we compute kerx,H^ BH. Take any(kuk^ BvkPkerx. Then (kukvk(1)^ p(vk(2))50.Applying id^e to the last equality, we find that(kukvk50, i.e., (kuk^vkPkerm. Any ( iwi8^ wi9 P kerm can be written as(kukSvk(1)^vk(2)PH^H, where;k, vkPker e anduk are lin-early independent. Thus
xS (iwi8^ Bwi9D 5xS (
kukSvk~1! ^ Bvk~2!D 5(
kuk^ p~vk!.
If ( iwi8 ^ Bwi9 P ker x then(kuk^ p(vk)50, thus for allk,p(vk)50. By assumptionvk5( jbkj vk
j ,wherebk
j Pker euB5kerpuB . Therefore,
2392 Tomasz Brzezinski: Quantum homogeneous and quotient spaces
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(iwi8^ Bwi95(
kukSvk~1! ^ Bvk~2!5(
j ,kuk~Svk~1!
j !Sbk~1!j
^ Bbk~2!j vk~2!
j
5(j ,k
e~bkj !ukSvk~1!
j^ Bvk~2!
j 50.
So kerx50, andx is a bijection as required. h
Therefore it is shown that an embeddableH space which is a quotient spaceB 5 HcoC asdescribed in Sec. III may be identified with a base manifold of the generalized quantum principalbundle, or equivalently thatH is aC-Galois extension ofB.
V. MANIN’S PLANE AS A QUANTUM QUOTIENT SPACE
In this section we show that Manin’s plane is a quotient space of the quantum general lineargroup GLq~2,C!. Recall that Manin’s planeCq
2 is defined for any nonzeroqPC as an associativepolynomial algebra overC generated by 1,x,y subject to the relationsxy5qyx. It is a quantumhomogeneous space of the quantum linear group GLq~2,C!. GLq~2,C! is defined as follows. Firstconsider an algebra generated by the matrixt5~g d
a b ! and the relations
ab5qba, ag5qga, ad5da1~q2q21!bg, ~5.1a!
bg5gb, bd5qdb, gd5qdg. ~5.1b!
The quantum determinantc5ad2qbg is central in the algebra~5.1!; thus it can be enlarged withc21. The resulting algebra is called GLq~2,C!. The quantum linear group GLq~2,C! is a Hopfalgebra of a matrix group type, i.e.,
Dt5t^ t, et51, St5c21S d 2q21b
2qg a D .The left coaction of GLq~2,C! on Cq
2 is given by
DLS xyD 5S a b
g d D ^ S xyD .Cq2 is not only a homogeneous space of GLq~2,C! but also an embeddable GLq~2,C! space. The
linear mapk:Cq2→C, k(xnym)5dm0,m,nPZ>0 is a character ofCq
2. By Proposition 3.2 it inducesan algebra mapi k :Cq
2→GLq~2,C!, which is explicitly given byi k(x)5a, i k(y)5g. The mapi k isclearly an inclusion. Thus the right idealJk is generated bya21 and g. The coalgebraC5GLq~2,C!/Jk may be easily computed. It is spanned byam,n5p(bmcn), mPZ.0, nPZ anda0,0515p~1!, wherep:GLq~2,C!→C is a canonical surjection. To see that theam,n really spanC,note that sinceJk is generated bya21 and g as a right ideal in GLq~2,C!, every a whichmultiplies any element of GLq~2,C! from the left is replaced by 1 and similarly anyg is replacedby 0 when the resulting element of GLq~2,C! is acted upon byp. Then we compute
p~akb lgmdncr !5p~b lgmdncr !5dm0p~b ldncr !5dm0qlnp~dnb lcr !5dm0q
ln~p~addn21b lcr !
2q21p~gbdn21b lcr !!5dm0qlnp~dn21b lcr11!5•••5dm0q
lnal ,r1n .
Therefore any element ofC5p~GLq~2,C!! may be expressed as a linear combination ofam,n .The coalgebra structure ofC is found from the coalgebra structure of GLq~2,C!, sinceDC
5 (p ^ p) + DGLq(2,C). Explicitly
2393Tomasz Brzezinski: Quantum homogeneous and quotient spaces
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Dam,n5 (k50
m Smk Dq
ak,n^am2k,n1k , e~am,n!5dm0 , ~5.2!
where the quantum binomial coefficients are defined by
Smk Dq
5@m#q!
@m2k#q! @k#q!, @m#q5
qm2q2m
q2q21 , @m#q!5)k51
m
@k#q , @0#q!51.
The next step in the identification ofCq2 as a quantum quotient space consists of computing the
fixed point subalgebraB 5 GLq(2,C)coC. For a general monomialakg lbmdncrPGLq~2,C!,
k,l ,m,nPZ>0, rPZ, we find
DR~akg lbmdncr !5akg lcr(i50
m
(j50
n
qj ~m2 i !Smi DqS nj D
q
am2 ib ign2 jd j^am1n2~ i1 j !,i1 j1r .
~5.3!
The right-hand side of Eq.~5.3! has the formu^1 for someuPGLq~2,C! if and only ifm5n5r50. ThusB is spanned by allakg l . ThereforeB, i k~Cq
2! and sincei k~Cq2!,B by Proposition 3.5
we conclude thatCq2 > GLq(2,C)
coC. By Example 4.4 GLq~2,C!~Cq2 ,C,r! is a quantum principal
r-bundle. The actionr0:C^GLq~2,C!→C is given explicitly by
r0~ai , j ,akb lgmdncr !5dm0q
i ~n2k!1 lnai1 l , j1n1r .
One can now proceed to define an algebra structure onC so that it becomes a Hopf algebra.Define the product inC by
ak,lam,n5qlm2knak1m,l1n .
First we notice thata0,051 is the unit element with respect to this product. Next we show that thisproduct is compatible with the coalgebra structure ofC. Compute
D~ak,l !D~am,n!5(i50
k
(j50
m S ki DqSmj D
q
ai ,laj ,n^ak2 i ,l1 iam2 j ,n1 j
5qlm2nk(i50
k
(j50
m
qim2k jS ki DqSmj D
q
ai1 j ,l1n^ak1m2~ i1 j !,l1n1 i1 j
5qlm2nk(r50
k1m S k1mr D
q
ar ,l1n^ak1m2r ,l1n1r5qlm2knD~ak1m,l1n!5D~ak,lam,n!.
The third equality is a consequence of the following property of theq-deformed binomial coef-ficients
;rP@0,k1m#, (i50
k
(j50
m
d i1 j ,rqim2k jS ki D
qSmj D
q
5S k1mr D
q
.
Clearly the counit ofC is an algebra homomorphism. Before defining an antipode, we show thatC is a polynomial algebra. Leta5a0,1, a
215a0,21, b5a1,0. Then for anymPZ.0, nPZ,
am,n5q2mnanbm, ab5q2ba, aa215a21a51.
2394 Tomasz Brzezinski: Quantum homogeneous and quotient spaces
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ThereforeC is a polynomial algebra indeed, and it is isomorphic toCq22
@x21#. The coalgebrastructure ofC written in terms ofa andb reads
Da615a61^a61, Db51^b1b^a, e~a61!51, e~b!50
and hence the antipode is defined asSa615a71, Sb52ba21.We have just shown thatC may be equipped with an algebra structure ofCq2
2@x21#, and then
the coalgebra structure ofC becomes a standard coalgebra structure of the latter. Therefore wehave proven
Theorem 5.1:
Cq25GLq~2,C!coCq2
2@x21#.
Notice that clearly neither p:GLq(2,C)→Cq22
@x21# nor DR5( id^ p)+D:GLq(2,C)→GLq(2,C) ^ Cq2
2@x21# are algebra maps. Still, following the proposal of Sec. IV the
generalized principal bundle GLq(2,C)(Cq2,Cq2
2@x21#,r,p) can be defined and analyzed. In par-
ticular one can truly develop a gauge theory, define connections and their curvature, closelyfollowing the quantum group gauge theory introduced in Ref. 3.
VI. PODLES’ SPHERE AS A QUANTUM QUOTIENT SPACE
In this section we prove that the quantum two-sphere is a quantum quotient space in the senseexplained in Sec. III. In the presentation of the quantum sphere the conventions of Ref. 10 arefollowed.
The general quantum two-sphereSq2~m,n! is a polynomial algebra generated by the unit and
x,y,z, and the relations
xz5q2zx, xy52q~m2z!~n1z!,
yz5q22zy, yx52q~m2q22z!~n1q22z!,
wherem, n, andqÞ0 are real parameters,mn>0, ~m,n!Þ~0,0!. The quantum sphere is a* algebrawith the * structurex*52qy, z*5z.
The quantum sphereSq2~m,n! is an SUq~2! homogeneous quantum space. SUq~2! is defined as
a quotient of GLq~2,C! by the relationc51, and has a*-structure given byd5a* , g52q21b* .The coaction of SUq~2! on Sq
2~m,n! is defined as follows. Let f25x,f05(11q22)21/2(m2n2(11q22)z), f15y. Then
DLS f2
f0
f1
D 5S a2 ~11q22!1/2ab b2
~11q22!1/2ag 11~q1q21!bg ~11q22!1/2bd
g2 ~11q22!1/2gd d2D ^ S f2
f0
f1
D .The quantum sphereSq
2~m,n! is not only a quantum homogeneous space but also an embeddableSUq~2! space. There is a* characterk:Sq
2~m,n!→C given by
k~x!5qAmn, k~y!52Amn, k~z!50.
Therefore there is also a*-algebra homomorphismi k :Sq2~m,n!→SUq~2!, which reads explicitly
i k~x!5Amn~qa22b2!1~m2n!ab,
i k~y!5Amn~qg22d2!1~m2n!gd,
2395Tomasz Brzezinski: Quantum homogeneous and quotient spaces
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i k~z!52Amn~qag2bd!2~m2n!bg,
and is clearly an inclusion. From now on we assume thatmÞn ~but also see Remark 6.5!. In thiscaseSq
2~m,n! depends on two real parameters only, namelyq andp 5 Amn/(m 2 n!. By Proposi-tion 3.4 the inclusioni k induces a coidealJk,SUq~2!, generated as a right ideal in SUq~2! by thefollowing three elements:
p~qa22b2!1ab2pq, p~qg22d2!1gd1p, p~qag2bd!1qbg.
Therefore we can construct the coalgebraC(p)5SUq(2)/Jk , and the corresponding quotientspaceB(p) 5 SUq(2)
coC(p) as described in Sec. III. At the end of this procedure we identifyB(p)with Sq
2~m,n!, mÞn. We start with the coalgebraC(p).Proposition 6.1: C(p) is a vector space spanned by15p~1!, xn5p(an) and yn5p(dn),
wherep:SUq(2)→C(p) is a canonical surjection and nPZ.0.Proof: For anyuPSUq~2! we use the explicit form of the generators ofJk and the relations in
SUq~2! to find that
p~bu!5p~badu!2qp~bgbu!5q21p~abdu!2qp~bgbu!52pp~a2du!1pq21p~b2du!
1pp~du!1pqp~agbu!2pp~bdbu!5pp~du!2pp~au!, ~6.1a!
and similarly
p~gu!5pp~du!2pp~au!. ~6.1b!
From Eq. ~6.1! it follows that for any uPSUq~2!, p(ubmgn)5p(ubm1n). Since SUq~2! isspanned by the monomialsambkg l ,dmbkg l ~cf. Lemma 7.1.2 of Ref. 11! it suffices to prove thatthe following elements ofC(p),
ak2~n!5p~dkbn2k!, ak1
~n!5p~akbn2k!, ~6.2!
where nPZ.0, k50,1,...,n can be expressed as linear combinations of 1,xm , ym . Clearlya02(n)5a01
(n) . Thus we simply write a0(n). Also, an1
(n)5xn and an2(n)5yn . For n51,
a0(1)5p(b)5p(y12x1). For a generaln we apply the rules~6.1! to ak6
(n) and express the latter interms ofal6
(m),m,n, andxn ,yn . We make the inductive assumption that for allm,n, al6(m) can be
written as linear combinations of 1,xr ,yr . Therefore, forn>2 we arrive at the system of equa-tions:
ak6~n!6pq6kak116
~n! 7pq7~k21!ak216~n! 56pq6kak216
~n22! ,~6.3!
a0~n!2pa12
~n!1pa11~n!50,
wherek51,2,...,n21. This is a system of 2n21 equations with 2n21 unknowns provided thatthe right-hand sides andxn ,yn are treated as known parameters. Obviously it has a solution if itsdeterminant is nonzero. The determinantDn of the system~6.3! may be easily computed. It doesnot depend onq and it can be reduced to the determinant of the following 2n2132n21 matrix:
2396 Tomasz Brzezinski: Quantum homogeneous and quotient spaces
J. Math. Phys., Vol. 37, No. 5, May 1996
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11 2p p 0 0 0 ... 0 0 0 0 0
p 1 0 2p 0 0 ... 0 0 0 0 0
2p 0 1 0 p 0 ... 0 0 0 0 0
0 p 0 1 0 2p ... 0 0 0 0 0
.... ..... .... ...... ... .... ...... .... .... ... ...... ..
0 0 0 0 0 0 ... 0 1 0 2p 0
0 0 0 0 0 0 ... 2p 0 1 0 p
0 0 0 0 0 0 ... 0 p 0 1 0
0 0 0 0 0 0 ... 0 0 2p 0 1
2 . ~6.4!
By the Laplace theoremDn can be further developed to give
Dn5A2n221p2~A2n231A2n24!1p4A2n25 ,
whereAm is zero for negativem, A051 and for anym,2n21, Am is the determinant of thematrix obtained from Eq.~6.4! by removing the first 2n212m rows and columns. The determi-nantsAm are the standard ones and we finally obtain the determinant of the system~6.3! as apolynomial
Dn5Pn21~p2
ckn5(
l5k
n
~21/4! l2kS 2n112 ll D S lkD .
For any n and any 0<k<n, ckn>c0
n5(n11)/4n and thus all the coefficientsckn are positive.
ThereforePn(x21/4)Þ0 for all realx>0. This implies that the determinantsDn of the proof ofProposition 6.1 are nonzero provided thatp2>21/4. Since for anyq, q1q21>2 one sees that theassertion of Proposition 6.1 holds for the exceptional quantum spheres too.
In a different context, polynomialsPn(x) appeared in Ref. 12. It was shown there that all thezeros ofPn are real and equal toxk521
4sec2(pk/2n12), k51,2,...,n. The numbers21/xk are the
discrete values of the index for subfactors of type II1 von Neumann algebras.Proposition 6.4: Let C(p) be a coalgebra described in Proposition 6.1 and let B(p)
5 SUq(2)coC(p).Then ik(Sq
2(m,n))5B(p) for all mÞn such that p5 Amn/(m 2 n).Proof: By Proposition 3.5, i k(Sq
2(m,n)),B(p), therefore one needs to show thatB(p), i k(Sq
2(m,n)). Introduce the gradingd:SUq~2!→Z by
d~a!5d~b!51, d~1!50, d~g!5d~d!521, d~uv !5d~u!1d~v ! ~6.5!
for any monomialsu,vPSUq~2!. A set of all elements of SUq~2! of degreekPZ forms a vectorsubspace of SUq~2!, which is denoted by SUq(2)
(k), and SUq(2) 5 % kPZSUq(2)(k). Moreover if
Du5 ( iui8 ^ ui9 for anyuPSUq(2)(k), then for alli ,d(ui8) 5 k. To see that the last statement is true
one can explicitly verify it fora, b, g, d and then use definition~6.5! of d to prove it for anySUq~2!. Therefored induces a grading ofB(p) andB(p) 5 % kPZB(p)
(k).Next, notice thatB(p) is contained in the subalgebra of SUq~2! spanned by monomials of
even degree. Therefore for anykPZ, B(p)(2k11)50.To prove the required inclusion observe that due to the form ofp and C(p), B(p) is a
deformation ofB~0!, i.e.,B~0!5lim p→0B(p). Denote byB(p)2n(2k) the vector space of homoge-
neous polynomialsuPB(p) of degree 2n such thatd(u)52k, uku<n. Notice thatB(p)2n(2k) and
B(p)2l(2k) need not be distinct forlÞn. B(0)2n
(2k) is spanned byambn1k2mgn2mdm2k, wherem5k,k11,...,n for k>0 andm50,1,...,n1k for k,0, and hence isn2uku11-dimensional. Thisis exactly the dimension ofi k(Sq
2(m,n))2n(2k). Suppose thatB(p)2n
(2k) is at leastn2uku12-dimensional. Then one can finduPB(p)2n
(2k) that does not contain any of the monomials spanningB(0)2n
(2k). If lim p→0uÞ0, then one would obtain thatB(0)2n(2k) is at leastn2uku12-dimensional,
hence contradiction. limp→0u is meant as the polynomial obtained fromu by replacing its coef-ficients with theirp50 limits. Assume that limp→0u50. The polynomialu may be written as alinear combination of monomials of degree 2n with coefficients that vanish as polynomials whenp tends to 0. Therefore there exists a positive integerm such that limp→0p
2mu exists, is finite andnonzero, and is an element ofB(0)2n
(2k). Thus we have a contradiction again. Since the aboveargument does not depend onn and k, and i k(Sq
2(m,n)),B(p) we conclude thati k(Sq
2(m,n))5B(p). h
Therefore we have shown that formÞn the quantum sphereSq2~m,n! is a quantum quotient
space. By Example 4.4 one has a principalr-bundle, SUq(2)(Sq2(m,n),C(p),r,p).
Remark 6.5:When m5nÞ0 the coidealJk is generated as a right ideal in SUq~2! by thefollowing elements:
qa22b22q, qg22d211, qag2bd.
Therefore for anyuPSUq~2!,
p~du!5p~au!, p~bu!5p~gu!, p~g2u!5q21p~a2u!2q21p~u!,
and hence the coalgebraC5SUq(2)/Jk is spanned by 15p~1!, xn5p(an), yn5p(an21g),nPZ>1. We conjecture that also for this caseSq
2(m,m) > SUq(2)coC.
2398 Tomasz Brzezinski: Quantum homogeneous and quotient spaces
J. Math. Phys., Vol. 37, No. 5, May 1996
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VII. CONCLUSIONS
In this article we have shown that certain embeddable quantum homogeneous spaces may beviewed as quantum quotient spaces. The examples of such quantum embeddable spaces includethe general quantum two-sphereSq
2~m,n! and the quantum planeCq2. The interpretation of quantum
embeddable spaces presented in this article seems specially interesting from the point of view ofquantum group gauge theory, the suitable generalization of which has also been proposed. Wethink that it would be interesting and indeed desirable to further develop this generalization ofquantum group gauge theory, and in particular, to construct connections on the quantum spacesdescribed in this article. For example this would allow for extending the construction of the Diracq monopole of Ref. 3 to general quantum spheres. This program of studying coalgebra gaugetheories, which will also incorporate braided group gauge theories, is currently being carried outand the first results may be found in Ref. 5~cf. Ref. 13!.
Note added in proof.After completing this article I have learned that the results similar tothose of Sec. III were also obtained in M. S. Dijkhuizen and T. H. Koornwinder, Geom. Dedicata.52, 291 ~1994!.
ACKNOWLEDGMENTS
The author would like to thank Shahn Majid for suggesting the proofs of Propositions 3.4 and3.5~2!.
Most of this article was written during my stay at Universite Libre de Bruxelles. At this timeI was supported by the European Union Human Capital and Mobility grant. This work is alsosupported by Grant No. KBN 2 P302 21706 p01 and the EPSRC Grant No. GR/K02244.
1P. Podles´, Lett. Math. Phys.14, 193 ~1987!.2P. Podles´, Commun. Math. Phys.170, 1 ~1995!.3T. Brzezinski and S. Majid, Commun. Math. Phys.157, 591 ~1993!; Erratum:167, 235 ~1995!.4A. Connes, Lecture given at the Conference on Noncommutative Geometry and Its Applications, Castle Trˇest, CzechRepublic, May 1995.
5T. Brzezinski and S. Majid, preprint DAMTP/95-74~1995!, q-alg/gS02022..6H.-J. Schneider, Israel J. Math72, 167 ~1990!; 72, 196 ~1990!.7Y. I. Manin, Montreal Notes, 1989.8M. E. Sweedler,Hopf Algebras~Benjamin, New York, 1969!.9N. Andruskiewitsch and J. Devoto, Algebra Analiz7, 22 ~1995!.10M. Nuomi and K. Mimachi, Commun. Math. Phys.128, 521 ~1990!.11V. Chari and A. Pressley,A Guide to Quantum Groups~Cambridge University, Cambridge, 1994!.12V. F. R. Jones, Invent. Math.72, 1 ~1985!.13T. Brzezinski, preprint DAMTP/96-15~1996!.
2399Tomasz Brzezinski: Quantum homogeneous and quotient spaces
J. Math. Phys., Vol. 37, No. 5, May 1996
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