Projective Quantum Spaces

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1 DAMTP/94-81 Projective Quantum Spaces U. MEYER D.A.M.T.P., University of Cambridge September 1994 arxiv:hep-th/ v1 6 Oct 1994 Abstract. Associated to the standard SU (n) R-matrices, we introduce uantum spheres, projective uantum spaces CP n 1, and uantum Grassmann manifolds G k (C n ). These S 2n 1 algebras are shown to be homogeneous spaces of standard uantum groups and are also uantum principle bundles in the sense of T. Brzeziński and S. Majid [1]. 1 Introduction This paper gives a non-commutative generalisation of projective and Grassmann manifolds in the framework of uantum groups. Quantum groups are Hopf algebras with and additional structure (uasitriangularity or its dual version) and can be employed to construct a very powerful type of non-commutative geometry. For an introduction to uantum groups and their numerous other applications see the review article [4] or the textbooks [7,2]. We use the notation and standard constructions from [4]. The standard SU (n) R-matrices were given in [9] as R n = 1/n i E ii E ii + E ii E jj + ( 1 ) E ij E ji, R. (1) i j j>i Here E ij denote the elementary matrices with components (E ij ) a b = δ a iδ j b. These R-matrices are invertible solutions of the n-dimensional uantum Yang-Baxter euation and are of real type, i.e. obey R ab n cd = R n dc ba. Furthermore, let ε a 1...a n = ε a 1...a n = ( ) l(σ), where l(σ) is the length of the permutation σ(1,..., n) = (a 1,..., a n ), and define uantum determinants as det,n = ν 1 n ε a1...a n t a 1 b1...t an b n ε bn...b 1. The normalisation factor is ν n = ε a1...a n ε an...a 1. In terms of these data, the uantum groups SU (n) were defined in [9] as the uotients A(R n )/ det,n=1. This algebra has well-known antipode S [9, Theorem 4] 1

2 and -structure t a b = Stb a such that tgt = g where t = St and g a b = 2(a n) 1 δ a b. We also define uantum groups U (n) as the uotients of the algebras A(R n ) by the relations det,n det,n = 1. For n = 1, one recovers the undeformed algebra of complex polynomial functions on U(1), i.e. U (1) is the associative C-algebra generated by elements z and z such that z = z 1. These uantum groups act covariantly on the algebra of uantum vectors V (R n ) with generators v i, i {1,...,n} and relations v i v j = λ 1 n R ij n kl vl v k. Here the normalisation factors are λ n = 1 1/n. The coaction is given by v i v k t i k. Similarly, the algebra of uantum covectors V (R n ) has generators x i, relations x i x j = x l x k λ 1 n R n kl ij, and coaction x i x k t k i. In terms of these algebras, we define C n = V (R n) V (R n ) as their braided tensor product (See [5] for an introduction to braided geometry). For the braiding between V (R n ) and V (R n ) we do not take the braiding induced by the coaction by SU (n) as given in [6], but Ψ(x i v j ) = v k lj x l λ n R n ik, which differs from the SU (n)-braiding by a factor λ n, i.e. is the braiding induced by a suitable dilatonic extension of SU (n). We euip this algebra with the -structure (1 x a ) = v a 1 and define the uantum norm N,n = v a g b ax b where g b bc a = λ n R n ca. The matrix R is defined as ((R t 2 ) 1 ) t 2, where t 2 denotes transposition in the second tensor component. Since R n is of real type, g is symmetric and the uantum norm is real. It is known from [6, Theorem 3.6] that for any invertible solution R of the uantum Yang-Baxter euation the uantum norm is also central. For the R-matrices (1), this general construction reproduces the -metric introduced above: one finds R n = 1/n 1 i E ii E ii + E ii E jj ( 1 ) 2(j i) E ij E ji, i j j>i and hence g a b = 2(a n) 1 δ b a. We will use the algebras C n to define uantum spheres and complex projective uantum spaces. These algebras are then recovered as base spaces of uantum principle bundles with U (n) and SU (n) as structure groups. The appropriate definition of a uantum principle bundle was given by T. Brzeziński and S. Majid in [1, Definition 4.1]: an algebra P is a uantum principle bundle with structure uantum group A and base B if (i) A is a Hopf algebra, (ii) P is a left A-comodule algebra with coaction β : P A P, and (iii) B = P A = {u P : β(u) = 1 u} is the subalgebra of A-invariant elements of P. Additionally, there is a freeness and an exactness condition. However, if B is a uantum homogeneous space, i.e. if there is a Hopf algebra surjection π : P A between Hopf algebras A and P and if additionally ker π (ker π B P) (2) 2

3 then the freeness and exactness condition are automatically satisfied [1, Lemma 5.2] and the uantum homogeneous space B = P A is a uantum principle bundle. In this case the map π is called exact. 2 Quantum spheres S 2n 1 and feature as base spaces of various uan- Quantum spheres are uotients of C n tum principle bundles. DEFINITION 2.1 Quantum spheres S 2n 1 are defined as the uotients C n / N,n=1. Due to the relation tgt = g, the uantum spheres S 2n 1 are covariantly coacted upon by both U (n) and SU (n). For the special cases where n euals 1 or 2, one finds: EXAMPLE 2.2 S 1 = U (1) and S 3 = SU (2) as -algebras. Proof. The first isomorphism is trivial and the second one is easily established by rescaling of the generators of S 3. This isomorphism only holds if one introduces the normalisation factor λ 2 in the definition of the braiding in C 2. PROPOSITION 2.3 There are n -algebra injections ı m : S 2n 1 U (n). Proof. In this proof, there is no summation over the variable m. Define the maps ı m by x a t m mm a. By inspection of (1) one finds Rn cd = 1 1/n δ m c δm d and hence the relations in A(R n ) imply 1 1/n t m at m b = Rn mm cd t c at d b = t m dt m crn cd ab, i.e. the generators t m a, a {1,..., n} obey the relations of V (R n ). Since R n is of real type, the generators t a m, a {1,...,n} then obey the V (R n )-relations. Next note that the relations in A(R n ) can be written as t a b R n cb de td g = tc d R n da gb t b e and hence with Rn am mg = 1 1/n δ a m δm g one finds tm i t j dj m = R n if t f b R n mb cm tc dj d = 1 1/n R n if t f m tm d. Thus the relations between t m i and t j m reproduce the braiding between V (R n ) and V (R n ) given above. This is the place where one needs the normalisation factor λ n in the definition of the braiding. Finally note, that t a m gb a tm b = λ nt a bi m R n ia tm b = t m i t i m = 1. Hence the maps ı m are -algebra injections. For the construction of uantum principle bundles over these uantum spheres we need the following lemma: 3

4 LEMMA 2.4 There are surjective -Hopf algebra morphisms α n : U (n) U (n 1) γ n : SU (n) SU (n 1). Proof. By inspection of (1) one finds that for i, j > 1, R ij n kl = R (i 1)(j 1) n 1 (k 1)(l 1) and also that Rn 11 ij = 1 1/n δ 1 iδ 1 j. Thus the elements t i j; i, j {2,..., n} of A(R n ) obey the relations of A(R n 1 ) and the uotient maps by the ideal generated by t 1 1 = 1 and t1 i = t i 1 = 0, i {2,..., n} are surjections A(R n) A(R n 1 ). These uotient maps also have the property det,n det,n 1 and thus descend to surjective -Hopf algebra maps U (n) U (n 1) and SU (n) SU (n 1), which we denote by α n and γ n, respectively. THEOREM 2.5 The Hopf algebra U (n) is a uantum principle bundle with base S 2n 1 and structure uantum group U (n 1): S 2n 1 = U (n) U(n 1) (3) The left covariant coaction of U (n 1) on U (n) is given by (α n id) in terms of the morphism α n from lemma 2.4. Proof. The subalgebra of U (n 1)-invariant elements of U (n) is generated by t 1 i, i {1,..., n} and their conjugates. Proposition 2.3 then implies (3). Since ker α n = ker α n S 2n 1 the maps α n are exact in the sense of (2) and the uantum homogeneous space is a uantum principle bundle. One can repeat this construction with the maps γ n : SU (n) SU (n) from lemma 2.4 to establish that SU (n) is also a uantum principle bundle over a uantum sphere: THEOREM 2.6 The Hopf algebra SU (n) is a uantum principle bundle with base S 2n 1 and structure uantum group SU (n 1): S 2n 1 = SU (n) SU(n 1) The left covariant coaction of SU (n 1) on SU (n) is given by (γ n id). 3 Complex projective uantum spaces We now turn our attention to complex projective spaces which are defined as U (1)-covariant subalgebras of uantum spheres S 2n 1. For this purpose note 4

5 that by virtue of lemma 2.4, the right covariant -coaction β : C n C n U (n) descends to a covariant U (1)-coaction defined as (id (α 2 α 3... α n )) β. This coaction is given explicitly by x i x i z and also descends to a covariant -coaction on S 2n 1. DEFINITION 3.1 Complex projective uantum space CP n is defined as CP n 1 = (S 2n 1 ) U(1), the subalgebra of U (1)-invariant elements of the uantum sphere S 2n 1. Since uantum projective spaces are merely subalgebras of uantum spheres S 2n 1, they are also covariantly coacted upon by SU (n) and U (n). Note also that their relations are independent of the normalisation factor λ n which we introduced in the definition of C n, i.e. at this stage one could work with the standard SU (n)- braiding. For the construction of uantum principle bundles on these complex projective uantum spaces we need the following lemma: LEMMA 3.2 There are a surjective -Hopf algebra morphisms δ n : SU (n) U (n 1). Proof. The proof is similar to the proof of lemma 2.4. The maps δ n are defined like β n with the only difference that δ n (t 1 1) = (det,n 1 ). THEOREM 3.3 The Hopf algebra SU (n) is a uantum principle bundle with base CP n 1 and structure uantum group U (n 1): CP n 1 = SU (n) U(n 1). The coaction of U (n 1) on SU (n) is given by (δ n id). Proof. In this case, invariant generators are products t i 1t 1 j, i, j {1,..., n} which generate the subalgebra of U (1)-invariant elements of S 2n 1, i.e. CP n 1. The kernel of δ n is generated by t i 1 and t 1 i, where i {2,...n} and the kernel of the restricted map ker δ n CP n 1 has generators t 1 i t j 1 where either i or j is larger that 1. The exactness of δ n is then established by the trivial observation t i 1 = (t i 1t 1 j t j 1) and the relation t 1 i = t 1 i t j 1g k j t1 k = λ dj n R n if t f 1t 1 d gk j t1 k = 5

6 (t f 1t 1 d λ dj n R n if gk j t1 k 2.3 and the fact that for i 1, ). Here we used the formulae from the proof of proposition is non-zero only if d 1. R dj n if As a corollary of theorem 2.5 one finds with definition 3.1 that complex projective uantum spaces CP n can also be regarded as homogeneous uantum spaces of U (n): THEOREM 3.4 The Hopf algebra U (n) is a uantum principle bundle with base CP n and structure uantum group U (1) U (n 1): CP n = U (n) U(1) U(n 1) The covariant coaction by U (1) U (n 1) is given explicitly in the next section, where we generalise this construction to uantum Grassmann manifolds. We now restrict our attention to the case of n = 2. In this case, the algebra SU (2) U(1) is a deformation of the two-sphere and was already discussed by P. Podle`s in [8], where it was denoted by S 2. As a corollary of theorem 3.3 we hence find: is isomorphic to Podle`s s uan- COROLLARY 3.5 Projective uantum space CP 1 tum sphere S 2. The n = 2 case is also in the undeformed case uite remarkable since all spaces involved are spheres: SU(2) = S 3, U(1) = S 1 and CP 1 = S 2. The corresponding principle bundle is the celebrated Hopf fibration of the three sphere. In the uantum case one finds with example 2.2 and theorem 3.3: COROLLARY 3.6 ( Hopf fibration of the uantum 3-sphere) There is a uantum principle bundle S 2 = (S 3 )S1 (4) The uantum homogeneous space (4) was already discussed in [11], although not as uantum principle bundles, an appropriate definition of which was only given a few years later. 4 Quantum Grassmann manifolds The construction encountered in theorem 3.4 can be generalised to give a definition of uantum Grassmann manifolds as uantum principle bundles. For this purpose, we prove the following lemma: 6

7 LEMMA 4.1 There are surjective -Hopf algebra morphisms α k,n : U (n) U (k) U (n k). Proof. Note that for i, j < n, R ij n kl = R ij n 1 kl, i.e. the elements ti j ; i, j {1,..., n 1} of A(R n ) obey the relations of A(R n 1 ). Similar to lemma 2.4 one can thus define surjective -Hopf algebra morphisms α n : U (n) U (n 1) by dividing by t n n = 1 (no summation!) and tn i = ti n = 0, i {1,...,n 1}. The maps ((α k+1 α k+2... α n ) (α n k+1... α n)) are then the reuired -Hopf algebra maps α k,n. The covariant coaction by U (1) U (n 1) on U (n) from theorem 3.4 is given by (α 1,n id). However, the real merit of this lemma is that it leads to a straightforward generalisation from uantum projective spaces to uantum Grassmann manifolds: DEFINITION 4.2 Quantum Grassmann manifolds are defined as the uantum principle bundles G k (C n ) = U (n) U(k) U(n k) where the covariant coaction is given by (α k,n id). This definition includes the case of projective uantum spaces as G 1 (C n ) = CPn. A useful set of generators of G k (C n ) is given by the elements (ε c1...c k t c 1 a1...t c k ak ) (ε d1...d k t d 1 b1...t d k bk ), where c i, d i {1,...,k} and a i, b i {1,..., n}. An earlier paper by E. Taft and J. Towber [10] (see also [3]) presented a related approach to uantum Grassmann manifolds and studied the algebra generated by elements ε d1...d k t d 1 b1...t d k bk, where the t a b are the generators of A(R n) and not of the uotient U (n). Thus a uotient of a subalgebra of the algebra from [10, Proposition 3.4] is the uantum Grassmann manifold G k (C n ). 7

8 References 1. T. Brzezinski and S. Majid, Comm. Math. Phys. 157, 591 (1993). 2. V. Chari and A. Pressley, A Guide to Quantum Groups, C.U.P., V. Lakshmibai and N. Reshetikhin, C. R. Acad. Sci. Paris I 313 (3), 121 (1991) 4. S. Majid, Int. J. Mod. Phys. A 5, 1 (1990). 5. S. Majid, in: M.-L. Ge and H. J. de Vega, Proc. 5th Nankai Workshop, World Scientific, S. Majid, J. Math. Phys. 34, 1176 (1993). 7. S. Majid, Foundations of Quantum Group Theory, C.U.P. (to be published) 8. P. Podles, Lett. Math. Phys. 14, 193 (1987). 9. N. Reshetikhin, L. Takhtadzhyan, and L. Faddeev, Leningrad Math. J. 1 (1), 193 (1990). 10. E. Taft and J. Towber, J. Algebra 142, 1 (1991). 11. S. L. Woronowicz, Publ. RIMS 23, 117 (1987). 8