Matrices, Bratteli Diagrams and Hopf-Galois Extensions
MMatrices, Bratteli Diagrams and Hopf-Galois Extensions
Ghaliah Alhamzi & Edwin Beggs Department of Mathematics and Statistics, College of Science,Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia College of Science, Swansea University, WalesSeptember 4, 2020
Abstract
We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf-Galoisextensions (quantum principle bundles) for certain abelian groups. The corresponding strong universalconnections are computed. We show that M n ( C ) is a trivial quantum principle bundle for the Hopfalgebra C [ Z n × Z n ]. We conclude with an application relating known calculi on groups to calculi onmatrices. We suppose that P is a unital algebra and that H is a Hopf algebra over C . We write the coproduct of H as ∆( h ) = h (1) ⊗ h (2) using the Sweedler notation. The idea of using a Hopf algebra in place of a groupin a principal bundle was given in [13]. Independently in [4, 5] this was described in terms of differentialcalculi. For the connection between classical Galois theory and Hopf-Galois extensions, see [7, 9, 11]. Definition . For an algebra P and a Hopf algebra H , we say that a right coaction ∆ R : P → P ⊗ H makes P into a right H -comodule algebra if ∆ R is an algebra map, i.e. for p, q ∈ P ( pq ) [0] ⊗ ( pq ) [1] = p [0] q [0] ⊗ p [1] q [1] , ∆ R (1) = 1 ⊗ R = p [0] ⊗ p [1] .This means that the invariants A = P coH = { p ∈ P : ∆ R ( p ) = p ⊗ } is a subalgebra of P . Definition . Let P be a right H -comodule algebra. P is a Hopf-Galois extension of A = P coH if thecanonical map ver (cid:93) : P ⊗ A P → P ⊗ H is a bijection, wherever (cid:93) ( p ⊗ q ) = pq [0] ⊗ q [1] . (2)The idea of Hopf-Galois extension can also be described as a quantum principle bundle for the case ofuniversal differential calculi [1]. We shall return to this later where we give an application to differentialcalculi on the matrices. If P is a Hopf-Galois extension then there are elements of P ⊗ A P mappingto 1 ⊗ h for all h ∈ H . For many practical purposes we seek an element h (1) ⊗ h (2) ∈ P ⊗ P (not ⊗ A )mapping to 1 ⊗ h under the canonical map. This is no longer unique, but we ask whether there is afunction ω (cid:93) : H → P ⊗ P given by ω (cid:93) ( h ) = h (1) ⊗ h (2) such that ω (cid:93) (1) = 1 ⊗ h (1) ⊗ h (2)[0] ⊗ h (2)[1] = h (1)(1) ⊗ h (1)(2) ⊗ h (2) ,h (1)[0] ⊗ h (1)[1] ⊗ h (2) = h (2)(1) ⊗ Sh (1) ⊗ h (2)(2) (3)in which case we say that ω (cid:93) is a strong universal connection (this name was given in [8]). In [2] it wasshown that if H has a normalised integral then ω (cid:93) always exists, using the next result, for which weprovide a framework of the proof as we shall require it later. a r X i v : . [ m a t h . QA ] S e p heorem 1.3. Suppose that H has normalised left-integral (cid:82) and bijective antipode, and that P is aright H -comodule algebra with ver (cid:93) surjective. Then ( P, H, ∆ R ) is a universal quantum principal bundleand admits a strong connection. To show this begin with a linear map h (cid:55)→ h (1) ⊗ h (2) ∈ P ⊗ P sothat (cid:55)→ ⊗ and ver (cid:93) ( h (1) ⊗ h (2) ) = 1 ⊗ h ∈ P ⊗ H , but not necessarily satisfying (3). Now define b : H ⊗ H → C by b ( h, g ) = (cid:82) ( hSg ) , and then a R : P ⊗ H → P and a L : H ⊗ P → P by a R ( p ⊗ h ) = p [0] b ( p [1] , h ) and a L ( h ⊗ p ) = b ( h, S − p [1] ) p [0] . Then we have a strong universal connection ω (cid:93) ( h ) = a L ( h (1) ⊗ h (2)(1) ) ⊗ a R ( h (2)(2) ⊗ h (3) ) (4)In 1972 Ola Bratteli introduced graphs for describing certain classes of C ∗ - algebras in terms of limitsof direct sums of matrices [3]. This is a graph split into levels, and an example of one level in a Brattelidigram is M ⊕ M = M • • M M • • M = M ⊕ M = P (5)which represents the map ( a ) ⊕ (cid:18) b cd e (cid:19) (cid:55)→ ( a ) ⊕ a a b c d e . In Section 2 we will show that such diagrams do not necessarily give Hopf-Galois extensions. However,we can split the level on the diagram into smaller pieces, each of which is a direct sum of Hopf-Galoisextensions. For example, we rewrite (5) as a composition of three stages • • • •• • •• • • and we refer to these as (from right to left) Case 1, Case 2 and Case 3. In each of these cases we shallexhibit a strong universal connection map. For our purposes it is sufficient to consider H = C [ G ], thecomplex valued functions on a finite group G . Then H has basis δ g , the function which is 1 at g ∈ G and zero elsewhere. The Hopf algebra operators are δ x δ y = (cid:40) δ x if x = y x (cid:54) = y and ∆ δ g = (cid:88) x,y : xy = g δ x ⊗ δ y (cid:88) x ∈ G δ x , (cid:15) ( δ x ) = δ x,e , S ( δ x ) = δ x − . The idea of a trivial quantum principle bundle was set down in [10]. In Section 6 we show that M n ( C ) isa trivial quantum principle bundle for the Hopf algebra C [ Z n × Z n ]. Note that in [6] it was shown that M n ( C ) was an algebra factorisation of two copies of C [ Z n ] satisfying a Galois condition. We conclude byan application relating differential calculi on M n ( C ) to differential calculi on C [ Z n × Z n ].The example of differential calculi shows the Hopf-Galois extensions described here have applications,and in general quantum principle bundles are an expanding era of interest in noncommutative geometry.In algebraic topology iterated fibrations (often called towers of fibrations) often occur, e.g. Postnikovsystems or Postnikov towers [12]. Here the mere existence of these iterated fibrations is very useful. A Bratteli Diagram which is not a Hopf-Galois Extension
We shall show that the Bratteli diagram M ⊕ M = M • • M M • • M = M ⊕ M = P (6)does not give an inclusion coming from a Hopf-Galois extension. In terms of matrices this is( a ) ⊕ ( b ) (cid:55)→ ( a ) ⊕ (cid:18) a b (cid:19) P has a linear basis (1) ⊕ ⊕ E ij for i, j ∈ { , } and A has a linear basis (1) ⊕ E and (0) ⊕ E .Note that multiplication by elements of A simply scales each of the given basis vectors in P by a number.Thus to find P ⊗ A P we only have to consider the first element of P to be a basis element. Note that in P ⊗ A P ((1) ⊕ ⊗ ( α ⊕ β ) = ((1) ⊕ ⊕ E ) ⊗ ( α ⊕ β ) = ((1) ⊕ ⊗ ((1) α ⊕ E β )so we have ((1) ⊕ ⊗ ((0) ⊕ E j ) = 0 . Next ((0) ⊕ E i ) ⊗ ( α ⊕ β ) = ((0) ⊕ E i )((1) ⊕ E ) ⊗ ( α ⊕ β ) = ((0) ⊕ E i )( α ⊕ E i β )so we have ((0) ⊕ E i ) ⊗ ((0) ⊕ E j ) = 0 . Next ((0) ⊕ E i ) ⊗ ( α ⊕ β ) = ((0) ⊕ E i )((0) ⊕ E ) ⊗ ( α ⊕ β ) = ((0) ⊕ E i ) ⊗ ((0) ⊕ E β )so we have ((0) ⊕ E i ) ⊗ ((1) ⊕
0) = ((0) ⊕ E i ) ⊗ ((0) ⊕ E j ) = 0 . We have shown that 12 tensor products of the basis of P with itself disappear, making P ⊗ A P P × dim H , which would bea multiple of 5. We consider two cases of subalgebras A of P = M m ( C ) and also a subalgebra of the direct sum of matrixalgebras, and show that they form quantum principle bundles. In the following section we count matricesfrom entry 0 , m arithmetic for the rows and columns of M m ( C ). We followon from the previous section by calculating P ⊗ A P in these three cases. Case 1:
We choose block decompositions of M m ( C ) with rows and columns being divided into intervals of nonzerolength l , l , . . . , l n − where l + l + . . . + l n − = m . Let A be the image of the nonzero diagonal embedding M l ( C ) ⊕ M l ( C ) ⊕ . . . ⊕ M l n − ( C ) −→ M m ( C ) . (7)For row or column j we take ( j ) ∈ Z n to be the block to which row or column j belongs. Thus for l = 2 , l = 1 , m = 3 we have a b c d
00 0 e ∈ A and (0) = 0, (1) = 0, (2) = 1. roposition 3.1. M m ( C ) ⊗ A M m ( C ) where A is the image of (7) is given by the isomorphism of M m ( C ) - M m ( C ) bimodules M m ( C ) ⊗ A M m ( C ) T (cid:47) (cid:47) M m ( C ) ⊗ C [ Z n ] which is given by T ( E ij ⊗ E ab ) = E ib δ ja ⊗ δ ( j ) Proof.
First we get, as E aa ∈ A , E ij ⊗ E ab = E ij ⊗ E aa E ab = E ij E aa ⊗ E ab = δ ja E ij ⊗ E ab . Next if ( k ) = ( j ) then E kj ∈ A so E ij ⊗ E jr = E ik E kj ⊗ E jr = E ik ⊗ E kj E jr = E ik ⊗ E kr . Case 2:
We take the embedding M k ( C ) −→ M m ( C ) where m = nk sending the matrix x to the block diagonalmatrix with all diagonal blocks being x , and let A be the image. E.g. for n = 3 we have x (cid:55)→ x x
00 0 x . (8) Proposition 3.2. M m ( C ) ⊗ A M m ( C ) where A is the image of (8) is given by the isomorphism of M m ( C ) - M m ( C ) bimodules R : M m ( C ) ⊗ A M m ( C ) −→ M m ( C ) ⊗ C [ Z n ] ⊗ C [ Z n ] which is given by R ( E ij ⊗ E ab ) = (cid:40) E ib ⊗ δ r ⊗ δ ( a ) if j = a + kr mod m, for ≤ r < n otherwise . Proof.
Setting y = (cid:88) ≤ r For unital algebra B , consider the replication map rep : B → B ⊕ n = P given byrep( b ) = b ⊕ · · · ⊕ b and call its image A . For s ∈ Z n we label b ,s as the element of B ⊕ n which is b ∈ B is the s th componentand 0 in the other positions. The elements of A are of the form a = (cid:88) s b ,s . Now for c ∈ Bc ,r ⊗ b ,t = c ,r ⊗ a · ,t = c ,r a ⊗ ,t = ( cb ) ,r ⊗ t so P ⊗ A P has an isomorphism u : P ⊗ A P → P ⊗ C ( Z n ), where t ∈ Z n u ( c ,r ⊗ b ,t ) = ( cb ) ,r ⊗ δ t . Block Matrices and Quantum Principle Bundles In this section we will show that the three cases in the previous section are actually examples of quantumprinciple bundles. In the definition of Hopf-Galois extension we only need the case where H = C [ G ]. If G acts on the algebra P on the left by algebra maps, i.e. g (cid:46) ( pq ) = ( g (cid:46) p )( g (cid:46) q ) then we have a right C [ G ] comodule algebra ∆ R : P → P ⊗ H by∆ R ( p ) = (cid:88) g ∈ G g (cid:46) p ⊗ δ g . (9)We set R n to be the group of complex n th roots of unity, and recall that we label matrices from row andcolumn zero. Case 1: We define the group G = g w = ω . . . ω n − : ω ∈ R n ⊂ GL m ( C ) (10)using the blocks of length l , · · · l n − . Now G acts on P = M m ( C ) by algebra maps g ω (cid:46) x = g ω xg − ω . For x purely in the st block for s, t ∈ Z n we have g ω (cid:46) x = ω s − t x so the fixed points of the G action are precisely the block diagonal subalgebra A . The canonical map iscan( E ij ⊗ E ab ) = δ ja (cid:88) ω ∈ R n E ib ω ( a ) − ( b ) ⊗ δ ω where a, b, i, j ∈ Z m . If the canonical map is surjective, then it is automatically injective since the dimension of M m ( C ) ⊗ A M m ( C )is the same as M m ( C ) ⊗ H by the previous section. Now for ξ ∈ R n can (cid:0) ξ ( i ) − ( j ) E ij ⊗ E ji (cid:1) = (cid:88) ω ∈ R n E ii (cid:18) ωξ (cid:19) ( j ) − ( i ) ⊗ δ ω so (cid:88) j ∈ Z m l ( j ) ξ ( i ) − ( j ) can (cid:0) E ij ⊗ E ji (cid:1) = (cid:88) q ∈ Z n ,ω ∈ R n E ii (cid:18) ωξ (cid:19) q − ( i ) ⊗ δ ω and by the formula for the sum of a geometric progression this sum is zero unless ωξ = 1, socan (cid:0) (cid:88) j,i l ( j ) ξ ( i ) − ( j ) E ij ⊗ E ji (cid:1) = n (cid:88) i E ii ⊗ δ ξ = n · I m ⊗ δ ξ (11)since the canonical map is a left P -module map we see that it is surjective, and we have a quantumprinciple bundle. We have proved the following Proposition Proposition 4.1. For ξ (cid:54) = 1 set δ (1) ξ ⊗ δ (2) ξ = 1 n (cid:88) j,i l ( j ) ξ ( i ) − ( j ) E ij ⊗ E ji , δ (1)1 ⊗ δ (2)1 = I m ⊗ I m − (cid:88) ξ (cid:54) =1 δ (1) ξ ⊗ δ (2) ξ . (12) Then (1) ⊗ (2) = I m ⊗ I m and for all η ∈ R n can (cid:0) δ (1) η ⊗ δ (2) η (cid:1) = I m ⊗ δ η . (13) ase 2: For ω ∈ R n and i ∈ Z n , define using blocks of length kg i,ω = . . . . . . . . . ω . . . . . . : : : : : : : ω n − i . . . ω n − . . . where the original 1 (in fact I k ) is in column i (counting from column 0). Note that g i,ω g j,η = η i g i + j,ωη , g − i,ω = ω i g − i, ω . (14)We take the group G of projective matrices G ⊂ P GL n ( C ) consisting of the g i,ω , so we get G ∼ = Z n × R n .Then define an action of G on M m ( C ) by ( i, ω ) (cid:46) x = g i,ω xg − i,ω which is not dependent on a scale factoron the g i,ω . We use F jt for j, t ∈ Z n to denote the identity matrix in the jt block and zero elsewhere.Now ( i, ω ) (cid:46) F jt = ω j − t F j − i,t − i , so ∆ R F jt = (cid:88) s,ω ω j − t F j − s,t − s ⊗ δ ( s,ω ) and the canonical map is can( F ab ⊗ F jt ) = (cid:88) ω ω j − t F a,t − j + b ⊗ δ ( j − b,ω ) and a particular case of this is, by setting b = j − i and t = i + a can( F a,j − i ⊗ F j,i + a ) = (cid:88) ω ω j − i − a F a,a ⊗ δ ( i,ω ) . Now, using the sum of powers of a root of unity,can (cid:0) (cid:88) j ξ i − j ( F a,j − i ⊗ F j,i + a ) (cid:1) = (cid:88) ω,j ω − a ( ωξ ) j − i F a,a ⊗ δ ( i,ω ) = n ξ − a F a,a ⊗ δ ( i,ξ ) so can (cid:0) (cid:88) j,a n ξ i − j + a ( F a,j − i ⊗ F j,i + a ) (cid:1) = (cid:88) a F a,a ⊗ δ ( i,ξ ) = I m ⊗ δ ( i,ξ ) . (15)Thus we have proved the following result Proposition 4.2. If we define, for ( i, ξ ) (cid:54) = (0 , δ (1)( i,ξ ) ⊗ δ (2)( i,ξ ) = 1 n (cid:88) j,a ξ i − j + a F a,j − i ⊗ F j,i + a , δ (1)(0 , ⊗ δ (2)(0 , = I m ⊗ I m − (cid:88) ( i,ξ ) (cid:54) =(0 , δ (1)( i,ξ ) ⊗ δ ( i,ξ )(2) . (16) Then (1) ⊗ (2) = I m ⊗ I m and for all ( j, η ) ∈ Z n × R n can (cid:0) δ (1)( j,η ) ⊗ δ (2)( j,η ) (cid:1) = I m ⊗ δ ( j,η ) . (17) Case 3: We use the group G = Z n acting on P = B ⊕ n by i (cid:46) b ,s = b ,s + i mod n . Nowcan(1 ,t ⊗ ,s ) = 1 ,t . (cid:88) i ∈ Z n ,s + i ⊗ δ i = 1 ,t ⊗ δ t − s (18)so we have the following Proposition. Proposition 4.3. If we define, for i ∈ Z n δ (1) i ⊗ δ (2) i = (cid:88) t ,t ⊗ ,t − i . (19) Then (1) ⊗ (2) = (cid:80) t ,t ⊗ (cid:80) s ,s and can (cid:0) δ (1) i ⊗ δ (2) i (cid:1) = (cid:88) t ,t ⊗ δ i . (20) Strong Universal Connection We find the strong universal connections corresponding to the cases in the previous section, starting withthe back maps h (cid:55)→ h (1) ⊗ h (2) given there. Note these have been defined so that 1 (1) ⊗ (2) = 1 ⊗ 1. Thisuses Theorem 1.3 and normalised integral on C [ G ] for G a finite group (cid:82) f = 1 | G | (cid:88) g ∈ G f ( g ). Proposition 5.1. In Case 1 ω (cid:93) ( δ η ) = 1 n (cid:88) i,j ∈ Z m :( i ) (cid:54) =( j ) l ( j ) η ( i ) − ( j ) E ij ⊗ E ji + 1 n I m ⊗ I m . (21) Proof. By Theorem 1.3 and Proposition 4.1 ω (cid:93) ( δ η ) = (cid:88) ω,ξ a L ( δ ω ⊗ δ (1) ξ ) ⊗ a R ( δ (2) ξ ⊗ δ ηω − ξ − )= (cid:88) ω,ξ (cid:54) =1 a L ( δ ω ⊗ δ (1) ξ ) ⊗ a R ( δ (2) ξ ⊗ δ ηω − ξ − ) + (cid:88) ω a L ( δ ω ⊗ I m ) ⊗ a R ( I m ⊗ δ ηω − ) − (cid:88) ω,ξ (cid:54) =1 a L ( δ ω ⊗ δ (1) ξ ) ⊗ a R ( δ (2) ξ ⊗ δ ηω − )= (cid:88) ω,ξ (cid:54) =1 a L ( δ ω ⊗ δ (1) ξ ) ⊗ a R ( δ (2) ξ ⊗ ( δ ηω − ξ − − δ ηω − )) + (cid:88) ω a L ( δ ω ⊗ I m ) ⊗ a R ( I m ⊗ δ ηω − )= 1 n (cid:88) ω,ξ (cid:54) =1 ,i,j l ( j ) ξ ( i ) − ( j ) a L ( δ ω ⊗ E ij ) ⊗ a R (cid:0) E ji ⊗ ( δ ηω − ξ − − δ ηω − ) (cid:1) + (cid:88) ω a L ( δ ω ⊗ I m ) ⊗ a R ( I m ⊗ δ ηω − ) . Then we calculate a R and a L for both sums by a R ( E ij ⊗ δ ξ ) = 1 n ξ ( j ) − ( i ) E ij and a L ( δ ξ ⊗ E ij ) = 1 n ξ ( i ) − ( j ) E ij and so ω (cid:93) ( δ η ) = 1 n (cid:88) ω,ξ (cid:54) =1 ,i,j l ( j ) ξ ( i ) − ( j ) ω ( i ) − ( j ) E ij ⊗ E ji (cid:0) ( ηω − ξ − ) ( i ) − ( j ) − ( ηω − ) ( i ) − ( j ) (cid:1) + (cid:88) ω,i n E ii ⊗ (cid:88) i n E ii = 1 n (cid:88) ω,ξ (cid:54) =1 ,i,j l ( j ) ξ ( i ) − ( j ) η ( i ) − ( j ) (cid:0) ξ ( j ) − ( i ) − (cid:1) E ij ⊗ E ji + (cid:88) ω n I m ⊗ I m = 1 n (cid:88) ξ (cid:54) =1 ,i,j l ( j ) η ( i ) − ( j ) E ij ⊗ E ji (cid:0) − ξ ( i ) − ( j ) (cid:1) + 1 n I m ⊗ I m . (22)Now for the first term if ( i ) − ( j ) (cid:54) = 0 we get (cid:88) ξ ξ ( i ) − ( j ) = 0 = 1 + (cid:88) ξ (cid:54) =1 ξ ( i ) − ( j ) (23)we split the last equation in (22) into an ( i ) = ( j ) part (the summand vanishes), and ( i ) (cid:54) = ( j ) part wheresumming over ξ and using (23) gives ω (cid:93) ( δ η ) = 1 n (cid:88) i,j :( i ) (cid:54) =( j ) l ( j ) η ( i ) − ( j ) E ij ⊗ E ji (cid:0) n − (cid:1) = 1 n (cid:88) i,j :( i ) (cid:54) =( j ) l ( j ) η ( i ) − ( j ) E ij ⊗ E ji . Proposition 5.2. In Case 2 ω (cid:93) ( δ ( k,η ) ) = 1 n (cid:88) b,s ∈ Z m η b − s F b,s ⊗ F s + k,b + k − n (cid:88) b,i ∈ Z m F b,b ⊗ F i + k + b,i + k + b . (24) roof. By Proposition 4.2 ω (cid:93) ( δ ( k,η ) ) = (cid:88) ( i,p,ω,ξ ) a L (cid:0) δ ( p,ω ) ⊗ δ (1)( i,ξ ) (cid:1) ⊗ a R (cid:0) δ (2)( i,ξ ) ⊗ δ ( k − p − i,ηω − ξ − ) (cid:1) = (cid:88) ( p,ω ) , ( i,ξ ) (cid:54) =(0 , a L (cid:0) δ ( p,ω ) ⊗ δ (1)( i,ξ ) (cid:1) ⊗ a R (cid:0) δ (2)( i,ξ ) ⊗ δ ( k − p − i,ηω − ξ − ) (cid:1) + (cid:88) ( p,ω ) a L (cid:0) δ ( p,ω ) ⊗ δ (1)(0 , (cid:1) ⊗ a R (cid:0) δ (2)(0 , ⊗ δ ( k − p,ηω − ) (cid:1) = (cid:88) ( p,ω ) , ( i,ξ ) (cid:54) =(0 , a L (cid:0) δ ( p,ω ) ⊗ δ (1)( i,ξ ) (cid:1) ⊗ a R (cid:0) δ (2)( i,ξ ) ⊗ ( δ ( k − p − i,ηω − ξ − ) − δ ( k − p,ηω − ) ) (cid:1) + (cid:88) ( p,ω ) a L (cid:0) δ ( p,ω ) ⊗ I m (cid:1) ⊗ a R (cid:0) I m ⊗ δ ( k − p,ηω − ) (cid:1) . Using a R ( F jk ⊗ δ ( r,ξ ) ) = 1 n ξ k − j F j + r,k + r and a L ( δ ( r,ξ ) ⊗ F jk ) = 1 n ξ j − k F j − r,k − r for both terms= 1 n (cid:88) ( p,ω ) , ( i,ξ ) (cid:54) =(0 , ,j,a a L (cid:0) δ ( p,ω ) ⊗ ξ i − j + a F a,j − i (cid:1) ⊗ a R (cid:0) F j,i + a ⊗ ( δ ( k − p − i,ηω − ξ − ) − δ ( k − p,ηω − ) ) (cid:1) = 1 n (cid:88) ( p,ω ) , ( i,ξ ) (cid:54) =(0 , ,j,a (cid:0) ξ i − j + a ω a − j + i F a − p,j − i − p (cid:1) ⊗ (cid:0) ( ηω − ξ − ) i + a − j F j + k − p − i,i + a + k − p − i − ( ηω − ) i + a − j F j + k − p,i + a + k − p (cid:1) = 1 n (cid:88) p,j,a, ( i,ξ ) (cid:54) =(0 , ( ξη ) i + a − j F a − p,j − i − p ⊗ (cid:0) ξ j − i − a F j + k − p − i,a + k − p − F j + k − p,i + a + k − p (cid:1) = 1 n (cid:88) p,j,a,i (cid:54) =0 ,ξ η i + a − j F a − p,j − i − p ⊗ (cid:0) F j + k − p − i,a + k − p − ξ i + a − j F j + k − p,i + a + k − p (cid:1) + 1 n (cid:88) p,j,a,ξ (cid:54) =1 η a − j F a − p,j − p ⊗ F j + k − p,a + k − p (cid:0) − ξ a − j (cid:1) . (25)The first part of equation (25) is= 1 n (cid:88) p,j,a,i (cid:54) =0 η i + a − j F a − p,j − i − p ⊗ (cid:0) F j + k − p − i,a + k − p − δ i + a − j, F j + k − p,i + a + k − p (cid:1) = 1 n (cid:88) p,j,a,i (cid:54) =0 η i + a − j F a − p,j − i − p ⊗ F j + k − p − i,a + k − p − n (cid:88) p,a,i (cid:54) =0 F a − p,a − p ⊗ F i + a + k − p,i + a + k − p = n − n (cid:88) p,s,a η a − s F a − p,s − p ⊗ F s + k − p,a + k − p − n (cid:88) p,a,i (cid:54) =0 F a − p,a − p ⊗ F i + a + k − p,i + a + k − p = n − n (cid:88) b,r η b − r F b,r ⊗ F r + k,b + k − n (cid:88) b,i (cid:54) =0 F b,b ⊗ F i + k + b,i + k + b (26)where we have relabelled s = j − i in the first term. Next relabelling b = a − p and r = s − p . Now let b = a − p and s = j − p in the second part of (25)1 n (cid:88) p,s,b,ξ (cid:54) =1 η b − s F b,s ⊗ F s + k,b + k (cid:0) − ξ b − s (cid:1) = 1 n (cid:88) s,b,ξ (cid:54) =1 η b − s F b,s ⊗ F s + k,b + k (cid:0) − ξ b − s (cid:1) = 1 n (cid:88) s,b η b − s F b,s ⊗ F s + k,b + k (cid:0) − δ b,s (cid:1) , (27)since (cid:88) ξ (cid:54) =1 (cid:0) − ξ b − s (cid:1) = n (1 − δ b,s ), and adding equations (26) and (27) gives the results. Proposition 5.3. In Case 3 ω (cid:93) ( δ i ) = (cid:88) s ∈ Z n ,s ⊗ ,s − i . (28) roof. From Theorem 1.3 and (19) ω (cid:93) ( δ i ) = (cid:88) j,k a L ( δ j ⊗ δ k (1) ) ⊗ a R ( δ k (2) ⊗ δ i − j − k ) = (cid:88) j,k,t a L ( δ j ⊗ ,t ) ⊗ a R (1 ,t − k ⊗ δ i − j − k )as ∆ R b ,t = (cid:88) s b t + s ⊗ δ s , then a L ( δ j ⊗ ,t ) = (cid:88) s b ( δ j ⊗ δ − s )1 ,t + s = (cid:88) s ( (cid:90) δ j δ s )1 ,t + s = 1 n ,t + j ,a R (1 ,t ⊗ δ j ) = (cid:88) s ,t + s b ( δ s ⊗ δ j ) = (cid:88) s ( (cid:90) δ s δ − j )1 ,t + s = 1 n ,t − j so ω (cid:93) ( δ i ) = 1 n (cid:88) j,k,t ,t + j ⊗ ,t − i + j = 1 n (cid:88) j,t ,t + j ⊗ ,t − i + j (29)and setting s = t + j gives the answer. We first recall that if Φ , Ψ are maps from a coalgebra (in our case H ) to an algebra then so is the convolution product (cid:12) defined by Φ (cid:12) Ψ = · (Φ ⊗ Ψ)∆and that Φ is convolution-invertible when there is an inverse Φ − such that Φ (cid:12) Φ − = Φ − (cid:12) Φ = 1 .(cid:15) . Proposition 6.1. [1] Let P be a right H -comodule algebra equipped with a convolution-invertible right-comodule map Φ : H → P with Φ(1) = 1 . Then P is a quantum principal bundle over A = P H . We callit a trivial bundle with trivialisation Φ . We show that the algebra M n ( C ) is a trivial Hopf-Galois extension of the group Z n × Z n , referringto the construction in Section 4 Case 2. We use the notation R n to be the multiplicative group of the n th roots of unity, generated by x = e πi/n with x n = 1. Proposition 6.2. The construction of Section 4 Case 2 gives a trivial quantum principle bundle withalgebra P = M n ( C ) and Hopf algebra H = C [ G ] for G = Z n × R n .Proof. We start by defining Φ( δ ( s,ω ) ) = (cid:88) ij Φ s,ω,i,j F ij to be a right comodule map, meaning the followingquantities are equal ∆ R Φ( δ ( s,ω ) ) = (cid:88) i,j,r,ξ Φ s,ω,i,j ξ i − j F i − r,j − r ⊗ δ ( r,ξ ) (Φ ⊗ id)∆( δ ( s,ω ) ) = (cid:88) t,η Φ( δ ( t,η ) ) ⊗ δ ( s − t, ωη ) = (cid:88) t,η,p,q Φ t,η,p,q F pq ⊗ δ ( s − t, ωη ) . (30)Using these we can show that there are β ij withΦ s,ω,i,j = ω j − i β i − s,j − s and for Φ(1) = 1 we need (cid:88) i β ii = 1 n . The inverse Ψ of Φ in Proposition 6.1 can be shown toobey ∆ R Ψ( h ) = Ψ( h (2) ) ⊗ Sh (1) . Writing Ψ as Ψ( δ ( s,ω ) ) = (cid:88) ij Ψ s,ω,i,j F ij we can show that Ψ s,ω,i,j = ω i − j γ i + s,j + s . The equations for convolution inverse reduce to η j − i β i − t,j − t ( ωη ) j − q γ j + s − t,q + s − t F iq = (cid:40) I n if s = 0 and ω = 10 otherwise o (cid:88) t,η,j ω j − q η q − i β i − t,j − t γ j + s − t,q + s − t = (cid:40) δ iq if s = 0 and ω = 1otherwisethe sum over η gives zero unless q = i , so we are left with n (cid:88) j ω j − i (cid:88) t β i − t,j − t γ j + s − t,i + s − t = (cid:40) s = 0 and ω = 10 otherwise . Now we use the result that evaluation gives an isomorphism from C [ ω ] 5] T. Brzezi´nski and S. Majid, Quantum group gauge theory on classical spaces, Phys. Lett. B.298, 339-343 (1993)[6] T. Brzezi´nski and S. Majid, Quantum geometry of algebra factorisations and coalgebra bundles,Commun. Math. Phys. 213, 491-521 (2000)[7] S.U. Chase and M.E. Sweedler, Hopf algebras and Galois theory, Springer, Berlin, Heidelberg,(1969)[8] P.M. Hajac, Strong connections on quantum principal bundles. Commun. Math. Phys. 182,579-617 (1996)[9] H.F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, IndianaUni. Math. Journal, 30, 675-692 (1981)[10] S. Majid, Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras,in eds. H.-D. Doebner, V.K. Dobrev and A.G. Ushveridze, Generalized Symmetries in Physics.World Sci. 13-41 (1994)[11] S. Montgomery, Hopf Galois theory: A survey, Geometry and Topology Monographs, 16, 367-400 (2009)[12] M.M. Postnikov, Determination of the homology groups of a space by means of the homotopyinvariants, Doklady Akademii Nauk SSSR. 76, 359-362 (1951)[13] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math 72,167-195 (1990)5] T. Brzezi´nski and S. Majid, Quantum group gauge theory on classical spaces, Phys. Lett. B.298, 339-343 (1993)[6] T. Brzezi´nski and S. Majid, Quantum geometry of algebra factorisations and coalgebra bundles,Commun. Math. Phys. 213, 491-521 (2000)[7] S.U. Chase and M.E. Sweedler, Hopf algebras and Galois theory, Springer, Berlin, Heidelberg,(1969)[8] P.M. Hajac, Strong connections on quantum principal bundles. Commun. Math. Phys. 182,579-617 (1996)[9] H.F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, IndianaUni. Math. Journal, 30, 675-692 (1981)[10] S. Majid, Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras,in eds. H.-D. Doebner, V.K. Dobrev and A.G. Ushveridze, Generalized Symmetries in Physics.World Sci. 13-41 (1994)[11] S. Montgomery, Hopf Galois theory: A survey, Geometry and Topology Monographs, 16, 367-400 (2009)[12] M.M. Postnikov, Determination of the homology groups of a space by means of the homotopyinvariants, Doklady Akademii Nauk SSSR. 76, 359-362 (1951)[13] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math 72,167-195 (1990)