On semisimple Hopf algebras with few representations of dimension greater than one

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(1)REVISTA DE LA ´ MATEMATICA ´ UNION ARGENTINA Volumen 51, N´ umero 2, 2010, P´ aginas 91–105. ON SEMISIMPLE HOPF ALGEBRAS WITH FEW REPRESENTATIONS OF DIMENSION GREATER THAN ONE V. A. ARTAMONOV Abstract. In the paper we consider semisimple Hopf algebras H with the following property: irreducible H-modules of the same dimension > 1 are isomorphic. Suppose that there exists an irreducible H-module M of dimension > 1 such that its endomorphism ring is a Hopf ideal in H. Then M is the unique irreducible H-module of dimension > 1.. Introduction Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either char k = 0 or char k > dim H. Semisimplicity of H means that H is a semisimple left H-module. In that case H has finite dimension. Throughout the paper we shall keep to notations from [M]. For example H ∗ stands for the dual Hopf algebra with the natural pairing h−, −i : H ∗ ⊗ H → k. The algebra H is a left and right H ∗ -module algebra with respect to the left and right actions f ⇀ x, x ↼ f of f ∈ H ∗ on x ∈ H, defined as follows, [M, Example 4.1.10]: if X ∆(x) = x(1) ⊗ x(2) (1) x. then. f ⇀x=. X. x(1) hf, x(2) i,. x↼f =. x. X. hf, x(1) ix(2) .. x. Denote by G = G(H ∗ ) the group of group-like elements in H ∗ . Elements of G are algebra homomorphisms H → k. Hence H as a k-algebra has a semisimple direct decomposition  H = (⊕g∈G keg ) ⊕ ⊕nj=1 Mat(dj , k) , (2). where {eg , g ∈ G} is a system of central orthogonal idempotents in H corresponding to one-dimensional direct summands. If h ∈ H and g ∈ G then heg = eg h = hg, hieg . The counit ε : H → k has the form ( δg,1 , ε(x) = 0,. x = eg , x ∈ Mat(di , k).. Research partially supported by grants RFBR 09-01-00058, 09-01-90416-Ucr-f-a.. 91. (3).

(2) 92. V. A. ARTAMONOV. Each one-dimensional H-module Eg = kug related to g ∈ G has a base ug and hug = hg, hiug ,. h ∈ H.. (4). 1 < d1 < d2 < · · · < dn .. (5). In this paper we consider the case when It just means that irreducible H-modules of the same dimension > 1 are isomorphic. The main result of the paper is Theorem 0.1. Let H be a semisimple Hopf algebra with decomposition (2), n > 1, such that (5) holds. Suppose that at least one single matrix constituent is a Hopf ideal in H. Then n = 1. Throughout the paper we shall use the following notations. By E (i) we shall (i) denote the identity matrix from Mat(di , k) and by Ers ∈ Mat(di , k) we shall denote corresponding matrix unit. The author thanks Prof. N. Andruskiewitsch and Prof. S. Natale for useful and very helpful discussions and comments. The author is grateful to the referee for many useful remarks and suggestions. Note that the similar classes of Hopf algebras were considered from another point of view in [N2, §3.4] and in [Ma], [N1], [N2], [T], [TY]. 1. The category of modules Let H be a semisimple Hopf algebra with direct sum decomposition (2) such that (5) is satisfied. The category H M of left H-modules is a monoidal category with respect to tensor products of modules. Since H is semisimple the abelian category is completely defined up to an isomorphism by its Grothendieck group K0 (H) = K0 (H M). Recall that the Abelian K-group K0 (H) is a free additive Abelian group with a base consisting of ismorphism clases of irreducible left H-modules Eg , g ∈ G, and modules Mi , i = 1, . . . , n, of dimension di > 1 corresponding to matrix constituents in (2). As we have already mentioned one-dimensional Hmodules Eg correspond to one-dimensional simple direct summands of H such that (4) is satisfied and elements g ∈ G. If M, N are left H-modules then M ⊗ N is also a left H-module under the action X h(x ⊗ y) = h(1) x ⊗ h(2) y, h ∈ H, x ∈ M, y ∈ N. (6) h. Tensor multiplication induces a structure of an associative ring on K0 (H). The next fact is well known.. Proposition 1.1. Let M be an irreducible H-module and Eg a one-dimensional H-module. Then M ⊗ Eg and Eg ⊗ M are irreducible H-modules. In particular if f, g ∈ G then Ef ⊗Eg ≃ Ef g . Under assumption (5) we have M ⊗Eg ≃ Eg ⊗M .  The module M ⊗ Eg is identified with M in which elements h ∈ H act as g ⇀ h. An isomorphism M ≃ M ⊗ Eg means that there exists an invertible linear operator ΦM (g) in M such that ΦM (g)hx = (g ⇀ h)ΦM (g)x for all h ∈ H and for all Rev. Un. Mat. Argentina, Vol 51-2.

(3) 93. SEMISIMPLE HOPF ALGEBRAS. x ∈ M . Note that we can identify the endomorphism algebra End M with an ideal in H. Namely, if M ≃ Mi then End M ≃ Mat(di , k). If g ∈ G then the action g ⇀ is an endomorphism of H. Because of the assumption (5) the ideal End M is −1 stable under the action g ⇀. Thus g ⇀ h = ΦM (g)hΦ in H. Note that we M (g)  −1 can always assume that ΦM (1) = 1 and ΦM g = ΦM (g)−1 for all g ∈ G. We denote by S the antipode of H.. Proposition 1.2. Let M be an irreducible H-module of dimension > 1. The map ΦM : G → PGL(M ) is a projective representation of the group G if M is an irreducible H-module. Moreover [ΦM (g), S (ΦM (v))] = 1 in PGL(M ) for all g, v ∈ G.. Proof. Since g ⇀ (v ⇀ h) = gv ⇀ h, conjugations by ΦM (g)ΦM (v) and by ΦM (gv) coincide in GL(M ). Hence ΦM (gv)−1 ΦM (g)ΦM (v) is a scalar matrix and therefore ΦM is a projective representation of G in M . By the assumption (5) the algebra End M is an ideal which is stable not only under the action g ⇀ where g ∈ G but also under the action ↼ g and under the antipode S since S is an involution of the algebra H. Similarly Eg ⊗ M can be identified with the vector space M in which h ∈ H acts as   h ↼ g = S (S(g) ⇀ S(h)) = S ΦM g −1 S(h)ΦM (g)  = S (ΦM (g)) hS ΦM g −1 . In order to prove the last statement it is necessary to recall that the actions ↼, ⇀ commute.  Proposition 1.3. Let M, N be irreducible H-modules and g ∈ G. Then HomH (M ⊗ N, Eg ) ≃ HomH (M ⊗ N, Eε ). Proof. Let φ ∈ Hom(M ⊗ N, Eg ). By Proposition 1.1 there is an isomorphism of H-modules ω : M ⊗ N → M ⊗ N ⊗ Eg−1 . Hence φ induces by Proposition 1.1 a homomorphism of H-modules M ⊗N. ω −1. / M ⊗ N ⊗ Eg−1. φ⊗1. / Eg ⊗ Eg−1. / Eε .. Denote by φ˜ this composition. It is easy to see that the correspondence φ 7→ φ˜ is an isomorphism between HomH (M ⊗ N, Eg ) and HomH (M ⊗ N, Eε ).  Corollary 1.4. If M, N are irreducible H-modules of dimensions > 1, then the multiplicity of each one-dimensional H-module Eg , g ∈ G, in M ⊗ N does not depend upon g.  If M is an irreducible left H-module then H ∗ = Homk (M, k) is again an irreducible left H-module. So we have Proposition 1.5. Let Mi , be the unique irreducible left H-module of dimension di > 1. Then Mi∗ ≃ Mi . In particular there exists a non-degenerate bilinear function hx, yi on Mi such that hhx, yi = hx, S(h)yi for all x, y ∈ Mi and for any h ∈ H.  Rev. Un. Mat. Argentina, Vol 51-2.

(4) 94. V. A. ARTAMONOV. If e∗ = (e1 , . . . , edi ) is a base of Mi as a vector space then we get an identification of End Mi with the full matrix algebra Mat(di , k) which is an ideal in H. Denote by Ui the square matrix of size di whose (α, β)th entry uαβ = heα , eβ i. It means that Ui is the Gram matrix of the bilinear function hx, yi. Theorem 1.6. The bilinear function hx, yi is (skew-)symmetric. If x ∈ Mat(di , k) from decomposition (2), then S(x) = Ui t xUi−1 . Using a canonical form of a Gram matrix of a (skew-)symmetric bilinear function we can always choose a base in Mi such that Ui is either an identity matrix in the symmetric case or the matrix   0 −E Ui = , E 0 in skew-symmetric (hyperbolic) case where E is the identity matrix of size both cases Ui−1 = ±t Ui .. di 2 .. In. Proof. Without loss of generality we can assume that h, S(h) ∈ Mat(di , k) have entries hζβ , rγλ , respectively. Then X X hheα , eβ i = heζ , eβ ihαζ = uζβ hαζ ; heα , S(h)eβ i =. ζ. ζ. X. X. rβγ heα , eγ i =. γ. rβγ uαγ .. γ. The equality hhx, yi = hx, S(h)yi means that t Ui · t h = S(h) t Ui . Finally since the antipode S has order 2, the matrix Ui and the form hx, yi are  (skew)-symmetric [LR]. So we can replace t Ui by Ui . We now study some properties of irreducible H-modules. The first one is a well-known, see [DNR, Lemma 7.5.10, p. 322]. Recall that Mi is an irreducible H-module of dimension di , 1 ≤ i ≤ n. Lemma 1.7. If 1 ≤ i, j ≤ n, then dim HomH (Mi ⊗ Mj , Eε ) = δij .. . Set A = ⊕g∈G Eg . Proposition 1.8. There is a direct sum decomposition  Mi ⊗ Mj = δij A ⊕ ⊕nt=1 mtij Mt ,. mtij = dimk HomH (Mi ⊗ Mj , Mt ) > 0.. (7). Hence di dj = δij |G| +. n X. mtij dt .. (8). t=1. In particular |G| 6 d21 .. Proof. Apply Lemma 1.7, Corollary 1.4 and compare dimensions of modules in (7). This gives (7), (8). We prove the last claim. Each coefficient mipq estimated by (8) as 1 0 6 mipq 6 dp dq − δpq |G|. di Rev. Un. Mat. Argentina, Vol 51-2.

(5) 95. SEMISIMPLE HOPF ALGEBRAS. In particular if p = q then 0 6 d2p − |G|.. . Proposition 1.5 implies that A ⊗ Ms = Ms ⊗ A = |G|Ms . Using the associativity conditions in the Grothendieck ring we obtain Theorem 1.9. The multiplicities mtij satisfy the equation (8) and the equations msij = mijs ,. δij δls |G| +. n X. mtij mlts = δjs δli |G| +. t=1. for all i, j, s, l = 1, . . . , n. In particular n X. δij δls |G| +. n X. mtjs mlit. t=1. msij. =. mijs. =. mjti mlts = δjs δli |G| +. t=1. mjsi n X. and mjst mlit .. t=1. . Corollary 1.10. If i, j, p = 1, . . . , n, then mpij 6 dmin(i,j,p) . Proof. Since mpij = mijp = mjpi we can assume that p > i, j. Then by (8) we get mpij dp 6 di dj or mpij 6. di dj 6 min(di , dj ) = dmin(i,j) = dmin(i,j,p) . dp  2. Special matrices. Consider special elements Ti =. di X. (i). (i). Eαβ ⊗ Eαβ ,. Ri =. di 1 X (i) (i) Eαβ ⊗ Eβα , di α,β=1. α,β=1. (9). Di = (1 ⊗ S) Ri. in Mat(di , k)⊗2 . These elements will be used in the rest of the paper. Direct calculations prove the first equality in Proposition 2.1. Let F, H ∈ Mat(di , k). Then     F ⊗ E (i) Ti E (i) ⊗ H = E (i) ⊗ t F Ti E (i) ⊗ t H , (F ⊗ H)Ri = Ri (H ⊗ F ). Here E (i) stands for the identity matrix of the size di . Moreover (1 ⊗ S) Ti = (S ⊗ 1) Ti ,. Di = (1 ⊗ S) Ri = (S ⊗ 1) Ri .. In particular for F, H as above (F ⊗ 1) · Dt · (1 ⊗ S(H)) = (1 ⊗ S(F )) · Dt · (H ⊗ 1) . Rev. Un. Mat. Argentina, Vol 51-2.

(6) 96. V. A. ARTAMONOV. Proof. The second equality was proved in [ACh, Proposition 1.1]. In order to prove the nest two equalities we shall use the first equality, (skew-) symmetry of Ui and the property Ui−1 = ±Ui . Hence     (1 ⊗ S)Ti = di E (i) ⊗ Ui Ri E (i) ⊗ Ui−1     = di Ui ⊗ E (i) Ri Ui−1 ⊗ E (i) = (S ⊗ 1)Ti ;    1  (i) (1 ⊗ S)Ri = E ⊗ Ui Ti E (i) ⊗ Ui−1 dt    1  = Ui ⊗ E (i) Ti Ui−1 ⊗ E (i) = (S ⊗ 1)Ri = Di . dt In order to prove the last equality apply (1 ⊗ S) to (9) and use previous equalities.  3. From module category to Hopf algebra In this section we shall apply previous results to a presentation of an explicit form of comultiplication and antipode in H. Let H have the direct sum decomposition (2) such that (5) holds. Denote by πg , g ∈ G, and by πi , 1 6 i 6 n, the projections of H onto Eg and onto Mat(di , k), respectively. Then the isomorphism Ef ⊗ Eg ≃ Ef g in Proposition 1.1 means that (πf ⊗ πg )∆(h) = hf g, hi(ef ⊗ eg ); (πg ⊗ πi )∆(h) = eg ⊗ (πi (h) ↼ g) ,. (10). (πi ⊗ πg )∆(h) = (g ⇀ πi (h)) ⊗ eg . Applying Proposition 1.8 we obtain (πi ⊗ πj )∆(πg (h)) = 0,. 1 6 i 6= j 6 n,. g ∈ G.. (11). Using (10), (11) we see that ∆(eg ) =. X. ef ⊗ ef −1 g +. f ∈G. X. Dg,i ,. (12). i=1,...,n. where {Dg,i , g ∈ G, 1 6 i 6 n}, is a system of orthogonal idempotents in Mat(di , k)⊗2 . Moreover Dg,i ∈ Mat(di , k) ⊗ Mat(di , k) ≃ Mat(d2i , k) ≃ Endk (Mi ⊗ Mi ). Lemma 3.1. Let g ∈ G, and Di from (9), i = 1, . . . , n. Then the H-module Dg,i (Mi ⊗ Mi ) is isomorphic to Eg and  Dg,i = 1 ⊗ g −1 ⇀ Di . Proof. Let h ∈ H and x, y ∈ Mi . Applying (3), (4), (6) and (12) we obtain hDg,i (x ⊗ y) = ∆(h)Dg,i (x ⊗ y) = ∆(h)∆(eg ) (x ⊗ y) = ∆(heg ) (x ⊗ y) = hg, hiDg,i (x ⊗ y) . Thus (4) implies that Dg,i (Mi ⊗ Mi ) ≃ Eg . Rev. Un. Mat. Argentina, Vol 51-2.

(7) 97. SEMISIMPLE HOPF ALGEBRAS. By (7)  and previous considerations it suffices to show that the element (1⊗ g −1 ⇀ Di is an idempotent and   ∆(h) 1 ⊗ g −1 ⇀ Di = hg, hi 1 ⊗ g −1 ⇀ Di .. Direct calculations show that     1 X (i) (i) (i) (i) D2i = 2 Eαβ Eγξ ⊗ S Eβα S Eξγ di α,β,γ,ξ   X 1 (i) (i) (i) (i) = 2 Eαβ Eγξ ⊗ S Eξγ Eβα di α,β,γ,ξ   X 1 (i) (i) = 2 di Eαξ ⊗ S Eξα = Di . di α,β=γ,ξ. Since g ⇀ is an algebra automorphism of H by Proposition 2.1 we have X     ∆(h) 1 ⊗ g −1 ⇀ S Ri = h(1) ⊗ h(2) g −1 ⇀ S Ri h. =. X. h(1) ⊗ g. h. =. X. −1.    ⇀ g ⇀ h(2) S Ri.    h(1) ⊗ g −1 ⇀ S Ri S g ⇀ h(2). h. = 1⊗ g. −1. " #   X   ⇀ S h1) ⊗ 1 Ri 1 ⊗ S g ⇀ h(2) h. = 1 ⊗ g −1 Finally X h. =. ".   ⇀ S Ri. 1⊗. X. h(1) S g ⇀ h(2). h. !# .  X  h(1) S g ⇀ h(2) = h(1) S h(2) hg, h(3) i h. X. h(1) S h(2) hg, h(3) i =. h. = hg,. . X h. X h.  ε h(1) h(2) i = hg, hi.  ε h(1) hg, h(2) i . Applying (10) for any x ∈ Mat(dr , k) we get ∆(x) =. X. [(g ⇀ x) ⊗ eg + eg ⊗ (x ↼ g)] +. g∈G. where ∆rij (x) ∈ Mat(di , k) ⊗ Mat(dj , k). Note that ∆rij (x) = (πi ⊗ πj )∆(x). It means that. n X. ∆rij (x),. (13). i,j=1. ∆rij : Mat(dr , k) → Mat(di , k) ⊗ Mat(dj , k) Rev. Un. Mat. Argentina, Vol 51-2.

(8) 98. V. A. ARTAMONOV. is an algebra homomorphism not necessarily preserving the unit element. Lemma 3.2. Let x ∈ Mat(di , k), g ∈ G and τ is the twist τ (a ⊗ b) = b ⊗ a for all a ∈ Mat(dq , k), b ∈ Mat(dp , k). Then τ (14) ∆ipq (x) = (S ⊗ S) ∆iqp (S(x)) , µ (1 ⊗ S) ∆ipq (x) = µ (S ⊗ 1) ∆ipq (x) = 0,. S (eg ) = eg−1 ,. µ (S ⊗ 1) Dg,i = µ (1 ⊗ S) Dg,i = δg,1 E (i) .. (15). Here µ : H ⊗ H → H is the multiplication map. Proof. The antipode S is an algebra and coalgebra involution. Hence each ideal Mat(di , k) in H is S-stable and hence (14) holds. The equalities (15) were proved in [A].  Note that Dg,i and ∆tij (Et ) are in fact projections of Mi ⊗ Mj onto Eg by Lemma 3.1 and onto mtij Mt , respectively. Hence Dg,i and ∆tij (Et ) are matrices in Mat(di dj , k) whose ranks are equal to dim Eg = 1 and to mtij dim Mt , respectively. In particular, (14) implies mipq = miqp for all i, p, q. Since ∆ is an algebra homomorphism for all i, j, p, q, r, s = 1, . . . , m and x ∈ Mat(dr , x), y ∈ Mat(ds , k) we have Dg,i · Dh,j = δgh δij Dg,i , Dg,i · ∆rpq (x) = ∆rpq (x) · ∆g,i = 0,. (16). ∆rij (x) · ∆spq (y) = δip δjq δrs ∆rij (xy). 4. Coassociativity conditions In this section we shall consider coassociativity conditions of comultiplication ∆ from (12), (13). Multiplying the equalities from [A, §2, Proposition 2.1] by ∆rij (E (r) )⊗ 1 from the left, by E (j) ⊗ ∆rij (E (r) ) from the right and the last equation by ∆tjp (E (t) ) ⊗ E (q) from the left and by E (j) ⊗ ∆rpq (E (r) ) from the right, respectively, we obtain using (16) that for any i, j, p, q = 1, . . . , n and all x ∈ Mat(di , k) the following five groups of identities are satisfied. The first group: [1 ⊗ (g ⇀ )] Dt,i = Dtg−1 ,i ; [( ↼ g) ⊗ 1] Dt,i = Dg−1 t,i ;. (17). [1 ⊗ ( ↼ g)] Dt,i = [(g ⇀ ) ⊗ 1] Dt,i . The second group: [( ↼ g) ⊗ 1] ∆ipq (x) = ∆ipq (x ↼ g) ; [1 ⊗ (g ⇀ )] ∆ipq (x) = ∆ipq (g ⇀ x) ; [1 ⊗ ( ↼ Rev. Un. Mat. Argentina, Vol 51-2. g)] ∆ipq (x). = [(g ⇀ ) ⊗. 1] ∆ipq (x).. (18).

(9) SEMISIMPLE HOPF ALGEBRAS. 99. The third group:    1 ⊗ ∆ipj Di = ∆jip ⊗ 1 Dj ;.   (Di ⊗ E (i) ) [x ⊗ Di ] = [Di ⊗ x] E (i) ⊗ Di .. The fourth group: i h    ∆rij (E (r) ) ⊗ E (j) [x ⊗ Dj ] = ∆rij ⊗ 1 ∆irj (x) (E (i) ⊗ Dj );  h i   Dj ⊗ E (i) (1 ⊗ ∆rji )∆ijr (x) = [Dj ⊗ x] E (j) ⊗ ∆rji (E (r) ) .. (19). (20). The last fifth group: h i   ∆tjp (E (t) ) ⊗ E (q) 1 ⊗ ∆rpq ∆ijr (x) i   h = ∆tjp ⊗ 1 ∆itq (x) E (j) ⊗ ∆rpq (E (r) ) .. (21). Proposition 4.1. Let X ∆irj (x) = Aτ (x) ⊗ A′τ (x),. (22). Also the equations (16) and (15) are satisfied. We shall apply the identities (17) — (21) to the study of the comultiplication and the antipode of H.. τ. ∆ijr (x) =. X. Bτ′ (x) ⊗ Bτ (x),. τ. where Aτ , Bτ ∈ Mat(dr , k) and A′τ , Bτ′ ∈ Mat(dj , k). Then the equalities (20) are equivalent to  X h i   ∆rij (Aτ (x)) E (i) ⊗ S (A′τ (x)) ; ∆rij E (r) x ⊗ E (j) = τ.     Xh i E (j) ⊗ x ∆rij E (r) = S (Bτ′ (x)) ⊗ E (i) ∆rji (Bτ (x)) τ. for all x ∈ Mat(di , k).. Proof. In the notations (22) using Proposition 2.1 we obtain in the right hand part of the first equation in (20)   r   ∆ij ⊗ 1 ∆irj (x) E (i) ⊗ Dj # " i h   X = ∆rij (Aτ (x)) E (i) ⊗ S (A′τ (x)) ⊗ E (j) E (i) ⊗ Dj . τ. By Proposition 2.1 the first equation in (20) has the form    i  Xh (j) (j) ∆rij E (r) x ⊗ Eαβ ⊗ S Eαβ α,β. =. X. τ,α,β.     (j) (j) ∆rij (Aτ (x)) E (i) ⊗ S (A′τ (x)) Eαβ ⊗ S Eβα Rev. Un. Mat. Argentina, Vol 51-2.

(10) 100. V. A. ARTAMONOV. which is equivalent to    X (j) ∆rij E (r) x ⊗ Eαβ. =. X. τ,α,β. α,β. h i (j) ∆rij (Aτ (x)) E (i) ⊗ S (A′τ (x)) Eαβ. for any α, β. Identifying α = β and taking a sum over all α we obtain the first required equality. The second equality is obtained in a similar way from the second  equality in (20). 5. Hopf ideals The next statement follows from the definition of a Hopf ideal and Theorem 1.9. Proposition 5.1. For a matrix constituent Mat(dt , k) in H from direct decomposition (2) the following are equivalent: (i) Mat(dt , k) is a Hopf ideal in H; (ii) ∆tij = 0 for i, j 6= t; (iii) mtij = 0 for i, j 6= t; (iv) if i, j 6= t then there are H-module isomorphisms   Mi ⊗ Mj ≃ δij (⊕g∈G Eg ) ⊕ ⊕l6=t mlij Ml Mj ⊗ Mt ≃ dj Mt ≃ Mt ⊗ Mj ,   Mt ⊗ Mt ≃ (⊕g∈G Eg ) ⊕ ⊕nl=1 mltt Ml .. . Proposition 5.2. If 1 6 i, j, p 6 n then the following are equivalent: (i) ∆ipj 6= 0, (ii) ∆pji 6= 0, (iii) ∆jip 6= 0. Proof. Let ∆ipj 6= 0. Then mipj 6= 0 by Proposition 5.1. But mipj = mpji = mjip .  Hence ∆pji , ∆jip 6= 0. Proposition 5.3. Assume that Mat(dt , k) is a Hopf ideal in H. If q 6= t, then ∆ttq (E (t) ) = E (t) ⊗ E (q) ,. ∆tqt (E (t) ) = E (q) ⊗ E (t) ,. and q < t. Thus t = n. Proof. Suppose that there exists an index j 6= t such that ∆jtq 6= 0. By Proposition 5.2 we have ∆tqj 6= 0 which contradicts Proposition 5.1. Hence    X j  ∆tq E (j) = ∆ttq E (t) . E (t) ⊗ E (q) = j. ∆tqt. The case is similar. The equalities mean that mttq = mtqt = dq . By Corollary 1.10 that dq = mttq 6 dmin(t,q) and therefore q 6 min(t, q) 6 t. Rev. Un. Mat. Argentina, Vol 51-2.

(11) 101. SEMISIMPLE HOPF ALGEBRAS. P Suppose that t < n. By (8) we get dt dt+1 = i mit,t+1 di . But mit,t+1 = mtt+1,i 6= 0 implies i = t since Mat(dt , k) is a Hopf ideal. Hence dt dt+1 = mtt,t+1 dt and mtt,t+1 = dt+1 . On the other from previous considerations it follows that mtt,t+1 = 0.  Consider in each matrix ideal Mat(di , k) the bilinear form hx, yi such that (i) (i) hEαβ , Eγτ i = δαγ δβτ . Then the dual space Mat(di , k)∗ can be identified with itself via this bilinear function. Proposition 5.4. Let Mat(dn , k) be a Hopf ideal in H. Suppose that x ∈ Mat(dn , k) and q < n. Then ∆nnq (x) =. dq  X. i,j=1. Proof. Let.  (q) (q) Eij ⇀ x ⊗ Eij , dq X. ∆nnq (x) =. ∆nqn (x) =. dq X. i,j=1. (q). aij (x) ⊗ Eij ,.   (q) (q) Eij ⊗ x ↼ Eij .. aij (x) ∈ Mat(dn , k).. i,j=1 (q). Then Ers ⇀ x =. Pdq. i,j=1. (q). (q). aij (x)hErs , Eij i = ars (x). The second case is similar. . Proposition 5.5. Let i < n and x ∈ Mat(di , k). Then dn h   i di X (n) (n) Eqp ↼ S t x ⊗ S Epq dn p,q=1    = di ↼ S t x ⊗ 1 Dn .. ∆inn (x) =. Proof. The first identity in (19) with p = j = n has the form  1 ⊗ ∆inn Di = (∆nin ⊗ 1) Dn .. (23). Applying Proposition 2.1 and Proposition 5.4 in the left and in the right hands sides of (23) we get the following   di di h  i   X  1 1 X (i) (i) (i) (i) 1 ⊗ ∆inn  Eab ⊗ S Eba  = Eab ⊗ ∆inn S Eba ; di di a,b=1 a,b=1 " # # " d dn n     1 X 1 X n (n) (n) n (n) (n) (∆in ⊗ 1) = E ⊗ S Eqp ∆ (E ) ⊗ S Eqp dn p,q=1 pq dn p,q=1 in pq     X 1 (i) (n) (i) (n) = Ers ⊗ Epq ↼ Ers ⊗ S Eqp . dn p,q=1,...,dn r,s=1,...,di. Hence (23) is equivalent to dn  h  i    di X (i) (i) (n) (n) = . ∆inn S Eba Epq ↼ Eab ⊗ S Eqp dn p,q=1 Rev. Un. Mat. Argentina, Vol 51-2.

(12) 102. V. A. ARTAMONOV. If x =.   (i) ∈ Mat(di , k) then x S E a,b ab ba  i   di X h (n) (n) (n) Eγλ ↼ xab Eab ⊗ S Eλγ ∆inn (x) = dn a,b,γλ   i di X h (n) (n) = Eγλ ↼ S t x ⊗ S Eλγ dn. P. a,b,γλ. . Proposition 5.6. The equations (20) with r = i = n, j < n hold in H if and only if E (j) ⇀ x = x = x ↼ E (j) for all x ∈ Mat(dn , k).  Proof. Note first that ∆nnq E (n) = E (n) ⊗ E (q) by Proposition 5.3. Using Propositions 5.4, 4.1 and 2.1 we can rewite the first equation in (20) with r = i = n, j < n in the form  h  i X (j) (j) x ⊗ E (j) = ∆nnj Eαβ ⇀ x E (n) ⊗ S Eαβ α,β.    X  (j) (j) (j) (j) = Eγλ ∗ Eαβ ⇀ x ⊗ Eγλ S Eαβ α,β,γ,λ. =.  i  X h (j) (j) (j) (j) ⇀ x ⊗ Eγλ Eαβ Eγλ ∗ S Eαβ. (24). α,β,γ,λ.   i X h (j) (j) (j) = Eγα ∗ S Eαβ ⇀ x ⊗ Eγβ α,β,γ. i   Xh (j) = δγβ E (j) ⇀ x ⊗ Eγβ = E (j) ⇀ x ⊗ E (j) β,γ. for any x ∈ Mat(dn , k). Thus (24) means that E (q) ⇀ x = x for all x. Similarly the second equality x ↼ E (q) is equivalent to the second one from  (20). Proof of Theorem 0.1. Suppose that n > 1. Then Mat(dn , k) is the unique component which is a Hopf ideal, Proposition 5.3. Let i < n. By Proposition 5.5 and Proposition 5.6 we have ∆inn (E (i) ) =. dn i h    di X (n) (n) Eqp ⊗ S t E (i) ⇀ S Epq dn p,q=1. dn dn h  i   di X di X (n) (i) (n) (n) (n) = Eqp ⊗ E ⇀ S Epq = Eqp ⊗ S Epq = di Dn . dn p,q=1 dn p,q=1. Suppose now that n > 2. Then e1 · E (1) = 0 in H and therefore   Dn · ∆1nn E (1) = 0.  On the other hand Dn · ∆1nn E (1) = d1 D2n = d1 Dn 6= 0, a contradiction. Rev. Un. Mat. Argentina, Vol 51-2. .

(13) SEMISIMPLE HOPF ALGEBRAS. 103. 6. The case n = 1 with ∆111 = 0 We shall remind some results in the case n = 1. For simplicity we put E (1) = (1) E, Eij = Eij , d1 = d, Φ1 = Φ, U = U1 . Theorem 6.1. [A, ACh] Suppose that n = 1 and char k is not a divisor of 2d2 . Let G = G(H ∗ ) be the group of group-like elements of the dual Hopf algebra H ∗ . Assume that ∆111 = 0 in (13), or equivalently |G| = d2 . Taking an isomorphic copy of H we can assume that the comultiplication in H has the form: ∆(eg ) =. X. eh ⊗ eh−1 g +. h∈G. ∆(x) =. X. d  1 X Eij ⊗ g −1 ⇀ S (Eji ) ; d i,j=1. [(g ⇀ x) ⊗ eg + eg ⊗ (x ↼ g)] ;. (25). g∈G. ε(eg ) = δg,1 ,. ε(x) = 0;. S(x) = U t xU −1. for all x ∈ Mat(d, k), where U ∈ GL(d, k) is a (skew-)symmetric matrix. There exists a faithful irreducible projective representation Φ : G → PGL(d, k) such that g ⇀ x = Φ(g)xΦ(g)−1 ,. x ↼ g = S (Φ(g)) xS (Φ(g))−1. (26). for any x ∈ Mat(d, k) and [Φ(g), S (Φ(v))] = 1. (27). in PGL(d, k) for all g, v ∈ G. It is shown in [F, J2] that G ≃ A × A is a direct product of two copies of an Abelian group A of order d. The next theorem shows the converse. If we start with a irreducible projective representation of a group G from Theorem 6.1 we can obtain Hopf algebra structure. Theorem 6.2. [ACh] Suppose that G is Abelian group of order d2 with direct decomposition G ≃ A × A for some Abelian group A of order d. The group G has a faithful irreducible projective representation Φ of degree d. There exists a (skew-) symmetric matrix U ∈ GL(d, k) such that (26), (27) holds where S(x) = U t xU −1 for any x ∈ Mat(d, k). Then an algebra H with direct decomposition (2) admits Hopf algebra structure defined in (26), (25). Each element g ∈ G can be identified with an element g ∈ H ∗ such that if X a= αh eh + x ∈ H, αh ∈ k, x ∈ Mat(d, k), h inG. then hg, ai = αg . The identification is an isomorphism of the group G and the group G(H ∗ ) of group-like elements in H ∗ . If d > 2, then these Hopf algebras are neither commutative nor cocommutative.. Note that semisimple Hopf algebras of dimension 2p2 for an odd prime p in the same case as in Theorem 6.1 were classified in [Ma] in other terms. All Hopf algebras of dimension 2p2 for an odd prime p were classified in [N0]. All Hopf algebra of dimension 2p2 for an odd prime p were classified in [HN]. In Theorem 6.1, 6.2 Rev. Un. Mat. Argentina, Vol 51-2.

(14) 104. V. A. ARTAMONOV. we expand these results of an arbitrary n using the language of faithful projective representations of the group G = G(H ∗ ). Note that by [J1], [J2] all projective faithful irreducible representations of the group G ≃ A × A as in Theorems 6.1, 6.2 have dimension d and obtained one from the other by a group automorphism of G. Any of these representations is monomial and it is induced by a one-dimensional representation of A.. References [A]. Artamonov V.A. On semisimple finite dimensional Hopf algebras, Math. Sbornik, 198(2007), N 9, 3–28. 15, 4, 6.1. [ACh] Artamonov V.A., Chubarov I.A., Properties of some semisimple Hopf algebras, Contemp. Math. 483, Algebras, representations and applications, A conference in honour of Ivan Shestakov’s 60th birthday, August 26–September 1, 2007, Maresias, Brazil. Edited by: Vyacheslav Futorny, Victor Kac, Iryna Kashuba and E. Zelmanov, Amer. Math. Soc., 2009, 23–36. 2, 6.1, 6.2 [DNR] D˘ asc˘ alescu S., N˘ ast˘ asescu C., Raianu S ¸ ., Hopf algebras, an introduction, Pure and Applied Mathematics, 235: a series of Monographs and Textbooks, Marcel Dekker, Inc, NY, Basel, 2001. 6 [F]. ¨ Frucht R., Uber die Darstekkung endlicher abeischer Gruppen durch Kollineationen, J. reine angew. Math., 166(1932), 16–29. 6. [HN]. Hilgemann M., Ng S. H., Hopf algebras of dimension 2p2 , J. London Math. Soc., 80(2009), 295–310. 6. [J]. Jmud E., On isomorphic irreducible projective representations of finite groups, Zapiski Math. Otdeleniya fiziko-matemt. faculteta Kharkov university and Kharkov math. Soc., 26(1960), N 4, 333–371.. [J1]. Jmud E., Simplectic geometry on finite abelian groups, Mat. Sbornik, 86 (1971), 9–34. 6. [J2]. Jmud E., Simplectic geometry and projective representations of finite abelian groups, Mat. Sbornik, 87 (1972), 3–17. 6, 6. [LR]. Larson R., Radford D., Finite-dimensional cosemisimple Hopf algebras in characteristic zero are semisimple. J. Algebra, 117(1988), 267–289. 6. [Ma]. Masuoka A., Some further classification results on semisimple Hopf algebras, Commun. Algebra, 24 (1996), 307–329. 5, 6. [M]. Montgomery S.: Hopf Algebras and Their Actions on Rings, Regional Conf. Ser. Math. Amer. Math. Soc., Providence RI, 1993. (document). [N0]. Natale S., On semisimple Hopf algebras of dimension pq 2 , J. Algebra, 221(1999), 242–278. 6. [N1]. Natale S., On group theoretical algebras and exact factorizations of finite groups, J.Algebra, 270 (2003), 190–211. 5. Rev. Un. Mat. Argentina, Vol 51-2.

(15) SEMISIMPLE HOPF ALGEBRAS. 105. [N2]. Natale S., Semisolvability of semisimple Hopf algebras of low dimension, Memoirs of AMS vol. 186, 2007. 5. [T]. Tambara D., Representations of tensor categories with fusion rules of self-duality for finite abelian groups, Israel J.Math., 118(2000), 29-60. 5. [TY]. Tambara D., Yamagami S., Tensor categories with fusion rules of self-duality for finite abelian groups, J.Algebra 209(1998), 692-707. 5. V. A. Artamonov Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University artamon@mech.math.msu.su. Recibido: 5 de marzo de 2010 Aceptado: 30 de mqyo de 2010. Rev. Un. Mat. Argentina, Vol 51-2.

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