DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS GIVEN BY THE CANONICAL NAKAYAMA AUTOMORPHISM

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1 DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS GIVEN BY THE CANONICAL NAKAYAMA AUTOMORPHISM M. GRAÑA, J.A. GUCCIONE, AND J.J. GUCCIONE Abstract. Every fte dmesoal Hopf algebra s a Frobeus algebra, wth Frobeus homomorphsm gve by a tegral. The Nakayama automorphsm determed by t yelds a decomposto wth degrees a cyclc group. For a famly of poted Hopf algebras, we determe ecessary ad suffcet codtos for ths decomposto to be strogly graded. 1. Itroducto Let k be a feld, A a fte dmesoal k-algebra ad DA the dual space Hom k (A, k), edowed wth the usual A-bmodule structure. Recall that A s sad to be a Frobeus algebra f there exsts a lear form ϕ: A k, such that the map A DA, defed by x xϕ, s a left A-module somorphsm. Ths lear form ϕ: A k s called a Frobeus homomorphsm. It s well kow that ths s equvalet to say that the map x ϕx, from A to DA, s a somorphsm of rght A-modules. From ths t follows easly that there exsts a automorphsm ρ of A, called the Nakayama automorphsm of A wth respect to ϕ, such that xϕ = ϕρ(x), for all x A. It s easy to check that a lear form ϕ: A k s aother Frobeus homomorphsm f ad oly f there exsts a vertble elemet x A, such that ϕ = xϕ. It s also easy to check that the Nakayama automorphsm of A wth respect to ϕ s the map gve by a ρ(x) 1 ρ(a)ρ(x). Let A be a Frobeus k-algebra, ϕ: A k a Frobeus homomorphsm ad ρ : A A the Nakayama automorphsm of A wth respect to ϕ. Defto 1.1. We say that ρ has order m N ad we wrte ord ρ = m, f ρ m = d A ad ρ r d A, for all r < m. Let G be a group. Recall that a G-graded algebra s a k-algebra A together wth a decomposto A = A g of A as a drect sum of subvector spaces, such that A g A g A gg for all g, g G. Whe A g A g = A gg for all g, g G, the gradg s called strog, ad the algebra s sad to be strogly graded. Assume that ρ has fte order ad that k has a prmtve ord ρ -th root of uty ω. For N, let C be the group of -th roots of uty k. Sce the polyomal X ordρ 1 has dstct roots ω (0 < ord ρ ), the algebra A becomes a C ordρ -graded algebra (1.1) A = A ω 0 A ω ordρ 1, where A z = {a A : ρ(a) = za}. As t s well kow, every fte dmesoal Hopf algebra H s Frobeus, beg a Frobeus homomorphsm ay ozero rght tegral ϕ H. Let t be a ozero rght tegral of H. Let α H be the modular elemet of H, defed by at = α(a)t 2000 Mathematcs Subject Classfcato. Prmary 16W30; Secodary 16W50. Ths work was partally supported by CONICET, PICT , UBA X294. 1

2 2 M. GRAÑA, J.A. GUCCIONE, AND J.J. GUCCIONE (otce that ths s the verse of the modular elemet α cosdered [R1]). For x H, let r x : H H be defed by r x (b) = bx = x(b (1) )b (2). The, as follows from [R1, Theorem 3(a)], we have ρ(b) = α 1 (b (1) )S 2 (b (2) ),.e. ρ = S 2 r α 1. Sce α s a group-lke elemet, S 2 r α 1 = r α 1 S 2 ad therefore (1.2) ρ l = S 2l r α l,.e. ρ l (h) = α l (h (1) )S 2l (h (2) ) = α l (S(h (1) ))S 2l (h (2) ), for all l Z. Now, α has fte order ad, by the Radford formula for S 4 (see [R2] or [S, Theorem 3.8]), the atpode S has fte order wth respect to composto. Thus, the automorphsm ρ has fte order, whch mples that fte dmesoal Hopf algebras are examples of the stuato cosdered above. Notce that by (1.2), f ρ l = d, the α l = ε ad the S 2l = d. The coverse s obvous. So, the order of ρ s the lcm of those of α ad S 2. I partcular, the umber of terms the decomposto assocated wth S 2 dvdes that the oe assocated wth ρ. Also, from (1.2) we get that ρ = S 2 f ad oly f H s umodular. The ma am of the preset work s to determe codtos for decomposto (1.1) to be strogly graded. Besdes the fact that the theory for algebras whch are strogly graded over a group s well developed (see for stace [M]), our terest o ths problem orgally came from the homologcal results [GG]. The decomposto usg S 2 stead of ρ was cosdered [RS]. We show below that f S 2 d, the ths decomposto s ot strogly graded. O the other had, as show [RS], uder sutable assumptos ts homogeeous compoets are equdmesoal. It s a terestg problem to kow whether a smlar thg happes wth the decomposto assocated wth ρ. For stace, all the lftgs of Quatum Lear Spaces have equdmesoal decompostos, as show Remark The umodular case Let H be a fte dmesoal Hopf algebra wth atpode S. I ths bref secto we frst show that the decomposto of H assocated wth S 2 s ot strogly graded, uless S 2 = d (ths apples partcular to decomposto (1.1) whe H s umodular ad ord ρ > 1). We fsh by gvg a characterzato of umodular Hopf algebras terms of decomposto (1.1). Lemma 2.1. Let H be a fte dmesoal Hopf algebra. Suppose H = H g s a graduato over a group. Assume there exsts g G such that ε(h g ) = 0. The the decomposto s ot strogly graded. Proof. Suppose the decomposto s strogly graded. The there are elemets a H g ad b H g 1 such that 1 = a b. The 1 = ε(1) = ε(a )ε(b ), a cotradcto. Corollary 2.2. Assume that S 2 d ad that H = z k H z, where H z = {h H : S 2 (h) = zh}. The ths decomposto s ot strogly graded. Proof. Sce ε S 2 = ε, the ε(h z ) = 0 for all z 1.

3 DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS 3 Let ow ϕ r ad Γ l, such that ϕ, Γ = 1, ad let α: H k be the H H modular map assocated wth t = S(Γ). Let ρ be the Nakayama automorphsm assocated wth ϕ. Assume that k has a root of uty ω of order ord ρ. We cosder the decomposto assocated wth ρ, as (1.1) (2.1) H = H ω 0 H ω ordρ 1. Corollary 2.3. If H s umodular ad S 2 d, the the decomposto (2.1) s ot strogly graded. Proposto 2.4. If h H ω, the α(h) = ω ɛ(h). Proof. By [R1, Proposto 1(e)], or the proof of [S, Proposto 3.6], ϕ, t = 1. The ɛ(h) = ɛ(h) ϕ, t = ϕ, th = ϕ, ρ(h)t = ϕ, ω ht = ω α(h). So, α(h) = ω ɛ(h), as we wat. Corollary 2.5. H s umodular f ad oly f H ω ker(ɛ), for all > 0 Proof. ): For h H ω, we have ɛ(h) = α(h) = ω ɛ(h) ad so ɛ(h) = 0, sce ω 1. ): For h H ω wth > 0, we have α(h) = ω ɛ(h) = 0 = ɛ(h) ad, for h H ω 0, we also have α(h) = ω 0 ɛ(h) = ɛ(h). 3. Bosozatos of Nchols algebras of dagoal type Let G be a fte abela group, g = g 1,..., g G a sequece of elemets G ad χ = χ 1,..., χ Ĝ a sequece of characters of G. Let V be the vector space wth bass {x 1,..., x } ad let c be the bradg gve by c(x x j ) = q j x j x, where q j = χ j (g ). We cosder the Nchols algebra R = B(V ) geerated by (V, c). We gve here oe of ts possble equvalet deftos. Let T c (V ) be the tesor algebra geerated by V, edowed wth the uque braded Hopf algebra structure such that the elemets x are prmtve ad whose bradg exteds c. The, R s obtaed as a colmt of algebras R = lm R, where R 0 = T c (V ) ad R +1 s the quotet of R by the deal geerated by ts homogeeous prmtve elemets wth degree 2. See [AG, AS] for alteratve deftos ad ma propertes of Nchols algebras. Assume that R s fte-dmesoal ad let t 0 R be a ozero homogeeous elemet of greatest degree. Let H = H(g, χ) = R#kG be the bosozato of R (ths s a alteratve presetato for the algebras cosdered by Nchols [N]). We have: (g ) = g g, (x ) = g x + x 1, S(g) = g 1, S 2 (g) = g, S(x ) = g 1 x, S 2 (x ) = g 1 x g = q 1 x. The elemet t 0 g s a o zero rght tegral H. Let α be the modular elemet assocated wth t. Sce, for all, the degree of x t 0 g s bgger tha the degree of t 0, we have that x t 0 g = 0, ad so α(x ) = 0. Moreover, α G s determed by gt 0 = α(g)t 0 g. Thus, ρ(g) = α(g 1 )g ad ρ(x ) = α(g 1 )q 1 x.

4 4 M. GRAÑA, J.A. GUCCIONE, AND J.J. GUCCIONE Ths mples that the ozero moomals x 1 x l g are a set of egevectors for ρ (whch geerate H as a k-vector space). I partcular, ρ s dagoalzable, whece k has a prmtve ord ρ -th root of uty. Cosder the subgroups L 1 = q 11,..., q, α(g) ad L 2 = α(g), of k. Sce ρ(x g 1 ) = q 1 x g 1 ad ρ(g) = α(g), the group L 1 s the set of egevalues of ρ. As the troducto, we decompose H = ω L 1 H ω, where H ω = {h H : ρ(h) = ωh}. Proposto 3.1. The followg are equvalet: (1) ω L 1 H ω s strogly graded, (2) L 1 = L 2, (3) Each H ω cotas a elemet G. (4) H s a crossed product H 1 kl 1. Proof. It s clear that (2) (3) ad (4) (1). The proof of (3) (4) s stadard. We sketch t for the readers coveece. For each ω L 1, pck g ω H ω G. Defe ρ : L 1 H 1 H 1 ad f : L 1 L 1 H 1 by ω a = ρ(ω, a) = g ω ag 1 ω ad f(ω, ω ) = g ω g ω g 1 ωω. Let H 1 f ρ kl 1 be the algebra wth uderlyg vector space H 1 kl 1 ad wth multplcato (a ω)(b ω ) = a(ω b)f(ω, ω ) ωω. It s easy to see that the map Ψ : H 1 f ρ kl 1 H gve by Ψ(a ω) = ag ω s a somorphsm of algebras. I partcular, H 1 f ρ kl 1 s assocatve wth ut 1 1, ad so ρ s a weak acto ad f s a ormal cocycle that satsfes the twsted module codto. We ow prove that (1) (2). Notce that H s also N 0 -graded by deg(g) = 0 for all g G, ad deg(x ) = 1. Call H = N H ths decomposto. Sce each H ω s spaed by elemets whch are homogeeous wth respect to the prevous decomposto, we have: H = Hω, where Hω = H ω H. 0, ω L 1 So, f ω L 1 H ω s strogly graded, the each H ω must cota ozero elemets H 0. Sce H 0 ω L 2 H ω, we must have L 1 = L 2. Quatum Lear Spaces. If the sequece of characters χ satsfes χ (g ) 1, χ (g j )χ j (g ) = 1 for j, the H(g, χ) s the Quatum Lear Space wth geerators G ad x 1,..., x, subject to the followg relatos: gx = χ (g)x g, x x j = q j x j x, x m = 0, where m = ord(q ). For these sort of algebras t s possble to gve a explct formula for ρ. I fact, the elemet t = x m1 1 1 x m 1 g s a rght tegral H. Usg ths tegral, t s easy to check that α(g) = χ m1 1 1 (g) χ m 1 (g).

5 DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS 5 I partcular, α(g ) = q m1 1 1 ρ(g) = α(g 1 )g ad ρ(x ) = α(g 1 ρ(x r1 1 xr g) = q m 1 1 <j )q 1 Proposto 3.1 apples to ths famly of algebras.. A straghtforward computato, usg that x, shows that q (1 mj)r (1 m)rj j α(g 1 )x r1 1 xr g. Example 3.2. Let k be a feld of characterstc 2 ad let G = {1, g}. Set g = g ad χ (g) = 1 for {1,..., }. The, q j = 1 for all, j, ad α(g) = ( 1). I ths case, the algebra H s geerated by g, x 1,..., x subject to relatos g 2 = 1, x 2 = 0, x x j = x j x, gx = x g. By Proposto 3.1, we kow that H s strogly graded f ad oly f s odd. 4. Lftgs of Quatum Lear Spaces I ths secto we cosder a geeralzato of Quatum Lear Spaces: that of ther lftgs. As above, G s a fte abela group, g = g 1,..., g G s a sequece of elemets G ad χ = χ 1,..., χ Ĝ s a sequece of characters of G, such that (4.1) (4.2) χ (g ) 1, χ (g j )χ j (g ) = 1, for j. Aga, let q j = χ j (g ) ad let m = ord(q ). Let ow λ k ad λ j k for j be such that λ (χ m ε) = λ j (χ χ j ε) = 0. Suppose that λ j + q j λ j = 0 wheever j. The lftg of the quatum affe space assocated wth ths data s the algebra H = H(g, χ, λ), wth geerators G ad x 1,..., x, subject to the followg relatos: (4.3) (4.4) (4.5) gx = χ (g)x g, x x j = q j x j x + λ j (1 g g j ), x m = λ (1 g m ). It s well kow that the set of moomals {x r1 1 xr g : 0 r < m, g G} s a bass of H. It s a Hopf algebra wth comultplcato defed by (4.6) (4.7) (g) = g g, for all g G, (x ) = g x + x 1. The cout ε satsfes ε(g) = 1, for all g G, ad ε(x ) = 0. atpode S s gve by S(g) = g 1, for all g G, ad S(x ) = g 1 that S 2 (g) = g ad S 2 (x ) = q 1 x. Let S be the symmetrc group o elemets. For σ S let Note that t σ 0. t σ = x mσ 1 1 σ 1 xσ g. Moreover, the x. We ote

6 6 M. GRAÑA, J.A. GUCCIONE, AND J.J. GUCCIONE Lemma 4.1. The followg hold: (1) λ j g g j les the ceter of H(g, χ, λ) for j. (2) λ g m les the ceter of H(g, χ, λ). (3) t σ g = t σ, for all g G. (4) t σ x σ = 0. Proof. For (1) t s suffcet to see that λ j g g j commutes wth x l. If λ j = 0 the result s clear. Assume that λ j 0. The, χ = χ 1 j, ad thus λ j g g j x l = λ j χ l (g )χ l (g j )x l g g j = λ j χ (g l ) 1 χ j (g l ) 1 x l g g j = λ j x l g g j. The proof of (2) s smlar to that of (1) ad (3) t s mmedate. For (4), we have: t σ x σ = x mσ 1 1 σ 1 xσ gx σ = x mσ 1 1 σ 1 x = λ σ x mσ 1 1 σ 1 σ ad the result follows by otcg that x σ χ σ (g)g x mσ 1 1 σ 1 (1 gσ mσ ) χ σ (g)g, (1 gσ mσ ) χ σ (g)g = χ σ (g)g gσ mσ χ σ (g)g = χ σ (g mσ σ g)gσ mσ g χ σ (g)gσ mσ g = 0, sce χ σ (gσ mσ ) = qσ mσ σ = 1. Proposto 4.2. t σ s a rght tegral. Proof. Let M = (m 1 1) + + (m 1). Let A = {f : {1,..., M} {1,..., } : #f 1 () = m 1 for all }. For f A, let x f = x f(1) x f(2) x f(m). We clam that f f, h A, the x f g = βx h g for some β k. To prove ths clam, t s suffcet to check t whe f ad h dffer oly, + 1 for some 1 < M, that s, whe h = f τ, where τ S M s the elemetary trasposto (, + 1). But, ths case, we have: x f g = x h τ g = q h(+1)h() x h g + λ h(+1)h() x h, b,î+1 (1 g h()g h(+1) ) = q h(+1)h() x h g where x h, b,î+1 = x h(1) x h( 1) x h(+2) x h(m). The secod equalty follows from relato (4.4) ad tem (1) the prevous Lemma. The Proposto follows ow usg tems (3) ad (4) the Lemma. g

7 DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS 7 Now we see that α(x ) = 0, α(g) = χ m1 1 1 (g) χ m 1 (g). For the frst asserto, by Proposto 4.2, we ca take σ such that σ 1 =. The, x t σ = x m = x mσ 2 1 σ 2 x mσ 2 1 σ 2 x xσ σ g λ (1 g m ) g = 0, where the secod equalty follows from tem (2) of Lemma 4.1. I partcular, α(g ) = q m1 1 1 q m 1. Sce ρ(h) = α(s(h (1) ))S 2 (h (2) ), we have: ρ(g) = α(g 1 )g, ρ(x ) = α(g 1 )q 1 x = q 1 mj j x. Thus, as ρ s a algebra map, 1 j j ρ(x r1 1 xr g) = q r1 11 q r = 1 <j 1 g r g 1 ) x r1 1 xr g α(g r1 q (1 mj)r (1 m)rj j α(g 1 ) x r1 1 xr g. So, the bass {x j1 1 xj g} s made up of egevectors of ρ. Cosder the groups k L 1 = q 11,..., q, α(g) L 2 = α(g). Usg that ρ(x g 1 ) = q 1 x g 1 ad ρ(g) = α(g 1 )g, t s easy to see that L 1 s the set of egevalues of ρ ad that the order of ρ s the l.c.m. of the umbers m 1,..., m ad the order of the character α G Ĝ ( partcular, k has a prmtve ord ρ -th root of uty). As before, we decompose H = H(g, χ, λ) as H = ω L 1 H ω, where H ω = {h H : ρ(h) = ωh}. The followg result s the verso of Proposto 3.1 for the preset cotext. Theorem 4.3. The followg are equvalet: (1) ω L 1 H ω s strogly graded, (2) L 1 = L 2, (3) Each compoet H ω cotas a elemet G, (4) H s a crossed product H 1 kl 1. Proof. Clearly (2) ad (3) are equvalet ad (4) (1). The proof of (3) (4) s the same as for Proposto 3.1. Next we prove that (1) (3). Let ω L 1. By Lemma 2.1, we kow that ε(h ω ) 0. Sce H ω has a bass cosstg of moomals x r1 1 xr g ad ε(x ) = 0, there must be a elemet g G sde H ω. Remark 4.4. We ext show that for lftgs of Quatum Lear Spaces, the compoets the decomposto H = ω L 1 H ω are equdmesoal. I fact, ths case we ca take the bass of H gve by {(x 1 g1 1 (x )r1 g 1 ) r g : 0 r < m, g G}.

8 8 M. GRAÑA, J.A. GUCCIONE, AND J.J. GUCCIONE Sce ρ(x g 1 ) = q 1 x g 1, the map θ : Z m1 Z m G k, takg (r 1,..., r, g) to the egevalue of (x 1 g 1 ) r1 (x g 1 ) r g wth respect to ρ, s a well defed group homomorphsm. From ths t follows mmedately that all the egespaces of ρ are equdmesoal. 5. Computg H 1 Assume we are the settg of the lftgs of QLS. Suppose H s a crossed product or, equvaletly, that L 1 = L 2. The, there exst elemets γ 1,..., γ G, such that α(γ ) = q. Set γ = g 1 γ 1 ad let y = x γ. It s mmedate that y H 1. Let N = ker(α G ) G. It s easy to see that H 1 has a bass gve by {y r1 1 yr N, y 1,..., y ad relatos gy = χ (g)y g, y y j = q j χ j ( γ )χ 1 y m g : g N}. Furthermore, H 1 ca be preseted by geerators = λ χ m(m 1)/2 ( γ j )y j y + χ j ( γ )λ j ( γ γ j γ 1 γ 1 ( γ )( γ m γ m ). j ), Notce that f λ 0, the χ m(m 1)/2 ( γ ) = ±1. We clam that λ j γ γ j, λ j γ γ j, λ γ m ad λ γ m belog to kn. It s clear that γ m N, sce α(γ m ) = q m = 1. We ow prove the remag part of the clam. Assume that λ j 0. The χ χ j = ε. Hece, If l, j, the χ l (g g j ) = q l q jl = q 1 l q 1 lj = χ χ j (g 1 l ) = 1. q = χ (g ) = χ j (g 1 ) = q 1 j = q j = q 1 jj. Thus, m = ord(q ) = ord(q jj ) = m j, ad the χ m 1 (g g j )χ mj 1 j (g g j ) = (q q j q j q jj ) m 1 = 1 ad α(γ γ j ) = q q jj = 1. It s ow mmedate that α(g g j ) = χ m1 1 1 (g g j ) χ m 1 (g g j ) = 1, ad so α( γ γ j ) = α(g 1 γ 1 It remas to check that λ γ m Thus, If l, the χ m l 1 l Sce χ m 1 ad so g 1 j γ 1 j ) = α(g j g ) 1 α(γ j γ ) 1 = 1. kn. Assume ow that λ 0. The χ m = ε. (g m ) = q (m l 1)m l = q (1 m l)m l (g m ) = q m(m 1) = 1, ths mples that α(g m ) = χ m1 1 1 (g m ) χ m 1 (g m ) = 1, α( γ m ) = α(γ g ) 1 = α(γ ) 1 α(g ) 1 = 1. Refereces = χ m (g 1 m l l ) = 1. [AG] N. Adruskewtsch ad M. Graña, Braded Hopf algebras over o-abela fte groups, Bol. Acad. Nac. Cec. (Córdoba), 63, (1999), [AS] N. Adruskewtsch ad H.-J. Scheder, Poted Hopf algebras, New drectos Hopf algebras, Math. Sc. Res. Ist. Publ., 43, 1 68, Cambrdge Uv. Press, Cambrdge, [GG] J.A. Guccoe ad J.J. Guccoe, Hochschld cohomology of Frobeus algebras, Proc. Amer. Math. Soc., 132, (2004), o. 5, (electroc). [M] A. Marcus, Represetato theory of group graded algebras, Nova Scece Publshers Ic., Commack, NY, 1999.

9 DECOMPOSITION OF SOME POINTED HOPF ALGEBRAS 9 [N] W.D. Nchols, Balgebras of type oe, Comm. Alg. 6 (1978), [R1] D.E. Radford, The trace fucto ad Hopf algebras, J. Algebra 163 (1994), o. 3, [R2] D.E. Radford, The order of the atpode of a fte dmesoal Hopf algebra s fte, Amer. J. Math. 98 (1976), o. 2, [RS] D. E. Radford ad H.-J. Scheder, O the eve powers of the atpode of a ftedmesoal Hopf algebra, J. Algebra 251 (2002), o. 1, [S] H-J. Scheder, Lectures o Hopf algebras, Trabajos de Matemátca, 31/95, Uversdad Nacoal de Córdoba, FaMAF, Córdoba, M.G., J.A.G., J.J.G.: Depto de Matemátca - FCEyN Uversdad de Bueos Ares Pab. I - Cudad Uverstara Bueos Ares - Argeta E-mal address, M.G.: matasg@dm.uba.ar E-mal address, J.A.G.: vader@dm.uba.ar E-mal address, J.J.G.: jjgucc@dm.uba.ar