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

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-02 12330, UBA X294. 1

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 4.4. 2. 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.

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 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).

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 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

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 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), 45 78. [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, 2002. [GG] J.A. Guccoe ad J.J. Guccoe, Hochschld cohomology of Frobeus algebras, Proc. Amer. Math. Soc., 132, (2004), o. 5, 1241 1250 (electroc). [M] A. Marcus, Represetato theory of group graded algebras, Nova Scece Publshers Ic., Commack, NY, 1999.

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