Construct non-graded bi-frobenius algebras via quivers

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1 Science in China Series A: Mathematics Mar., 2007, Vo. 50, No. 3, Construct non-graded bi-frobenius agebras via quivers Yan-hua WANG 1 &Xiao-wuCHEN 2 1 Department of Appied Mathematics, Shanghai University of Finance Economics, Shanghai , China; 2 Department of Mathematics, University of Science Technoogy of China, Hefei , China (emai: yhwgm@fudan.edu.cn, xwchen@mai.ustc.edu.cn Abstract Using the quiver technique we construct a cass of non-graded bi-frobenius agebras. We aso cassify a cass of graded bi-frobenius agebras via certain equations of structure coefficients. Keywords: quivers, bi-frobenius agebras, graded agebras. MSC(2000: 16W30, 16G20 1 Introduction Frobenius agebra is a very important cass of finite-dimensiona agebras. Typica exampes are semi-simpe agebras finite-dimensiona Hopf agebras. Recenty, Frobenius agebras are found to have a cose reation with the soutions of Yang-Baxter equation the topoogica quantum fied theory, see [1]. Bi-Frobenius agebras were introduced by Doi Takeuchi [2]. They are certain natura generaizations of finite-dimensiona Hopf agebras. Roughy speaking, a bi-frobenius agebra is a Frobenius agebra as we as a Frobenius coagebra together with an (mutipicative comutipicative anti-automorphism. However up to now there are few exampes of bi-frobenius agebras which are not Hopf agebras besides 2.5 in [2] 2.1 in [3]. It is we-known that the quiver is a fundamenta too to construct study agebras coagebras [4]. The appication of quivers to Hopf agebras quantum groups can be found in [5 7], where Hopf agebras some quantum groups are constructed by quivers. An advantage for this construction is that a natura basis consisting of paths is avaiabe. Thus a natura probem arises: can we construct bi-frobenius agebras via quivers? The first try is shown in [8], where a cass of graded bi-frobenius agebras which are not Hopf agebras are given on basic cyces. Inspired by the exampe in 1.7 of [5], where one can define non-graded Hopf agberas on basic cyces, we wi construct a cass of non-graded bi-frobnius agebras on basic cyces over the fieds of characteristic 2, see Theorem 2.4. Let us remark that bi-frobenius agebras in Theorem 2.4 can be viewed as the deformations of the obtained bi-frobenius agebras in [8] (in the sense of [9]. More precisey, they are graded agebra deformations [10] of the bi-frobenius agebras in [8] which preserve the coagebra structure. Recenty, the cassification of Hopf agebras is a quite active topic in the theory of Hopf Received March 19, 2006; accepted Juy 18, 2006 DOI: /s Corresponding author This work was supported in part by the Nationa Natura Science Foundation of China (Grant Nos , ,

2 Construct non-graded bi-frobenius agebras via quivers 451 agebras [11]. Using quivers, some Hopf agebras can be cassified [5,12]. In sec. 3, we cassify a cass of graded bi-frobenius agebras on basic cyces with a given coagebra structure (over arbitrary fieds, which turns out to depend on some equations of structure coefficients, see Theorem 3.1. Throughout, et K be a fied. A agebras coagebras are over K. LetV be a K-space, denote V Hom K (V,K. 2 Non-graded bi-frobenius agebras Let A be a finite-dimensiona agebra, A Hom K (A, K. Then A has a natura A-Abimodue structure given by (af(b f(ba, (fa(b f(ab, f A, a, b A. We say that A is a Frobenius agebra provided that A A (A A as the eft A-modues, or equivaenty A A (A A as the right A-modues. Let A be a Frobenius agebra with a fixed eft A-modue isomorphism Φ : A A (A A.Then φ : Φ(1 A is a cycic generator of A A. Note that φ is aso a cycic generator of A A. We ca the pair (A, φ a Frobenius agebra (if φ is needed to be specified, ca φ a Frobenius homomorphism. For more about Frobenius agebras, see [1, 2, 4]. Let C be a finite-dimensiona coagebra with a comutipication a counit ɛ, C its dua agebra. Then C is a C -C -bimodue via fc c 1 f(c 2, cf f(c 1 c 2 f C, c C. Apair(C, t witht C is caed a Frobenius coagebra if C tc or equivaenty C C t. We refer to [13] for the notation of coagebras. Let us reca the definition of bi-frobenius agebras introduced by Doi Takeuchi (see [2] 2.2 in [14]. Definition 2.1. Let A be a finite-dimensiona agebra coagebra with t A φ A. Suppose that (i The counit ɛ is an agebra map 1 A is a group-ike eement; (ii (A, φ is a Frobenius agebra, (A, t is a Frobenius coagebra with a comutipication ; (iii the inear map ψ : A A given by ψ(a φ(t 1 at 2, a A (1 is an anti-agebra map an anti-coagebra map, where (t t 1 t 2. Then we ca (A, φ, t, ψ a bi-frobenius agebra the map ψ theantipodeofa. In [8], a cass of (graded bi-frobenius agebras arising from quivers are studied. Reca some notation. Let Z n be the basic cyce of ength n, i.e. anorientedgraphwithn vertices e 0,...,e n 1, a unique arrow a i from e i to e i+1 for each 0 i n 1. Take the indices moduo n, i.e. the indices are in the group Z/nZ. Setγi : a i+ 1 a i+1 a i to be the path of ength starting at the vertex e i.notethatγi 0 e i γi 1 a i. For n, d N d 2, et C d (n be the subcoagebra of the path coagebra KZn c with the basis of the set of a paths of ength stricty ess than d, see [5]. In other words, the coagebra

3 452 Yan-hua WANG & Xiao-wu CHEN C d (n has a basis γi,i Z/nZ, 0 <d} with (γ i s0 γ s i+s γs i, ɛ(γ iδ 0,, where the δ i,j is the Kronecker symbo for i, j Z/nZ. Denote by (γi the dua basis of C d (n. The foowing resut can be found in [8]. Lemma 2.2. (C d (n, φ,t,ψ is a bi-frobenius agebra (which is not a Hopf agebra, where φ (γ0 d 1, t n 1 i0 γd 1 i, ψ(γi γ i the agebra structure is given by γ i γs j γ, γ i γs j 0, + s d. Note that the above mutipication is graded with the ength grading. In what foows, we wi deform it obtain a cass of new bi-frobenius agebras. Let μ K. Define a mutipication m : C d (n C d (n C d (n as foows: γ i γj s γ, γ i γs j μ(γ d + γ d +d, + s d. Note that if μ 0, m is just the mutipication in Lemma 2.2. Lemma 2.3. (C d (n, m is a Frobenius agebra. Proof. First we caim that the mutipication m is associative. Indeed, given γi,γs j,γm k,i,j, k Z/nZ, 0, s, m d 1inC d (n, we have m( m Id(γi γj s γk m m(γ γm k, m(μ(γ d γ +m, μ(γ +m d μ(γ +m d μ 2 (γ +m 2d On the other h, m(id m(γi γj s γk m m(γ i γ s+m j+k, m(γ i μ(γs+m d j+k γ +m, μ(γ +m d μ(γ +m d μ 2 (γ +m 2d + γ d +d γm k, + s d, + s + m<d, + γ +m d +d, + s<d + s + m d, + γ +m d +d, +2γ +m 2d +d + s d + s + m<2d, + γ +m 2d +2d, + s + m 2d. s+ m<d, + γ s+m d j+k+d γm k, s+ m d, + s + m<d, + γ +m d +d, s+ m<d + s + m d, + γ +m d +d, +2γ +m 2d +d s+ m d + s + m<2d, + γ +m 2d +2d, + s + m 2d.

4 Construct non-graded bi-frobenius agebras via quivers 453 The above two identities impy that m is associative. Here γ0 0 is the unit eement. Next we caim that (C d (n, m, 1γ0 0 is a Frobenius agebra. In fact, for γi,γs j C d(n, we have (γ d 1 0 (γ, (γ i(γ d 1 0 (γ s j (γ d 1 0 (γ s j γ i δ s+, d 1 δ, 0, μ(γ d 1 0 (γ d + γ d +d, + s d, where δ i,j is the Kronecker symbo with the one moduo n for i, j Z/nZ, hence γi (γd 1 0 (γn i d 1. Therefore (C d (n C d (n(γ0 d 1 C d (n is a Frobenius agebra (with Frobenius homomorphism (γ0 d 1. This competes the proof. The main resut in this section is as foows. Theorem 2.4. Let K be a fied of characteristic 2. (Use the notation in Lemma 2.3. Then (C d (n, φ (γ0 d 1, t n 1 u0 γd 1 u, ψ is a bi-frobenius agebra with the mutipication m. Proof. First note that the counit ɛ is an agebra map. In fact, we have ɛ(γ i γs j ɛ(γ, ɛ(μ(γ d + γ d +d, + s d, δ, 0, μ(δ d, 0 + δ d, 0, + s d, δ, 0 δ s, 0 ɛ(γi ɛ(γs j, for γi, γs j C d(n. Here we use the assumption that chark 2. Obviousy, 1 γ0 0 is a group-ike eement of C d (n. By Lemma 1.2 in [14], to prove that (C d (n, φ (γ0 d 1, t n 1 u0 γd 1 u, ψisa(nongraded bi-frobenius agebra, we just need to show that ψ is a bijective anti-agebra anticoagebra map. By (1, we have ψ(γi ( n 1 φ(t 1 γi t 2 (γ0 d 1 ( n 1 (γ0 d 1 γ i, u0 s γ d 1 s+ u+s+i + d 1 u0 s0 n 1 u0 >s γu+s d 1 s γi γu s μ(γ s 1 u+s+i + γ s 1 u+s+i+d γ s u for 0 d 1. That is, the antipode of C d (n isgivenbyψ(γi γ i. Ceary, it is bijective. Next, we wi show that ψ is an anti-agebra anti-coagebra morphism. For any i, j Z/nZ, 0, s d 1, we have ψ(γ i γs j ψ(γ, μ(ψ(γ d γ i j s, +ψ(γ d +d, + s d, μ(γ d i j s+d + γ d i j s, + s d

5 454 Yan-hua WANG & Xiao-wu CHEN γ s+ j s i, s+ <d, ψ(γj s ψ(γiγ j sγ s i μ(γ s+ d j s i + γs+ d j s i +d, s+ d. Thus ψ is an anti-agebra map. To see that ψ is an anti-coagebra map, just note that ( (ψ(γi (γ i γ s i γs i s0 (ψ ψt (γi (ψ ψ(γi s γ s s0 where T is the fip. This competes the proof. i+s (γ i s s γ s s0 i, 3 A cass of graded bi-frobenius agebras It is known that there are many mutipications on the coagebra C d (n such that it becomes a bi-frobenius agebra, see [5, 8]. Thus it is an interesting probem to ask, for a given coagebra, how many agebra structures one can assign to the coagebra C d (n suchthatitbecomesa bi-frobenius agebra, or some simiar probems under more restrictions. In this section, we sti consider the coagebra C d (n, cassify a cass of graded agebra structures on it such that (C d (n, φ (γ0 d 1, t n 1 i0 γd 1 i, ψ(γi γ i becomes a bi-frobenius agebra. Moreover, we require that γ0 0 is the unit. Reca that C d (n isgradedby the ength grading, see sec. 2. Let K be an arbitrary fied. Denote A C d (n. Assume that the graded agebra structure of (A, φ isgivenbyγi γs j P (i,, j, sγ F (i,,j,s, for i, j Z/nZ, 0, s d 1, P(i,, j, s K, F (i,, j, s Z/nZ. (Here we assume that γi γs j 0if d, thatis,p (i,, j, s 0 whenever + s d. Note that 1 γ0 0 is the unit eement of the agebra A, hence γ i γ0 0 P (i,, 0, 0γ F (i,,0,0 γ0 0 γ i P (0, 0,i,γ F (0,0,i, γ i. Thus we obtain F (i,, 0, 0 F (0, 0,i,i (2 P (i,, 0, 0 P (0, 0,i,1. (3 By the associativity of the mutipication, we get (γ iγ s j γ m h P (i,, j, sγ F (i,,j,s γm h P (i,, j, sp (F (i,, j, s,+ s, h, mγ +m F (F (i,,j,s,,h,m γ i(γ s j γ m h γ i P (j, s, h, mγs+m F (j,s,h,m P (j, s, h, mp (i,, F (j, s, h, m,s+ mγ +m F (i,,f (j,s,h,m,s+m.

6 Construct non-graded bi-frobenius agebras via quivers 455 Then we obtain F (F (i,, j, s,+ s, h, m F (i,, F (j, s, h, m,s+ m (4 P (i,, j, sp (F (i,, j, s,+ s, h, m P (j, s, h, mp (i,, F (j, s, h, m,s+ m. (5 By the definition of ψ, wehave γ i ψ(γi ( n 1 φ(t 1 γit 2 (γ0 d 1 ( n 1 (γ0 d 1 d 1 j0 m0 d 1 j0 m0 γ d 1 m j+m γ i γj m P (j + m, d 1 m, i, γ d 1 m+ F (j+m,d 1 m,i, n 1 P (j +, d 1, i, δ 0,F (j+,d 1,i, γj. j0 γ m j Hence F ( i, d 1, i, 0, P( i, d 1, i, 1 (6 F ( i,d 1, i, 0, P( i,d 1, i, 1, i i. (7 By the condition that ψ is an anti-agebra morphism, we have ψ(γiγ j s P(i,, j, sψ(γ F (i,,j,sp(i,, j, sγ s F (i,,j,s ψ(γ s j ψ(γ i γs j s γ i P ( j s, s, i, γs+ F ( j s,s, i,. This impies that s F (i,, j, s F ( j s, s, i, (8 P (i,, j, s P ( j s, s, i,. (9 In summary, combining a the above conditions, we get F (i,, 0, 0 F (0, 0,i,i, F (F (i,, j, s,+ s, h, m F (i,, F (j, s, h, m,s+ m, (i F ( i, d 1, i, 0, F ( i,d 1, i, 0, i i, s F (i,, j, s F ( j s, s, i, (ii P (i,, 0, 0 P (0, 0,i,1, P (i,, j, sp (F (i,, j, s,+ s, h, m P (j, s, h, mp (i,, F (j, s, h, m,s+ m, P ( i, d 1, i, 1, P ( i,d 1, i, 1, i i, P (i,, j, s P ( j s, s, i,.

7 456 Yan-hua WANG & Xiao-wu CHEN Note that i, j, h Z/nZ 0, s, m d 1. We note that ψ is aways an anti-coagebra morphism, see the proof of Theorem 2.4. By comparing the above argument with Definition 2.1, using Lemma 1.2 in [14], we have the main theorem of this section. Theorem 3.1. Assume that the mutipication of the coagebra C d (n is γiγ j s P (i,, j, sγ F (i,,j,s, i, j Z/nZ, 0, s d 1, where P (i,, j, s K, F (i,, j, s Z/nZ with P (i,, j, s 0whenever + s d. Then (C d (n, φ(γ0 d 1,t n 1 i0 γd 1 i,ψ(γi γ i becomes a (graded bi-frobenius agebra if ony if F P satisfy (i (ii respectivey. Remark 3.2. (1 Note that F (i,, j, s i + j, P (i,, j, s 1(fori, j, h Z/nZ, 0, s, m d 1 + s<d are the soutions of the equations above, see [7]. (2 Assume that char K 0. Suppose that the order of q is d d n. Let ( m q m! q (m! q! q be Gauss coefficients. Then F (i,, j, s, P (i,, j, s q j( m are aso the soutions of the q equations above, see [5]. Here we note that a finite-dimensiona Hopf agebra is bi-frobenius. (3 By a simiar argument, one can aso obtain the cassification of non-graded bi-frobenius agebras on C d (n via finding the equations of structure coefficients. Acknowedgements We woud ike to thank the referees for their hepfu comments. References [1] Chen X W, Huang H L, Wang Y H. A note on modues, comodues cotensor products over Frobenius agebras. Chin Ann Math Ser B, 27(4: (2006 [2] Doi Y, Takeuchi M. Bi-Frobenius agebras. Contemp Math, AMS, 267: (2002 [3] Doi Y. Bi-Frobenius agebras group-ike agebras. In: Hopf Agebras. Lecture Notes in Pure App Math, Vo New York: Dekker, [4] Auser M, Reiten I, Smaø S O. Representation Theory of Artin Agebras. Cambridge Studies in Adv Math 36. Cambridge: Cambridge Univ Press, 1995 [5] Chen X W, Huang H L, Ye Y, Zhang P. Monomia Hopf agebras. J Agebra, 275: (2004 [6] Chen X W, Huang H L, Zhang P. Dua Gabrie Theorem with appications. Sci China Ser A-Math, 49(1: 9 26 (2006 [7] Chin W, Montgomery S. Basic coagebras, Moduar interfaces (Reverside, CA. 1995, 41 47, AMS/IP Stud Adv Math 4. Providence: Amer Math Soc, 1997 [8] Wang Y H, Zhang P. Construct bi-frobenius agebras via quivers. Tsukuba J Math, 28(1: (2004 [9] Gerstenhaber M. Deformations of rings agebras. Ann Math, 79(2: (1964 [10] Braverman A, Gaitsgory D. Poincaré-Birkhoff-Witt theorem for quadratic agebras of Koszu type. J Agebra, 181: (1996 [11] Schneider H J. Lectures on Hopf agebras. Córdoba: University of Argentina, [12] Liu G X, Ye Y. Monomia Hopf agebras over fieds of positive characteristic. Sci China Ser A-Math, 49(3: (2006 [13] Dǎscǎescu S, Nǎstǎsescu C, Raianu S. Hopf Agebras: An Introduction. New York-Base: Marce Dekker, 2000 [14] Doi Y. Substructures of bi-frobenius agebras. J Agebra, 256: (2002