A Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"

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1 Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: /s z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules, and Cotensor Products over Frobenus Algebras" aowu CHEN Hualn HUANG Yanhua WANG Abstract Ths s a note on Abrams paper \Modules, Comodules, and Cotensor Products over Frobenus Algebras, Journal of Algebras" (1999). Wth the applcaton of Frobenus coordnates developed recently by Kadson, one has a drect proof of Abrams characterzaton for Frobenus algebras n terms of comultplcaton (see L. Kadson (1999)). For any Frobenus algebra, by usng the explct comultplcaton, the explct correspondence between the category of modules and the category of comodules s obtaned. Moreover, wth ths we gve very smpled proofs and mprove Abrams results on the Hom functor descrpton of cotensor functor. Keywords Frobenus coordnates, Cotensor, Hochschld cohomology 2000 MR Subject Classcaton 17A60, 18G15 1 Introducton Recently Abrams provded n detal the proof of the equvalence of the category of twodmensonal topologcal quantum eld theores and the category of commutatve Frobenus algebras (see [1, 8]). Hs characterzaton for Frobenus algebras n terms of comultplcaton played an mportant role. Ths observaton made the correspondence of the categores of twodmensonal topologcal quantum eld theores and commutatve Frobenus algebras hghly ntutve. Later on he generalzed the characterzaton to the case of noncommutatve Frobenus algebras, and then he could gve an alternatve descrpton of Elenberg and Moore s cotensor product functor and ts derved functors va the usual hom functor and the Hochschld cohomology functors respectvely (see [2, 3]). Ths s a note on Abrams work [2] va the so-called Frobenus coordnates. These notons were recently developed by Kadson [5] and have turned very useful n the study of Hopf algebras and other related topcs (see [6, 7, 10]). Let A be a nte-dmensonal algebra over an arbtrary eld K: By A we denote ts dual lnear space. The regular bmodule structure on A naturally nduces a A-bmodule structure Manuscrpt receved January 18, Department of Mathematcs and Shangha Insttute for Advanced Studes, Unversty of Scence and Technology of Chna, Hefe , Chna. E-mal: xwchen@mal.ustc.edu.cn Department of Mathematcs and Shangha Insttute for Advanced Studes, Unversty of Scence and Technology of Chna, Hefe , Chna; Mathematcal Secton, The Abdus Salam ICTP, Strada Costera 11, Treste 34014, Italy. Insttute of Mathematcs, Fudan Unversty, Shangha , Chna. Project supported by AsaLnk Project \Algebras and Representatons n Chna and Europe" ASI/B7-301/98/ and the Natonal Natural Scence Foundaton of Chna (No ).

2 420. W. Chen, H. L. Huang and Y. H. Wang on A : An algebra A s sad to be a Frobenus algebra f there s a left A-modules somorphsm : A A! A A ; or equvalently a rght A-modules somorphsm A A = A A : If A = A as A- bmodules, then A s called a symmetrc algebra. There are many other equvalent dentons of Frobenus algebras and symmetrc algebras. We refer the reader to [4] for more nformaton. Abrams provded a characterzaton of Frobenus algebras n terms of comultplcaton: an algebra A s a Frobenus algebra f and only f t has a coassocatve countal comultplcaton : A! A A whch s a map of A-bmodules. The comultplcaton was dened as follows. Assume now A s a Frobenus algebra and : A A! A A s an somorphsm. Let : AA! A denote the multplcaton and : A A! A A denote the canoncal twst map. Denote = and ts dual. The comultplcaton map : A! A A s dened to be the composton ( 1 1 ) : Let " = (1 A ): Then (A; ; ") s a coalgebra. Applyng the Frobenus coordnates, one obtans an explct formula for the comultplcaton (see Lemma 2.1). Moreover, we observe that the coalgebra (A; ; ") s nothng but A cop.e., the coopposte coalgebra of the dual coalgebra of the Frobenus algebra A (see Proposton 2.1). Hence t s straghtforward that the category of left (resp. rght) modules over a Frobenus algebra A s somorphc to the category of left (resp. rght) comodules over the correspondng coalgebra (A; ; "); we also gve the explct correspondence between modules and comodules (see Theorem 2.1 and Remark 2.1). Wth the explct correspondence we are able to descrbe for any Frobenus algebras the cotensor functor and ts derved functors usng the Hom functor and the Hochschld cohomology (see Theorem 3.1 and Corollary 3.1). For unexplaned notatons on coalgebras, we refer the reader to [9]. 2 Frobenus Algebras and Frobenus Coordnates Let K be a eld and A a Frobenus algebra over K wth unt 1 A. Assume : A A! A A s an somorphsm of left A-modules. Denote := (1 A ): Then s a cyclc generator of A A ; and the somorphsm s gven by (a) = a; for all a 2 A: Also s a cyclc generator of A A ; and the somorphsm A A = A A s gven by a 7! a: Suppose x 2 A; f 2 A form dual bases,.e., for each a 2 A; P f (a)x = a: Let y 2 A such that f = y : Then we have for all a 2 A; x (y a) = a = (ax )y : We refer to as a Frobenus homomorphsm, (x ; y ) as dual bases, P x y as the Frobenus element, and (; x ; y ) as Frobenus coordnates after Kadson and Stoln [6] (also see [10]). The followng lemma s Abrams characterzaton of Frobenus algebras n terms of comultplcaton. Va Frobenus coordnates, Kadson gave a smpled proof (see [5]). For completeness and later use, we nclude a proof. Lemma 2.1 An algebra A s a Frobenus algebra f and only f t has a coassocatve countal comultplcaton : A! A A whch s a map of A-bmodules. Proof Assume that the algebra A has a coassocatve countal comultplcaton : A! A A whch s a map of A-bmodules. Let " : A! K be the count. We clam that the

3 A Note on \Modules, Comodules, and Cotensor Products over Frobenus Algebras" 421 left A-module morphsm gven by (a) = a" s an somorphsm and hence A s a Frobenus algebra. It suces to verfy that s njectve. Assume (1 A ) = x y : Thus (a) = ((1 A ))a = x y a: If a" = 0; then we have a = x "(y a) = x (a")(y ) = 0; snce the comultplcaton s countal. Conversely, assume that A s a Frobenus algebra wth left A-module somorphsm : Let (; x ; y ) be the Frobenus coordnates. Dene : A! A A to be the lnear map gven by (a) = ax y = x y a: Note that s an A-bmodule map by the denton. Let " = : Now t s easy to check that (A; ; ") s a coalgebra. By drect calculaton we show that the coalgebra we construct above concdes wth Abrams coalgebra (see [2]). Note that for any a; x; y 2 A; (a)(x y) = (a)(x y) = (a)(yx) = (yxa) = y (xax )y = (yy )(xax ) = (ax )(x)(y )(y): Now t follows that = ( 1 1 ) : From the proof of Lemma 2.1, we obtan a satsfactory understandng of the element (1 A ) and the submodule (A) of A A generated by t. The element (1 A ) s just the Frobenus element P x y ; and the module n o n o (A) = ax y a 2 A = x y a a 2 A : It s obvous that (A) = A as bmodules snce s njectve. Moreover, by the explct comultplcaton map, we observe that the coalgebra (A; ; ") s actually somorphc to A cop ; the coopposte coalgebra of the dual coalgebra of A: Proposton 2.1 The lnear map : A! A cop gven by (a) = a s an somorphsm of coalgebras. Proof It remans to check that s a coalgebra map snce t s bjectve. Recall that for any f 2 A cop ; (f) = f 1 f 2

4 422. W. Chen, H. L. Huang and Y. H. Wang f and only f f(ab) = f 1 (b)f 2 (a) for all a; b 2 A: Applyng agan the Frobenus coordnates, we have (a)(x y) = (a)(x y) = (yxa) = y (xax )y = (yy )(xax ) = (ax )(x)(y )(y) = ( ) (a)(x y) for all a; x; y 2 A: The vercaton of " = " s mmedate. Ths completes the proof. Now by ths observaton, the followng Abrams theorem s straghtforward. See Theorem 3.3 n [2]. Before statng the theorem, we x some notatons. Assume that C s a coalgebra and A s an algebra. By C M we denote the category of left C-comodules and A M the category of A-modules. Smlarly we use M C and M A : Theorem 2.1 The category of left (resp. rght) modules over a Frobenus algebra A s somorphc to the category of left (resp. rght) comodules over A: Proof It s well known that for a nte-dmensonal algebra A; A M = M A and that for a coalgebra C; Ccop M = M C : Now by Proposton 2.1, the theorem follows mmedately. Remark 2.1 It s nterestng to wrte down the explct correspondence between A M and A M. Ths wll be used n Secton 3. Let N be a left A-module. Then by Theorem 2.1, N also has a correspondng left A-comodule structure. We denote the comodule map by Then for any n 2 N; N : N! A N: N (n) = n 1 n 0 () a:n = n 0 (an 1 ) for all a 2 A: And smlarly, let M be a rght A-module and the correspondng rght A-comodule structure s denoted by (M; M ): Then for any m 2 M; M (m) = m 0 m 1 () m:a = m 0 (am 1 ) for all a 2 A: 3 Cotensor Product and Its Derved Functors Let A e = A A op be the tensor product of the algebra A and the opposte algebra A op : Thus an A-bmodule s exactly an A e -module and vce versa. The am of ths secton s to understand the cotensor product over a Frobenus algebra A usng the functor Hom A e(a; ): Let M be a rght A-module and N a left A-module. Thus by Theorem 2.1, M and N are rght and left A-comodule respectvely. Denote ther comodule structure maps as M and N respectvely. The cotensor product MN of M and N over A s dened to be the kernel of : M N! M A N;

5 A Note on \Modules, Comodules, and Cotensor Products over Frobenus Algebras" 423 where = M Id N Id M N : Theorem 3.1 There s a vector space somorphsm MN = Hom A e(a; N M): Proof Recall our explct correspondence of modules and comodules n Remark 2.1. It s not hard to observe that m n 2 MN () m:a n = m a:n for all a 2 A: In fact, suppose P m n 2 MN: Then m0 m 1 n = m n 1 n 0 : Applyng Id M a Id N to both sdes, we have m:a n = m a:n: Smlarly we have the converse part. Now note that any f 2 Hom A e(a; N M) s unquely determned by f(1 A ): Snce f s an A e -map, we have f(1 A ) = n m () n m:a = a:n m for all a 2 A: Hence Map f to f(1 A ); the theorem follows. f(1 A ) 2 MN: Let Cot A (; ) be the rght derved functors of the cotensor product and H (; ) be the Hochschld cohomology functors (see [4]). The followng corollary s a drect consequence of Theorem 3.1. Corollary 3.1 Let A; M; and N be as above. We have lnear somorphsms Cot A(M; N) = Ext A e(a; N M) = H (A; N M): Remark 3.1 Note that n Theorem 4.5 of [2], Abrams used the A e -module D; whch s generated by the twst of the Frobenus element (1 A ) = y x : Actually, ths s not true. The reason s that the correspondence between rght modules and comdules over A n [2] was not rght.

6 424. W. Chen, H. L. Huang and Y. H. Wang References [1] Abrams, L., Two-Dmensonal topologcal quantum eld theores and Frobenus algebras, J. Knot Theory and Its Ramcatons, 5, 1996, 569{587. [2] Abrams, L., Modules, comodules and cotensor products over Frobenus algebras, J. of Algebras, 291, 1999, 201{213. [3] Abrams, L. and Webel, C., Cotensor products of modules, Trans. Amer. Math. Soc., 354, 2002, 2173{ [4] Curts, C. and Rener, I., Representaton Theory of Fnte Groups and Assocatve Algebras, Interscence Publshers, New York, [5] Kadson, L., Frobenus Extensons, Unversty Lecture Seres, Vol. 14, A. M. S., Provdence, RI, [6] Kadson, L. and Stoln, A. A., An approach to Hopf algebras va Frobenus coordnates I, Betrage Algebra Geom., 42(2), 2001, 359{384. [7] Kadson, L. and Stoln, A. A., An approach to Hopf algebras va Frobenus coordnates II, J. Pure Appl. Algebra, 176(2{3), 2002, 127{152. [8] Kock, J., Frobenus Algebras and 2D Topologcal Quantum Feld Theores, Avalable at unce.fr/kock/tqft.html. [9] Montgomery, S., Hopf Algebras and Ther Actons on Rngs, CBMS Lectures, Vol. 82, A. M. S., Provdence, RI, [10] Wang, Y. H. and Zhang, P., Construct b-frobenus algebras va quvers, Tsukuba J. Math., 28(1), 2004, 215{221.