hal , version 2-3 Dec 2009

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1 SKEW GROUP ALGEBRAS OF PATH ALGEBRAS AND PREPROJECTIVE ALGEBRAS LAURENT DEMONET Abstract. We compute explctly up to Morta-equvalence the skew group algebra of a fnte group actng on the path algebra of a quver and the skew group algebra of a fnte group actng on a preprojectve algebra. These results generalze prevous results of Reten and Redtmann [RR] for a cyclc group actng on the path algebra of a quver and of Reten and Van den Bergh [RV, proposton 2.3] for a fnte subgroup of SLCX CY actng on C[X, Y ]. hal , verson 2-3 Dec 29. Introducton and man results Let k be an algebracally closed feld and G be a fnte group such that the characterstc of k does not dvde the cardnalty of G. If Λ s a k-algebra and f G acts on the rght on Λ, the acton beng denoted exponentally, the skew group algebra of Λ under the acton of G s by defnton the k-algebra whose underlyng k-vector space s Λ k k[g] and whose multplcaton s lnearly generated by a ga g = aa g gg for all a,a Λ and g,g G see [RR]. It wll be denoted by ΛG. Identfyng k[g] and Λ wth subalgebras of ΛG, an alternatve defnton s ΛG = Λ,k[G] g,a G Λ,g ag = a g k-alg Let now Q = I,A be a quver where I denotes the set of vertces and A the set of arrows. Consder an acton of G on the path algebra kq permutng the set of prmtve dempotents {e I} and stablzng the vector space spanned by the arrows. Note that ths s more general than an acton comng from an acton of G on Q snce an arrow may be sent to a lnear combnaton of arrows. We now defne a new quver Q G. We frst need some notaton. Let Ĩ be a set of representatves of the classes of I under the acton of G. For I, let G denote the subgroup of G stablzng e, let Ĩ be the representatve of the class of and let κ G be such that κ =. For,j Ĩ2, G acts on O O j where O and O j are the orbts of and j under the acton of G. A set of representatves of the classes of ths acton wll be denoted by F j. For,j I, defne M j kq to be the vector space spanned by the arrows from to j. We regard M j as a rght k[g G j ]-module by restrctng the acton of G. The quver Q G has vertex set I G = e I{} rrg where rrg s a set of representatves of somorphsm classes of rreducble representatons of G. The set of arrows of Q G from,ρ to j,σ s a bass of ρ κ G Gj,σ κ j G Gj k M j,j F j Hom mod k[g G j ] where the representaton ρ κ of G s the same as ρ as a vector space, and ρ κ g = ρ κ gκ for g G = κ G κ. Table gves two examples of quvers Q G. A detaled example s also computed n secton 2. We can now state the two man results of ths paper and two corollares : Theorem. There s an equvalence of categores mod k Q G mod kqg.

2 Theorem was proved by Reten and Redtmann n [RR, 2] for cyclc groups. The followng theorem deals wth the case of preprojectve algebras. The defnton of the preprojectve algebra Λ Q of a quver Q s recalled n secton 3.2. Theorem 2. If G acts on kq, where Q s the double quver of Q, by permutng the prmtve dempotents e and stablzng the lnear subspace of kq spanned by the arrows, and f, for all g G, r g = r where r s the preprojectve relaton of ths quver, then Q s of the form Q G for some quver Q and Λ Q G s Morta equvalent to Λ Q. One can always extend an acton on kq to an acton on kq and ths yelds : Corollary 3. The acton of G on a path algebra kq permutng the prmtve dempotents and stablzng the lnear subspace of kq spanned by the arrows nduces naturally an acton of G on kq and Q G s somorphc to the double quver of Q G. Moreover, there s an equvalence of categores mod Λ QG mod Λ Q G. hal , verson 2-3 Dec 29 Theorem 2 and corollary 3 wll be used n [Dem] for constructng 2-Calab-Yau categorfcatons of skew-symmetrzable cluster algebras. Another corollary, whch s an easy consequence of the defnton of the McKay graph, s : Corollary 4. Let Q be the quver and G a fnte subgroup of SLCα Cα. Then α α There s an dentfcaton of Q G wth a double quver Q such that the non orented underlyng graph of Q s somorphc to the affne Dynkn dagram correspondng to G through the McKay correspondence. 2 We have kq/αα α α k[α,α ] and there s an equvalence of categores mod k[α,α ]G modλ Q. Corollary 4 was proved by a geometrcal method n [RV, proof of proposton 2.3] see also [CBH, theorem.]. Table. Examples of computatons of Q G Q G Q G 2... n 2... n α β n Z/2Z G SL Cα Cβ type A n n n 2... n n + n

3 2. An example Suppose that k = C and that Q s the followng quver α β β α γ γ 2 δ δ hal , verson 2-3 Dec 29 Let also 3 G = a,b a 3 = b 2,b 4 =,aba = b be the bnary dhedral group of order 2. One lets G act on kq by : e e e 2 e 3 α α β β γ γ δ δ a e e 2 e 3 e ζ α ζα γ γ δ δ β β b e e e 3 e 2 α α β β δ δ γ γ where ζ s a prmtve sxth root of unty. Usng the notaton of the ntroducton, one can choose Ĩ = {,}, κ = κ =, κ 2 = a, κ 3 = a 2. One has G = G, G = b Z/4Z, G 2 = ba Z/4Z, G 3 = ab Z/4Z. One can also choose F, = {,}, F, = {,}, F, = {,} and F, = {,,,2,2,}. The rreducble representatons of Z/4Z wll be denoted by θ α where α {,,,} s the scalar acton of a specfed generator b, ba or ab when Z/4Z s realzed as G, G 2 or G 3. The group G has sx rreducble representatons : four of degree of the form a α 2, b α for each α {,,,}, whch wll be denoted by λ α, and two of degree 2 : and ρ : a σ : a ζ ζ ζ 2 ζ 2 b b One checks easly that λ ρ σ, ρ ρ σ λ λ, ρ σ ρ λ λ, σ σ σ λ λ. The other product formulas are deduced from these. One computes M, = ρ, M, = M, = λ and M, = M,2 = M 2, =. The vertces of Q G are then,,,, ρ, σ, and where we wrte α =,λ α and α =,θ α for smplcty. One has and therefore there s one arrow from ρ to σ, and therefore there s no arrow from to σ, and therefore there s one arrow from to σ, Hom G ρ,σ ρ Hom G ρ,ρ λ λ C Hom G λ,σ ρ Hom G λ,σ ρ C Hom Z/4Z θ,σ Z/4Z θ Hom Z/4Z θ,θ θ and therefore there s no arrow from to σ, Hom Z/4Z θ,σ Z/4Z θ Hom Z/4Z θ,θ θ C and therefore there s one arrow from to σ, Hom Z/4Z θ,λ Z/4Z θ C 3

4 and therefore there s one arrow from to. All the other computatons can be done n the same way. Fnally, Q G s the followng quver : ρ σ hal , verson 2-3 Dec 29 where one can remark that the full subgraph havng vertces {,,,, ρ, σ } s the affne Dynkn dagram correspondng to G n the McKay correspondence, as expected. Hence modcqg mod CQ G. Moreover, t s easy to check that the preprojectve relaton αα α α + ββ β β + γγ γ γ + δδ δ δ s stable under the acton of G and therefore there s an equvalence of Morta between Λ Q G and Λ Q2 where Q = Q, Q 2 = Q G, and Λ Q, Λ Q2 are the preprojectve algebras of Q and Q Proofs of the man propostons 3.. Proof of theorem. We retan the notaton of secton. Thus Q s a quver, G acts on kq by stablzng the set of prmtve dempotents correspondng to vertces and the vector space spanned by the arrows, Ĩ s a fxed set of representatves of the G-orbts of I, etc.. Let R be the subalgebra of kq generated by the prmtve dempotents and M kq be the lnear subspace spanned by the arrows, seen as an R-bmodule. Defne T = R and for every postve nteger n, T n = T n R M. Then, recall that the tensor algebra TR,M s T endowed wth the canoncal product. It s clear that kq s canoncally somorphc to TR,M on whch the acton of G s graded. As G stablzes R and M, one can defne the skew-group algebra RG whch s a subalgebra of kqg. Thus, MG = M k k[g] s a sub-rg-bmodule of kqg and one gets easly a canoncal somorphsm kqg TR,MG TRG,MG whch maps m m 2... m n g to m m 2... m n m n g = m m 2... m n g m g n = = m m 2 g... m g n m g n = m g m g 2... m g n m g n. Let now e = I ee R RG whch s an dempotent. Then, accordng to [RR, proposton.6] and ts proof, mod RG mod erge N en RGe erge N N s a Morta equvalence between RG and erge. Hence, t s a classcal and easy fact that there s an equvalence of categores modkqg mod TRG,MG mod TeRGe,eMGe N en RGe erge N N 4

5 hal , verson 2-3 Dec 29 usng the somorphsm MG RGe erge emge erge erg. One has erge I ek[g ] where, for each Ĩ, G s the stablzer of e. As G s sem-smple, one can fx, for each Ĩ and ρ rrg, ẽ ρ to be a prmtve dempotent of k[g ] correspondng to ρ. Let ẽ = e I ρ rrg ẽρ whch satsfes eẽe = ẽ. Then we have a Morta equvalence between erge and ẽergeẽ = ẽrgẽ, and, as before, between TeRGe,eMGe and TẽRGẽ,ẽMGẽ. Moreover, usng the notaton I G of the ntroducton, ẽrgẽ,ρ I G kẽ ρ and therefore, t s enough to compute ẽ jσ MGẽ ρ = ẽ jσ e j MGe ẽ ρ for each,ρ and j,σ n I G to fnsh the proof of theorem. Remark now that e j MGe = G j κ j M j κ G = G j κ j M j κ,j O O j,j F j and therefore, f one denotes G j = G G j for every,j I, ẽ jσ e j MGe ẽ ρ Hom k k,ẽjσ G j κ j M j κ G ẽ ρ,j F j Hom k[g ] ρ,σ k[gj ] G j κ j M j κ G,j F j Hom k[g ] ρ κ,σ κ j k[gj ] G j M j G,j F j Hom k[g ] ρ κ,σ κ j k[gj ] k[g j ],j F j k[g j ]G j M j G j k[g j ] k[g ] and, because of the relatons defnng kqg, the multplcaton n kqg nduces an somorphsm of G j,g j -bmodules k[g j ] k M j G j M j G j. Note that the acton of G j,g j on k[g j ] k M j s defned here by gv mh = gvh h mh = gvh m h recall that we use the multplcaton notaton for the product n kqg and the exponental notaton for the acton of G on kq fxed at the begnnng. Therefore ẽ jσ e j MGe ẽ ρ Hom k[g ] ρ κ,σ κ j,j F j k[g j ] k[g j ] k M j k[g j ] k[g ] Hom k[g ] ρ κ, σ κ j G j k M j k[g j ] k[g ],j F j Hom k[g j ] ρ κ G j,σ κ j G j k M j,j F j where the representaton ρ κ of G s the same as ρ as a vector space and, f g G = κ G κ, ρ κ g = ρ κ gκ. It concludes the proof of theorem. Note that we go from the penultmate lne to the last one by the classcal adjuncton between nducton and restrcton of representatons. G 3.2. Proof of theorem 2. We retan the notaton of secton 3.. Defne the R-bmodule M = M M the R-bmodule structure on M s the natural one : afbm = fbma for a,b R, f M and m M. The tensor algebra TR,M s the path algebra of the double quver Q of Q. The non-degenerate skew-symmetrc blnear form defned on M by m + f,m + f = f m fm where m,m M and f,f M satsfes, for every a,b R and m,n M, amb,n = m,bna. If {x } S s a k-bass of M, then denote by {x } S ts left dual bass for the blnear form, that s the one satsfyng x,x j = δ j for every 5

6 ,j S. Then the element r = S x x M R M s ndependent of the choce of the bass. By defnton, r s the preprojectve relaton correspondng to Q and the preprojectve algebra Λ Q of Q s TR,M/r. For more detals about preprojectve algebras, see for example [DR] or [Rn]. Suppose now that the group G acts on kq by stablzng the set of prmtve dempotents correspondng to vertces and the k-subspace spanned by the arrows. Then t stablzes r f and only f t stablzes the blnear form, ndeed, for g G, r g = r mples that {x g } S s the left dual bass of {x g } S snce for every S, Id M, r g x g = Id M, r xg = x g. Extend now, to a blnear form on MG by settng { m,n m g,n h = h f gh = else hal , verson 2-3 Dec 29 whch s clearly skew-symmetrc and non-degenerate. Moreover, for a,b RG and m,n MG, one has easly amb,n = m,bna. If {x } S s a bass of M and {x } S s ts left dual bass, then {x g g },g S G s the left dual bass of the bass {x g},g S G of MG. Hence, the preprojectve relaton r G correspondng to, n MG RG MG s r G = x g x g g =,g S G =,g S G,g S G x g x g g = x g x g g,g S G x x = #G r where the preprojectve relaton r of TR,M s mapped by the canoncal ncluson from M R M to MG RG MG. As #G s nvertble n k, one gets TR,M/r G TRG,MG/r = TRG,MG/rG. Moreover, for,, ẽm Gẽ and ẽm G + M G ẽ are orthogonal supplementary subspaces. Thus,, restrcts to a skew-symmetrc non-degenerate blnear form on ẽm Gẽ whch satsfes, for every a,b ẽrgẽ and m,n ẽmgẽ, amb,n = m,bna. By takng a bass of MG whch s the unon of a bass of ẽmgẽ and a bass of ẽmg+mg ẽ, t s clear that the equvalence of categores of secton 3. restrcts to an equvalence modkqg/r G mod TRG,MG/r G mod TẽRGẽ,ẽMGẽ/r e N ẽn RGẽ erge N N where r e s the preprojectve relaton n TẽRGẽ,ẽMGẽ k Q G kq. It completes the proof of theorem 2 t s enough to take a maxmal sotropc subspace of ẽmgẽ to fnd arrows of Q whch s of course non unque Proof of corollary 3. We retan the prevous notaton. If G acts on kq by stablzng the set of prmtve dempotents and M, then ts acton can be extended to an acton on M = M M usng the contragredent representaton on M. Then MG = MG M G as RGbmodules. Moreover, M G s clearly maxmal sotropc for the blnear form, extended on MG as before. Hence Q G = Q G and t proves corollary 3. Acknowledgments The author would lke to thank hs PhD advsor Bernard Leclerc for hs advces and correctons. He would also lke to thank Chrstof Geß, Bernhard Keller and Idun Reten for nterestng dscussons and comments on the topc. He s also grateful to the referee for helpful suggestons for shortenng and readablty of ths artcle. 6

7 References [CBH] Wllam Crawley-Boevey and Martn P. Holland. Noncommutatve deformatons of Klenan sngulartes. Duke Math. J., 923:65 635, 998. [Dem] Laurent Demonet. Categorfcaton of skew-symmetrzable cluster algebras. arxv: [DR] Vlastml Dlab and Claus Mchael Rngel. The preprojectve algebra of a modulated graph. In Representaton theory, II Proc. Second Internat. Conf., Carleton Unv., Ottawa, Ont., 979, volume 832 of Lecture Notes n Math., pages Sprnger, Berln, 98. [Rn] Claus Mchael Rngel. The preprojectve algebra of a quver. In Algebras and modules, II Geranger, 996, volume 24 of CMS Conf. Proc., pages Amer. Math. Soc., Provdence, RI, 998. [RR] Idun Reten and Chrstne Redtmann. Skew group algebras n the representaton theory of Artn algebras. J. Algebra, 92: , 985. [RV] Idun Reten and Mchel Van den Bergh. Two-dmensonal tame and maxmal orders of fnte representaton type. Mem. Amer. Math. Soc., 848:v+72, 989. LMNO, Unversté de Caen, Esplanade de la Pax, 4 Caen E-mal address: Laurent.Demonet@normalesup.org hal , verson 2-3 Dec 29 7