Chapter 18 Coxeter Transformations associated with

Progress in Mathematics, Vol. 173, © 1999 Birkhliuser Verlag BaseVSwitzerland

Chapter 18 Coxeter Transformations associated with Finite Dimensional Algebras Helmut Lenzing Abstract

Let A be a finite dimensional algebra of finite global dimension. On the K-theoretic level the Auslander-Reiten translation on the bounded derived category of A-modules induces the Caxeter transformation which is an important invariant, preserved under derived equivalence, and displaying a lot of interesting homological information. The paper deals with the ramifications of this concept, stresses its homological nature and the computational aspects with a focus on hereditary and canonical algebras.

18.1

Introduction

For a finite dimensional k-algebra A of finite global dimension the Coxeter transformation is the automorphism A of the Grothendieck group K o (A) which is induced by the Auslander-Reiten translation TV of the bounded derived category V of A-modules. The Coxeter transformation preserves the homological bilinear form (Euler form) on K o (A). Its characteristic polynomial, the Coxeter polynomial XA, further controls the growth behaviour of A, hence of TV. The Coxeter transformation (and associated polynomials) form important invariants llllder derived equivalence but also provide natural links between the representation theory of finite dimensional algebras and other theories, notably the theory of Lie algebras (classical and Kac-Moody) [4], the theory of C*-algebras [15], the spectral theory of graphs [6] and also to number theory and computing [24, 5]. The Coxeter-data are moreover easy to calculate from the Cartan matrix of A, whose entries are the dimensions of the Hom-spaces between the indecomposable projective A-modules. On the other hand, except for particular classes of algebras, fairly little is known on the general structure of the Coxeter transformation and its characteristic polynomial. Throughout we work over an algebraically closed field k. The algebras, we consider, are associative with a unit element, and moreover finite dimensional over k if not stated otherwise. For a finite dimensional k-algebra A we consider usually finite dimensional right modules and denote by mod (A) the category of such modules. The ordinary k-duality Homk (-,k) will be denoted by D. 287

288

18. Coxeter Transformations associated with Finite Dimensional Algebras

The stable module category mod(A) modulo projectives (resp. mod (A) modulo injectives) has the same objects as mod (A), its morphism spaces are given as the factor spaces Hom(X, Y) (resp. Hom (X, Y)) of HomA (X, Y) of A-linear maps modulo morphisms factorizing through a projective (resp. an injective) A-module. Important for the finite dimensional representation theory is the concept of an almost-split sequence 0 ---. A ---. B ---. G ---. 0 in mod (A) and the associated concept of the Auslander-Reiten translation(s) TA and TAl relating the end terms G = TA A and A = TAG of almost-split sequences. Actually, TA is a functor from mod (A) to mod (A) with inverse TAl: mod (A) ---. mod (A). The two functors are uniquely determined by the validity of Auslander-Reiten duality

) = DExt A I (X, Y) = -HomA (Y,TAX) HomA (TA I Y,X to be interpreted as isomorphisms which are functorial in X E mod (A) and Y E mod (A). As general references to the subject we refer to 11], [Il] and [36]. By Jordan-Halder's Theorem, the Grothendieck group K o (A) of mod (A) modulo short exact sequences is the free abelian group on the classes [SI], [8 2 ], •.. , [8 n ] of simple A-modules. The class [M] of a A-module equals [M] = L~I [M : Si] lSi], where [M : 8 i ] denotes the multiplicity ofthe simple module Si in a composition series of M. In order to achieve a sensible K-theory, we assume for the rest of the paper that A has finite global dimension, in which case the classes [PI], [P2 ], .•• ,[Pn ] of indecomposable projective A-modules form another natural basis of Ko (A). A similar assertion holds for the classes [QI], [Q2], ... , [Qn] of indecomposable injective A-modules. We arrange the numbering in such a way that Pi (resp. Qi) is the projective cover (resp. injective envelope) of 8 i . The (usually non-symmetric) Euler form (-, -) on K o (A) is the homological bilinear form given on classes of A-modules by the formula 00

([X],[Y]) = L(-lrdimkExt~(X,y). i=O

We always consider K o (A) to be equipped with this additional structure, resulting in a bilinear lattice in the sense of [25]. Accordingly, isomorphisms between Grothendieck groups are assumed to preserve the respective Euler forms. By Schur's Lemma we get the dual basis formula ([Pi], [Sj]) = Dij, relating the classes of indecomposable projectives and simples, respectively, as dual bases with respect to the Euler form. In particular, the Euler form is non-degenerate and has determinant ±1. With respect to the basis of classes of indecomposable projective modules the Euler form is given by the Garlan matrix GA = (Cij), where Cij = ([Pi] , [Pj]) = dimk HomA (Pi, Pj ). Without going into technical details we review some basic facts on the bounded derived category Db (mod (A)) of A-modules; see [16] and [17] for further details. The main function of the derived category is to form the natural environment for the homological algebra for mod A. This can be seen, in particular, from the fact that V = Db (mod (A)) contains for each integer n E Z a copy (modA)[n] of mod (A), with objects written X[n], such that

Helmut Lenzing

Homv(X[n], Y[m])

= Ext~-n(x, Y)

289

for all X, Y E mod (A).

The derived category V carries the structure of a triangulated category, that is V is equipped with a translation functor T : V --> V such that T (X [n]) = X [n + 1] for each A-module X; moreover V disposes of a distinguished system of triangles X --> Y --> Z --> X [1]. Finally, a triangle X ~ Y ~ z !!:.. X [1] in V has all the terms X, Y and Z in mod (A) if and only if f1 : 0 --> X ~ Y ~ Z --> 0 is an exact sequence of A-modules. This latter fact implies that the inclusion mod (A) '----' V, X f---> X [0], induces an isomorphism between the Grothendieck group K o (A) and the Grothendieck group K o (V) of the triangulated category V modulo distinguished triangles. Moreover, this isomorphism maps the Euler form of K o (A) to the Euler form on K o (V) given on classes of objects from V by the expression ([X], [Y]) = LnEZ (-It dimk Homv (X, Y [n]). We are thus going to identify K o (A) and K o (V) with their structure of Euler forms from nowon. A distinguished triangle A --> B --> C --> A [1] is called an Auslander-Reiten triangle if its "end terms" A and C are indecomposable, and further each nonisomorphism A --> X into an indecomposable object X of V extends to B or, equivalently, each non-isomorphism Y --> C from an indecomposable object Y of V lifts to B. The Auslander-Reiten translation TV of V carries C to A. If A has finite global dimension, the bounded derived category Db(mod A) of A-modules has Auslander-Reiten triangles [19]. Moreover, the corresponding Auslander Reiten-translation TV : Db(mod A) --> Db(mod A) is an equivalence of categories preserving the triangulated structure. Auslander-Reiten duality, in the context of the derived category, takes the form DHomv (X, Y [1]) = Homv (Y, TVX) for all X, Y E V. Note that this adjunction formula determines the functor TV. We are now in a position to introduce the object of our study properly. The Coxeter transformation is an automorphism

-->

(v,w) - (V,W)Sl' we get a commutative

V

----+

0

17' V

----+

0

(7: T~)' Now pass to characteristic polynomials.

o

Lemma 18.3. Assume that V2 is a bilinear lattice of rank 2 with non-zero radical and having a root. Then V2 = Za EB Zw, where a is a root and W generates the radical of V. The bilinear form is given with regard to the basis a, W by the matrix (_\ 6), accordingly T(a) = a - 2w, T(W) = w. Moreover v E V is a root if and only if v = ±(a + nw), n E Z. Definition 18.1. A bilinear lattice V arising from V2 , by an attachment of tubes T 1 , ... ,Tt with identical axes for a I-tube Z. w of V is called a canonical bilinear lattice with of type PI, ... ,Pt, where Pi denotes the rank of T;.

The next statement provides a normal form for canonical bilinear lattices. Proposition 18.5. If V is a canonical bilinear lattice of type PI, ... ,Pt, then

a,

T

j

Si (1 ::; i ::; t, 0::; j ::; Pi - 2), w,

(18.10)

is a basis of V, where (a, a) and (a,Tjsi)

= 1,

(a, w)

= 1,

= 0 for 0 < j:S. Pi -

(a, Si)

= 1,

(Si' Si)

= 1,

(w, w)

= 0,

2.

The next statement (cf. [29]) implies that the type of a canonical bilinear lattice is actually an invariant, and shows that the spectrum of its Coxeter polynomial consists of roots of unity. Proposition 18.6. Let V be a canonical bilinear lattice of type PI, ... ,Pt, then the Coxeter polynomial of V is given by t

Xv(X) = (X -

I? IT

1 X-I .

XPi _

i=l

In particular, rad(V) has rank one or two. Proof. Immediate from Corollary 18.3.

o

Proposition 18.7. A bilinear lattice is canonical if and only if it arises as the Grothendieck group Ko (A) of a canonical algebra A (or an algebra E derivedequivalent to a canonical algebra).

297

Helmut Lenzing

Proof. Assume that A is a canonical algebra. Then mod (A) has a separating tubular family (Tx ) of standard stable tubes, indexed by lP'1 (k) [36, 12]. Let p(x) denote the rank of Tx , i.e. the number of indecomposables in the mouth of the tube Ix, let S be a quasi-simple module from a tube with p(x) = 1, and put w = [5]. Since K o (A) as a Z-module has finite rank, only for finitely many points x, say for XI,'" ,Xt, we can have p(x) > 1. Next we pick an indecomposable module L of rank ([L] ,w) = 1, and for each Ix i a quasi-simple 5 i with Hom (L, 5i ) =I- O. Putting a = [L], Si = lSi], Pi = p(Xi) does the job in view of Proposition 18.5. 0 To dispose of an explicit form for the Coxeter polynomial is also interesting in view of a recent result of Happel [20], stating (for an algebra A of finite global dimension) that the alternating sum of the dimensions of the Hochschild cohomology groups W(A)) equals the trace of the Coxeter transformation. In particular, the above formula for Xv yields that a derived canonical algebra has H2 (A) =I- 0 if t 2 4. 18.4.2 Riemann-Roch Formula and Genus V denotes a canonical lattice, and we keep the previous notation. We define the rank function as rk = (-, w) and introduce an average on the bilinear form setting p-I

((x,Y))

=

p-I

~)TjX,y) = ~)X,T-jy). j=O

j=O

Fixing a root a of rank one from V2 C V, we define the degree function through degx = ((a,x)) - rk(x) ((a,a)). Note that degw = 0,

degw

= p,

. p degTJsi = - , Pi

in particular, deg is T-stable on Vo = ker[V ~ Z]. Lemma 18.4.

(i) ((a,TX)) - ((a,x))

= 8 [V] rk (x)

for each

X

E V.

(ii) 2((a,a)) - p8[V]. Theorem 18.2 (Riemann-Roch). For all x, y from a canonical bilinear lattice V we get

1 . 1 -((x,Y))=ll-g[V])rkxrky+p p

where g[V] is given by g[V]

=

1 + ~8[V]

= 1+ ~

I

drkx egx

drky egy

((t - 2) -

I'

2:;=1 *).

298

18. Coxeter Transformations associated with Finite Dimensional Algebras

Proof. In view of the definition of the degree and of Lemma 18.4, the formula is satisfied for x = a. For x, y from the set formed by wand all T j Si both sides 0 evaluate to zero. Observing ((y,x)) = -((X,TY)) finishes the proof.

Accordingly g[V] is called the genus of V or its corresponding quadratic form. For g[V] < 1 (according to 8[V] < 0) resp. g[V] = 1 (according to 8[V] = 0) we call V domestic (resp. tubular). If x is in V and degx =/=- 0 or rkx =/=- 0, we call f1(x) = degxlrkx the slope of x. Proposition 18.8. Let V be a canonical bilinear lattice. (i) If 8[V] < 0, then the radical of V has rank one, and qv is positive semidefinite. (ii) If 8[V] = 0, then the radical of V has rank two, and qv is positive semidefinite. (iii) If 8[V] > 0, then the radical of V has rank one, and qv is indefinite. Proof. Assume first that 8[V] =/=- 0, and x E rad(V) has non-zero rank, so that the slope f1(x) = deg x Irk x is defined. Then f1( TX) = f1(x) + 8[V], contradiction. Hence rad(V) belongs to the subgroup Va of elements of rank zero. Since rad(V) n Va = Z.w, rad(V) has rank one in cases (i) and (iii). If 8[V] = 0, then T P = 1, hence u = I:j:6 T j a is a non-zero T-stable element of rank p. In view of Proposition 18.6 rad(V) thus has rank two. Concerning the definiteness of qv, we only deal with (iii), i.e. assume 8[V] > O. In view of the Riemann-Roch formula, the element u = I:j:6 Tj a satisfies qv(u) 1 than the Coxeter polynomial

of the wild hereditary tree [2,3,7], whose Mahler measure agrees with the spectral radius P[2,3,7] ~ 1.1762808 ... of its Coxeter transformation. Note in this context that computer calculations [5] have confirmed the conjecture for such polynomials up to degree 20. Further C.C. Xi has shown [40] that P[2,3,7] is a lower bound for the spectral radii, hence for the Mahler measures, of all wild hereditary algebras (arbitrary base fields allowed). It would be interesting to confirm Lehmer's conjecture for the subclass of Coxeter polynomials of algebras or bilinear lattices with spectral radius> 1.

306

Bibliography

Bibliography [1] M. Auslander, 1. Reiten, S. O. Smal0: Representation Theory oj Arlin Algebras, Cambridge Studies in Advanced Mathematics. 36, Cambridge University Press, Cambridge (1995). [2] D. Baer, W. Geigle, H. Lenzing: The preprojective algebrn oj a tame hereditary Arlin algebrn, Comm. Algebra, 15, 425-457 (1987). [3] A. Boldt: Methods to determine Coxeter polynomials, J. Linear Algebra Appl., 230, 151-164 (1995). [4] N. Bourbaki: Groupes et Algebres de Lie, chapitres 4,5 et 6. Hermann, Paris (1968).

[5] D.W. Boyd: Reciprocal polynomials having small measure II, Math. Comput., 53, 355-357 (1989). [6] D. Cvetkovic, M. Doob, H. Sachs: Spectrn oj grnphs, Academic Press, 1980.

[7] V. Dlab, C. M. Ringel: Indecomposable Representations oj Grnphs and Algebrns, Mem. Am. Math. Soc. 173 (1976). [8] V. Dlab, C. M. Ringel: Eigenvalues oj Coxeter trnnsJormations and the GelJand-Kirillov dimension oj the preprojective algebrns, Proc. Am. Math. Soc., 83, 228-232 (1981). [9] P. Dowbor, T. Hubner: A Computer Algebrn approach to sheaves over weighted projective lines, these Proceedings. [10] P. Driixler, R. Norenberg (eds.): CRE? Manual Parl II, Ergiinzungsreihe 97-009 des SFB 343, Universitiit Bielefeld (1997).

[11] P. Gabriel, A.V. Roiter: Algebra VIII: Representations oj Finite-dimensional Algebrns, Encyclopaedia of Mathematical Sciences 73, SpringerVerlag, Berlin (1992). [12] W. Geigle, H. Lenzing: A class oj weighted projective curves arising in representation theory oj finite dimensional algebras, in: Singularities, representations of algebras, and vector bundles, Lect. Notes Math., 1273, 265-297 (1987). [13] W. Geigle, H. Lenzing: Perpendicular categories with applications to representations and sheaves, J. Algebra, 144, 273-343 (1991). [14] 1. M. Gelfand, V. A. Ponomarev: Model algebras and representations oj grnphs, Funct. Anal. Appl., 13,157-166 (1979). [15] F.M. Goodman, P. de la Harpe, V.F.R. Jones: Coxeter Grnphs and Towers oj Algebras, Springer-Verlag New York-Berlin-Heidelberg, 1989.

Helmut Lenzing

307

[16] P.P. Grivel: Categories derivees et foncteurs derives, in: Algebraic Dmodules, Perspectives in Math., 2, Academic Press 1987. [17] D. Happel: Triangulated Categories in the Representation Theory of finite dimensional Algebras, LMS Lecture Note Series 119, Cambridge 1988. [18] D. Happel: Hochschild cohomology of finite-dimensional algebras, in: Seminaire d'algebre P. Dubreil et M.-P. Malliavin. Lect. Notes Math.,; 1404, 108-126 (1989). [19] D. Happel: Auslander-Reiten triangles in derived categories of finite dimensional algebras, Proc. Amer. Math. Soc., 112, 641-648 (1991). [20] D. Happel: The trace of the Coxeter matrix and Hochschild cohomology, Linear Algebra App!., 258,169-177 (1997). [21] D. Happel, I. Reiten, S. Smal0: Tilting in Abelian Categories and Quasitilted Algebras, Mem. Am. Math. Soc. 575, (1996). [22] D. Happel, C. M. Ringel: Tilted Algebras, Trans. Am. Math. Soc., 274, 399-443 (1982). [23] O. Kerner: Tilting wild algebras, J. London Math. Soc., 39, 29-47 (1989). [24] D.H. Lehmer: Factorization of certain cyclotomic functions, Ann. Math., 34,461-479 (1933). [25] H. Lenzing: A K-theoretic study of canonical algebras, CMS Conf. Proc., 18, 433-454 (1996). [26] H. Lenzing: Hereditary noetherian categories with a tilting complex, Proc. Am. Math. Soc., 125, 1893-1901 (1997). [27] H. Lenzing, H. Meltzer: Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, Representations of algebras, Sixth International Conference, Ottawa 1992. CMS Conf. Proc., 14, 313-337 (1993). [28] H. Lenzing, H. Meltzer: Tilting sheaves and concealed-canonical algebras, Representations of algebras, Seventh International Conference, Cocoyoc (Mexico) 1994. CMS Conf. Proc., 18,455-473 (1996). [29] H. Lenzing, J. A. de la Peiia: Wild canonical algebras, 403-425 (1997).

Math. Z., 224,

[30] H. Lenzing, J. A. de la Peiia: Concealed-canonical algebras and algebras with a separating tubular family, Proc. London Math. Soc. (to appear). [31] H. Lenzing and A. Skowronski: Quasi-tilted algebras of canonical type, Colloq. Math., 71,161-181 (1996).

308

Bibliography

[32] F. Lukas: Elementare Moduln tiber wilden erblichen Algebren, Dissertation Dusseldorf 1993. [33] H. Meltzer: Tubular mutations, Colloq. Math., 74, 267-274 (1997). [34] J. A. de la Pena: Coxeter transformations and the representation theory of algebras, in: V. Dlab and L.L. Scott (eds.), Finite Dimensional Algebras and Related Topics, 222-253, Kluwer 1994. [35] J. A. de la Pena, M. Takane: Spectral properties of Coxeter transformations and applications, Arch. Math., 55, 120-134 (1990). [36] C. M. Ringel: Tame Algebras and Integral Quadratic Forms, Lect. Notes Math. 1099, Springer, Berlin-Heidelberg-New York (1984). [37] C. M. Ringel: The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Math. Ann., 300, 331-339 (1994). [38] M. Takane: On the Coxeter transformation of a wild algebra, Arch. Math., 63,128-135 (1994). [39] G. Wilson: The Cartan map on categories of graded modules, J. Algebra, 85, 390-398 (1983). [40] Ch. Xi,: On wild hereditary algebras with small growth numbers, Commun. Algebra, 18, 3413-3422 (1990).

[4:1.] D. Zacharia: On the Cartan matrix oj an algebra oj global dimension two, J. Algebra, 82, 353-357 (1983). [42] Y. Zhang: Eigenvalues oj Coxeter transformations and the structure of regular components oj an Auslander-Reiten quiver, Comm. Algebra, 17, 23472362 (1989).