.
TAME AND WILD WORKSHOP
List of Participants, Schedule and Abstracts of Talks
Department of Mathematics Uppsala University . TAME AND WILD WORKSHOP
Department of Mathematics Uppsala University Uppsala, Sweden November 26-28, 2004
Supported by • The Swedish Foundation for International Cooperation in Research and Higher Education (STINT) • The Swedish Research Council
Within the framework of the joint research project “Representation theory of algebras and applications” of the • Department of Functional Analysis, Institute of Mathematics of the Ukrainian Academy of Science; • Department of Algebra and Mathematical Logics, Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University; • Department of Mathematics, Gothenburg University and Chalmers University of Technology; • Department of Mathematics, Uppsala University.
1 Organizers:
• Volodymyr Mazorchuk, Department of Mathematics, Uppsala University;
• Lyudmila Turowska, Department of Mathematics, Gothenburg University and Chalmers University of Technology.
2 List of Participants:
1. Jan Adriaenssens, University of Antwerp, Belgium
2. Raf Bocklandt, University of Antwerp, Belgium
3. Lesya Bodnarchuk, University of Kaiserslautern, FRG
4. Igor Burban, Institut de Mathematiques de Jussieu, France
5. Michael C R Butler, University of Liverpool, UK
6. Paolo Casati, II University of Milano, Italy
7. Ernst Dieterich, Uppsala University, Sweden
8. Vlastimil Dlab, Carleton University, Canada
9. Elena Drozd, University of California at Berkeley, USA
10. Yuriy Drozd, Kyiv University, Ukraine/ Uppsala University, Sweden
11. Karin Erdmann, Oxford University, UK
12. Anders Frisk, Uppsala University, Sweden
13. Gert-Martin Greuel, University of Kaiserslautern, FRG
14. Martin Herschend, Uppsala University, Sweden
15. Josef Jirasko, Czech Technical University, Prague, Czech Republic
16. Peter Jorgensen, Leeds University, UK
17. Ekaterina Jushchenko, Kyiv University, Ukraine
18. Dirk Kussin, University of Paderborn, FRG
19. Steffen K¨onig, University of Leicester, UK
20. Helmut Lenzing, University of Paderborn, FRG
21. Viktor Levandovskiy, University of Kaiserslautern, FRG
22. Lars Lindberg, Uppsala University, Sweden
23. Volodymyr Mazorchuk, Uppsala University, Sweden
3 24. Sergiy Ovsienko, Kyiv University, Ukraine
25. Claus Michael Ringel, Bielefeld University, FRG
26. Sibylle Schroll, Oxford University, UK
27. Vladimir Sergeichuk, Institute of Mathematics, Kyiv, Ukraine
28. Sergei Silvestrov, Lund Institute of Technology, Sweden
29. Daniel Simson, Nicholas Copernicus University, Torun, Polen
30. Sverre Smalø, NTNU Trondheim, Norway
31. Alexander Stolin, Gothenburg University, Sweden
32. Stijn Symens, University of Antwerp, Belgium
33. Lyudmila Turowska, Chalmers University of Technology, Gteborg, Sweden
34. Leonid Vainerman, University of Caen, France
35. Geert Van de Weyer, University of Antwerp, Belgium
36. Adrian Williams, Imperial College, London, United Kingdom
4 Schedule of Talks
Saturday, November 27th, 2004.
09.00-09.50 Greuel On the classification of vector bundles on curves of arithmetic genus 1 10.00-10.50 Lenzing Tubular and elliptic curves 11.00-11.30 Coffee-break 11.30-12.20 Erdmann Tame self-injective algebras 12.30-14.00 Lunch 14.00-14.50 Smalø Some local Ext-limits do not exist 15.00-15.50 K¨onig Polynomial functors and Schur algebras 16.00-16.30 Coffee-break 16.30-17.20 Vainerman On operator-algebraic quantum groups of different types 17.30-18.20 Ovsienko On exact categories
Sunday, November 28th, 2004.
09.00-09.50 Burban Derived category of an irreducible projective curve of arithmetic genus one 10.00-10.50 Ringel Controlled Wildness 11.00-11.30 Coffee-break 11.30-12.20 Dieterich Some normal form problems arising in the classification theory of real division algebras 12.30-14.00 Lunch
5 ABSTRACTS
Derived category of an irreducible projective curve of arithmetic genus one
Igor Burban
In my talk based on my joint work with Bernd Kreussler I shall discuss properties of the derived category of coherent sheaves on a projective curve of arithmetic genus one, comparing common features and pointing out some principal differences between the case of a smooth and a singular curve and giving in particular some ideas about the behavior of the derived category in families. I am going to describe the group of exact auto-equivalences, the set of all t-structures, the set of spherical objects of the bounded derived category of a singular Weierstrass curve. The technique of Fourier-Mukai transforms will be then applied to obtain a combinatorial description of indecomposable semi-stable sheaves of degree zero on a nodal Weierstrass curve. Moreover, it gives another point of view on the classification of indecomposable objects of the derived category of coherent sheaves on a nodal Weierstrass curve, which was obtained in my joint work with Yuriy Drozd.
Institut de Mathematiques de Jussieu, France
Some normal form problems arising in the classification theory of real division algebras
Ernst Dieterich
By a real division algebra we mean a real vector space A, endowed with a bilinear multiplication A × A → A, (x, y) 7→ xy, such that 0 < dim A < ∞ and xy = 0 only if x = 0 or y = 0. Famous theorems assert that (R, C, H) classifies all associative real division algebras (Frobenius 1878), (R, C, H, O) classifies all alternative real division algebras (Zorn
6 1931), and that each real division algebra has dimension 1,2,4 or 8 (Hopf 1940, Kervaire, Bott and Milnor 1958). The problem of classifying all real division algebras of fixed dimension d ∈ {1, 2, 4, 8} is trivial in case d = 1, has lately been solved in case d = 2 by various mathematicians (Burdujan 1985, Gottschling 1998, H¨ubnerand Petersson 2002, Dokovi´cand Zhao 2004, Dieterich 2004), and attracted increasing interest in the cases d = 4 and d = 8, where to date only partial solutions are known. In my talk I will indicate how some parts of this classification problem conceptually can be reduced to normal form problems which are strongly reminiscent of wild matrix problems, and yet admit explicit solutions in terms of n-parameter families, where n = 4, 9, 12,... .
Uppsala universitet, Matematiska institutionen, Box 480, SE-751 06 Uppsala, Sweden
Tame self-injective algebras
Karin Erdmann
Let A be a finite-dimensional tame self-injective algebra over an algebraically closed field. We discuss the structure of the stable Auslander-Reiten quiver and consequences for classifying some families of such algebras. As a new result, we deduce from existing litera- ture that the stable Auslander-Reiten quiver of A does not have a component isomorphic to ZA∞.
Oxford University, UK
7 On the classification of vector bundles on curves of arithmetic genus 1
Gert-Martin Greuel
I shall reviev some results about the classification of vector bundles on reduced pro- jective curves. This problem becomes immedeately wild. It is however tame for a bigger class of curves if we restrict ourself to simple vector bundles. We report on new results about the classification of vector bundles on curves of arithmetic genus 1, mainly obtained by Lesya Bodnarchuk, a joint student of Yuriy Drozd and myself.
University of Kaiserslautern, Germany
Polynomial functors and Schur algebras
Steffen K¨onig
Polynomial functors have been used in topology, in K-theory and in group cohomology of (finite) general linear groups. They are closely related to polynomial modules of (infinite) general linear groups, that is, to modules over Schur algebras. Few of the rings, whose module categories are categories of polynomial functors, have been described explicitly. Drozd was the first to give a complete description of non-trivial examples. He conjectured a similar description to be true for a larger class of examples, and this conjecture has been verified (in joint work with Alexander Zimmermann).
University of Leicester, UK
8 Tubular and elliptic curves
Helmut Lenzing
Let k be an algebraically closed field. Let T be a tubular curve, that is, a weighted projective line of weight type (2,2,2,2), (3,3,3), (4,2,2) or (6,3,2). It is known for a long time that the category of coherent sheaves over T is very similar to the category of coherent sheaves over an elliptic curve.
Theorem. The category of coherent sheaves on a tubular or an elliptic curve is a heredi- tary, noetherian, Hom-finite k-category with Serre duality such that the Auslander-Reiten translation τ has finite period. Conversely, each such category is equivalent to the category of coherent sheaves over a tubular or an elliptic curve, where the period of τ decides which of the cases happens. Moreover, we establish a natural map from (isomorphism classes) of tubular curves to (isomorphism classes) of elliptic curves. This map is surjective and generically bijective, and we relate the categories of coherent sheaves for corresponding curves.
University of Paderborn, Germany
On exact categories
Serge Ovsienko
In our talk we present a technique of calculation in some class of exact categories. Under some condition on the exact category C we construct the derived categories D±(C) and endow it with a structure of an A(∞)-category. The constructed derived categories are connected with some dualities, which generalize Ringel duality. We show, how to extend this notion to the wide class of associative algebras. As an application we consider exact categories, associated with stratifying systems of modules. In the talk we emphasize an important role of ideas of Yuriy Drozd for the development of our theory.
[BS] P. Balmer M. Sclichtling, Idempotent completion of triangulated categories, J. of Al- gebra, 236, No. 2 (2001), pp.819-834. [BB] W. B. Burt, M. C. R. Butler, Almost split sequences for Bocses, Canadian Math. Soc., Conference Proceedings, vol.11, 1991, 89–121.
9 [K] B. Keller, A(∞)-algebras in rerpesentation theory. Contribution to the Proceedings of ICRA IX, Beijing. [O1] S. A. Ovsienko, Bimodule and matrix problems, Progress in Mathematics, Vol. 173, Birkhaeuser, 1999, 325-357. [O2] S. A. Ovsienko, Boxes and quasi-hereditary algebras, Third international Algebraic Conference in Ukraine. Sumy, July 2-8, 2001, 84-87.
Kyiv University, Ukraine
Controlled Wildness Claus M. Ringel
The aim of the lecture is to compare different notions of wildness.
Bielefeld University, Germany
Some local Ext-limits do not exist Sverre Smalø
In this lecture it will be shown that for k a field, the four dimensional algebra Λ = khx, yi/hx2, y2, xy + qyxi when qn 6= 1, 0 for all n, ( which is a tame algebra) there exist a two dimensional module M and a family of two dimensional modules Mi, i = 1, 2, .. such i that dimk ExtΛ(M,Mj) = 1 for i = 0, j and j + 1 and zero otherwise. This is probably the easiest example giving a negative answer to a question raised by Maurice Auslander. Also connections between this question and some other homological conjecture will be given.
NTNU Trondheim, Norway
10 On operator-algebraic quantum groups of different types
Leonid Vainerman
We give an overview of operator-algebraic quantum groups obtained by the bicrossed product construction. Our aim is, in particular, to show concrete examples of these objects having all possible types of algebras in the sense of the classification of von Neumann algebras - from relatively simple which can be viewed as ”tame” to quite complicated which can be viewed as ”wild”.
Universite de Caen, France
11 POSTERS
Hyper-desingularization through Brauer-Severi Varieties
Raf Bocklandt, Stijn Symens, Geert Van de Weyer
The setting...
• X is a normal projective variety and A is a OX -order.
π • BS(A) ³ X is the Brauer-Severi scheme of A over X.
The loci in X...
Azu(A) the Azumaya locus of X: all fibers here are isomorphic to Pn−1 _ Flat(A) the flat locus of X for π: all fibers here are equidimensional _ Smooth(A) _ the smooth locus of X X
Two questions...
When is the flat locus near a sin- What are the fibers in points of gularity in X equal to the Azu- the flat locus near a singularity in maya locus? X?
The answers...
• The local structure around p ∈ X and the fiber of π in p is governed by the #(Qp)0 quiverP datum (Qp, αp, γp), with (Qp, αp) a quiver setting and γp ∈ N such that γ (v) = n. v∈(Qp)0 p
12 ˜ 1. construct the extended quiver setting (Qp, α˜p):
new vertex v0 original quiver setting (Qp, αp)
αGFED@ABC(v ) hhh 08 p 1 hhhhh γp(v1) arrowshhhh hhhhh hhhhh hhhhh hhhhh 1 VhV . . VVVVVV . . VVVVVV . . VVVγVp(vN ) arrows VVVVV VVVVV VVVVV VVV &. αGFED@ABCp(vN )
˜ 2. Assign to (Qp, α˜p) the character θp with θp(v0) = −n and θp(vi) = γp(vi), then a theorem of Le Bruyn yields
−1 ˜ ˜ π (p) = (Null(Qp, α˜p) ∩ ressθp (Qp, αp))/GLα˜p ˜ ˜ with ressθp (Qp, αp)) the variety of θp-semistable representations, Null(Qp, α˜p)
the variety of nilpotent representations and the action of GLα˜p on both varieties given by basechange.
• The possible quiver settings for a point p ∈ Flat(A) are a connected sum of sub- quivers of the form below, joined in a single vertex with dimension 1
1. either cyclic quiver settings A˜ and dimension vector 1; 2. a quiver setting of the form
'&%$ !"# / '&%$ / !"# z< d ··· d DD ; zz DD zz D" '&%$ !"#d bDo 1 o 2 DD zz D z| z 76540123d−1 o ··· o 76540123d−1
3. a quiver setting of the form '&%$ !"#C d ; ×× ×× '&%$ % !"#× . 1 e d [77 . 77 7 '&%$ !"#d 4. a quiver setting of the form 76540123 d@ −1 ÑÑ ÑÑ '&%$ % !"#Ñ . 1 e d ^== . == = 76540123d−1
13 5. or a connected sum of two cyclic quivers with dimension vector 2.1
• For p a singularity, Flat(A) = Azu(A) unless (Qp, αp) can be obtained from a con- nected sum C of cyclic quivers by replacing each vertex v with a quiver setting (Sv, αv) such that the arrows a from C have αv(h(a)) = αv(t(a)) = 1. For example
* 1 j / 1 (S , α ) O 1L 1
* 1 jo 1 (S , α ) O 2L 2
o * 1 j 1 (S3, α3)
In case Flat(A) 6= Azu(A) we have that q ∈ Flat(A) − Azu(A) has as local quiver setting Qq a connected sum of cyclic quivers with dimension vector 1. ˜ • Let p ∈ Flat(A) with (Qp, αp) a connected sum of k cyclic quivers Ani and αp = 1, then Null(Q, α) has (n1 + 1).....(nk + 1) irreducible components Ci, each of which −1 ˜ is a tree Ti. The irreducible components of π (p) then correspond to mossθp (Ti, 1).
3; 1 ooo oooo oooo oooo oo +3 1;;; 1 ;;; ;;; ;;; ;;; ;! 1 ˜ ;; An example of a (Ti, 1) ;; ;; ;; ;!) 1
• The fiber π−1(p) can now be described as iterated fiber bundles of graphs of projective
14 maps. Assign to each rooted subtree of T the series of projection maps
PDv Dv P (v) A ~ A ϕv ~ ϕv 1 A rv π1 ~ ... A ~ ~ A Q D D Dw 1 w1 P P w1 × ... × P rv w∈Q (v) ,D (w)=1 1 1 p 0 v (w ) (wr ) 1 < w1 v 1 1 ; w Õ < ϕr 1 w1 ; ϕ1 Õ 1 π2 ... < ... ; Õ < ; ÒÕ D 2 ; Q D wr Dw 2 w1 ; P P w1 × ... × P 1 ...... 2 2 w∈Qp(v)0,Dv(w)=2 (w ) (w ) 1 rw1 D { 1 D π3 { ... D { D { D ... ...... }{ ! ... x E x E x E πhv x ... E x{ E" Q D h D h Dw w v w v P P i × ... × P N w∈Qp(v)0,Dv(w)=hv hv hv (wi ) (wN )
Assign to each vertex w ∈ T the projective space PDw with X Dw = du − 1. '&%$ !"#w o o/ o/ o/ u and let Y w πi = πi
w∈Qp(v)0,Dv(w)=i−1 with Yrw w w πi = ϕj j=1
w Pj−1 where ϕ is the projection on the projective coordinates numbered from 1+ (D w + j k=1 ak Pj 1) to (D w + 1). We will denote the graph of this collection of rational maps k=1 ak by gr(v). Identify a rooted subtree of T with its root and let the rooted subtrees of T be
15 connected as follows:
v0 w HH ww HH ww HH ww HH ww H v01 ... v0r y GG 0 GG y GG GG yy GG GG yy GG GG yy G GG v ... v ...... 011 01r01J HH ww uu JJ HH ww uu JJ HH ww uu JJ HH ww uu JJ HH ...... w u ...... t JJ tt JJ tt JJ tt JJ tt J v...1 ... v...r...
That is, v0 has common subgraphs with v01 for 1 ≤ i ≤ r0 but not with any other w
for w 6∈ {v01, . . . , v0r0 }, likewise for v01, and so on. Now let
F = (... ((gr(v ) × gr(v )) × gr(v ) ... ) × gr(v )), 0 0 gr(v0∩v01) 01 gr(v0∩v02) 02 gr(v0∩v0r0 ) 0r0
F1 = (... ((F0 ×gr(v01∩v011) gr(v011)) ×gr(v01∩v012) gr(v012) ... )
×gr(v0r ∩v0r r )gr(v0r0r0r )), 0 0 0r0 0 ...
Fn = (... ((Fn−1 ×gr(v01...1∩v01...11) gr(v01...11)) ×gr(v01...1∩v01...12) gr(v01...12) ... ) × gr(v )), gr(v0r0 ∩v0r0r0r ...r0r r ... ) 0r0r0r0 ...r0r0r0r ... 0 0 0r0 0 −1 then the irreducible component of π (p) corresponding to T is equal to Fn. An example... Consider the point p with local quiver datum given by the quiver setting
local quiver setting γ
1J 2
1J 2
1 1
Then the fiber π−1 has four irreducible components, corresponding to
16 9 ? 1 9 ? 1O /* /* 1 ? 1 1 ? 1O ?? ?? ?? ?? ?? ?? ? ? 1 1 gr(P4 → P3 → P1) gr(P4 → P2 → P0) 9 ? 1 9 ? 1O /* /* 1 ? 1O 1 ? 1 ?? ?? ?? ?? ?? ?? ? ? 1 1
3 1 0 3 2 gr(P → P × P ) gr(P ×P1 P )
University of Antwerp, Belgium
On unitary representations and enveloping C∗-algebras of the group *-algebras of affine Coxeter groups.
Ekaterina Jushenko
Introduction. In the presented talk we study the unitary representations of semidirect products of d d the form Z o Gf , where Gf is a finite group and the action of Gf on Z is supposed to be faithful. The basic examples of the groups specified above are affine Coxeter groups.S If S = {s1, s2, ..., sn} is the set of generators and msi,sk : S × S −→ N {∞} such that
msi,si = 1, i = 1..n,
msi,sj > 1, for all distinct i and j,
ms ,s then W = G < s1, s2, ..., sn|(sisj) i j = e > is called a Coxeter group. A Coxeter group can be associated with the matrix K = (kij), where π kij = − cos , i, j = 1..n msi,sj
17 K is called Cartan’s matrix. Its known, see [Bou], that the Coxeter group W is finite if and only if K is positive definite. If all principal minors of K are positive and det K = 0, n−1 n−1 than W is a semidirect product of Z and finite group Gfin, W = Z o Gfin, in this case the group W is called affine Coxeter group. Definition. Let A be a ∗-algebra. C∗-algebra Ae with a ∗-homomorphism ϕ : A 7→ Ae is called enveloping C∗-algebra of the algebra A if for every representation π : A → B(H) of the algebra A there exists unique representation πe : Ae → B(H) of the algebra Ae such that the following diagram is commutative:
A ϕ @ π ? @ Ae -@R B(H) πe
The enveloping C∗-algebra of ∗-algebra C[G], where G is a group, is called a group C∗- algebra. In this paper we construct the general position representation, see definition below, of d d G = Z o Gf , Gf is finite and an action on Z is faithful. We show that family of general position representations has the following property, any irreducible representation of G is either general position representation or is its irreducible component. It will also be shown that family of general position representations determines a family R of continuous unitary d n × n matrix-functions defined on some compact F(Gf ) ⊂ T , where n = |Gf |. Using the property mentioned above we prove that the group C∗-algebra C∗(G) is isomorphic to the C∗-algebra generated by the family R. Finally we describe C∗(G) as algebra of continuous |Gf | × |Gf | matrix-functions on F(Gf ) satisfying some boundary conditions.
Induced representations. First we recall the construction of induced representations of a semidirect products G = H o Gf , where H is commutative and Gf is finite. b b Let H be a dual space of H. For any g ∈ Gf there exists automorphism of the group H defined by the rule: χg(h) = χ(hg), where hg = ghg−1. Fix a character χ ∈ Hb, denote via
Gχ = StGf (χ) the stabilizer of χ with respect to the action of Gf . Let π : Gχ → GL(V ) χ be irreducible representation of Gχ. Further, construct the representation π of H o Gχ by the following rule:
χ π (h, g) = χ(h)π(g), where h ∈ H, g ∈ Gχ. (1)
g1 g2 gk Let Oχ be an orbit of χ under the action of Gf , Oχ = {χ , χ , ..., χ }, k = |Gχ\Gf | and g1, ..., gk are representatives of the right cosets. Now define a representation T of G = H o Gf acting on the space of functions on Oχ with values in V by the following way:
gi gi −1 gig (T(h,g)f)(χ ) = χ (h)π(giggl )f(χ ), (2)
18 gi g gl where (χ ) = χ , g ∈ Gf , h ∈ H. The representation T of the group G constructed by (2) is called induced representation and will be denoted by Indπχ.
Theorem 1. (G. W. Mackey, see [M]) If G = H oGf , where H - abelian, Gf is finite, then every irreducible representation πe of the group G has the form πe = Indπχ, for some b character χ ∈ H and irreducible representation π of the group Gχ.
Let χ1 and χ2 be any fixed characters. Then the next proposition holds (see [M]):
χ1 χ2 Proposition 1. Representations Ind(π1 ) and Ind(π2 ) are equivalent, iff there exists 0 g0 g0 g0 0 0−1 g ∈ Gf such that χ2 = χ1 and π1 ∼ π2, where π1 (g) = π1(g gg ), ∀g ∈ Gχ2 .
Representation of the general position. Now we give the construction of general position representation. d bd d Denote via y1, y2,..., yd the generators of the group Z . Let Z ' T be the group of the d characters of Z , every character χ corresponds to the vector (ϕ1, ϕ2, ..., ϕd) = ϕ, where iϕj d d χ(yj) = e , j = 1..d. An action of Gf on Z induces an action on T by evident rule. d d d g P g g P g P g Namely, for every g ∈ Gf let yi = mjiyj, then ϕ = ( mj1ϕj, ..., mjdϕj). j=1 j=1 j=1 d Consider ϕ ∈ T as variable with independent over Z mod 2π components ϕi, i = 1, .., d. g Let O(ϕ) = {ϕ , g ∈ Gf }. Evidently, StGf (ϕ) =< e > and |O(ϕ)| = |Gf |. d cd Fix some ϕ0 ∈ T and denote by χϕ0 the corresponding character of Z . Further, consider the space, F , of C-valued functions defined on O(ϕ) and construct a representation d Tϕ0 of G = Z o Gf by the following formula: (T (h, g)f)(ϕs) = χs (h)f(ϕsg), g, s ∈ G , h ∈ H. ϕ0 ϕ0 f
d Note, that ∀ϕ0 ∈ T one has dim Tϕ0 = |Gf |. We call Tϕ0 the representation of the general χ position. If St (ϕ ) = G =< e >, then T is irreducible (in this case T = Ind(id ϕ0 ), Gf 0 χϕ0 ϕ0 ϕ0 where id is identity representation of < e >). Otherwise Tϕ0 is reducible and in the following
Theorem we present the decompositions of Tϕ0 on irreducible components.
Theorem 2. Let χ = χϕ0 and {πi, i = 1..s} be full system of irreducible representations χ d of Gχ with dimensions dim πi = ni. Let πi be an irreducible representation of Z o Gχ, defined by the rule: χ d πi (h, g) = χ(h)πi(g), h ∈ Z , g ∈ Gχ
Then Tϕ0 is unitary equivalent to the following direct sum of irreducible representations:
χ χ χ Tϕ0 ' n1Ind(π1 ) ⊕ n2Ind(π2 ) ⊕ ... ⊕ nsInd(πs ). (3)
Remark. It is easy to see that Tϕ0 (h, g) with fixed (h, g) ∈ G determines a continuous matrix-function on Td.
Enveloping C∗-algebra.
19 In this section we discuss a realization of the group C∗-algebra C∗(G) as a C∗-algebra of continuous matrix-functions on some compact satisfying a boundary conditions. The f results are illustrated by example G = C2. d Let F(Gf ) be a fundamental region of the finite group Gf in the space T . Namely, bd F(Gf ) ⊆ T is an open set satisfying the following properties:
1. If e 6= g ∈ Gf , then F(Gf ) ∩ gF(Gf ) = ∅.
d S 2. T = {gF(Gf ) | g ∈ Gf }.
The Mackey’s Theorem and Proposition 1 implies that every irreducible representation d χ πe of the group G = Z o Gf has the form πe = Indπ , for some character χ ∈ F(Gf ) and irreducible representation π of the group Gχ.
If ϕ0 ∈ F(Gf ) then StGf (ϕ0) =< e > and Tϕ0 is irreducible. For ϕ0 ∈ ∂F(Gf ) =
F(Gf )\F(Gf ) the decomposition of Tϕ0 is described in the Theorem 2. Denote by V (ϕ),
V (ϕ): ∂F(Gf ) → M|Gf |, the unitary matrix-function such that for any (h, g) ∈ G Ms ∗ χϕ V (ϕ)Tϕ(h, g)V (ϕ) = πi (h, g), see Theorem 2. i=1
χϕ Put ni(ϕ) = dim πi . d Theorem 3. Let G = Z o Gf , Gf is a finite group, then
∗ ∗ C (G) = {f ∈ C(F(G) → M|Gf |(C)) : V(ϕ)f(ϕ)V(ϕ) ∈
χ χ χ M ϕ (C) ⊗ 1n1(ϕ)×n1(ϕ) ⊕ M ϕ (C) ⊗ 1n2(ϕ)×n2(ϕ) ⊕ ... ⊕ M ϕ (C) ⊗ 1ns(ϕ)×ns(ϕ), |Ind(π1 )| |Ind(π2 )| |Ind(πs )| ∀ϕ ∈ ∂F(G)},
Example. Now we describe a group C∗-algebra of affine Coxeter group
e 4 4 2 2 2 C2 =< s1, s2, s3|(s1s2) = (s2s3) = s1 = s2 = s3 = e >= 2 4 2 2 2 = Z o < s1, s2|(s1s2) = s1 = s2 = e >= Z o C2.
2 The action of C2 on T is defined as follows:
s1 s2 (ϕ1, ϕ2) = (−ϕ1, ϕ2), (ϕ1, ϕ2) = (−ϕ2, −ϕ1).
2 Evidently, the fundamental region of the action C2 on the space T is 0 < ϕ1 < ϕ2 < π. Below we present the stabilizers, Gχ, of the points (ϕ1, ϕ2) ∈ F(C2), dimensions of the full system of non-equivalent irreducible representations of Gχ, the length of the orbit O(ϕ) and the decomposition of the general position representation onto irreducible components.
20 1. 0 < ϕ1 < ϕ2 < π: < e >; [1]; |O(ϕ1, ϕ2)| = 8; χ Tϕ = Ind(π11);
2. 0 = ϕ1 < ϕ2 < π: < s1 >; [1, 1]; |O(ϕ1, ϕ2)| = 4; χ χ Tϕ = Ind(π12) ⊕ Ind(π22);
3. 0 < ϕ1 < ϕ2 = π: < s2s1s2 >; [1, 1]; |O(ϕ1, ϕ2)| = 4; χ χ Tϕ = Ind(π13) ⊕ Ind(π23);
4. 0 < ϕ1 = ϕ2 < π: < s1s2s1 >; [1, 1]; |O(ϕ1, ϕ2)| = 4; χ χ Tϕ = Ind(π14) ⊕ Ind(π24);
5. ϕ1 = 0, ϕ2 = π: < s1, s2s1s2 >; [1, 1, 1, 1]; |O(ϕ1, ϕ2)| = 2; χ χ χ χ Tϕ = Ind(π15) ⊕ Ind(π25) ⊕ Ind(π35) ⊕ Ind(π45);
6. ϕ1 = ϕ2 = 0: C2; [1, 1, 1, 1, 2]; |O(ϕ1, ϕ2)| = 1. χ χ χ χ χ Tϕ = Ind(π16) ⊕ Ind(π26) ⊕ Ind(π36) ⊕ Ind(π46) ⊕ 2Ind(π56);
7. ϕ1 = ϕ2 = π: C2; [1, 1, 1, 1, 2]; |O(ϕ1, ϕ2)| = 1; χ χ χ χ χ Tϕ = Ind(π17) ⊕ Ind(π27) ⊕ Ind(π37) ⊕ Ind(π47) ⊕ 2Ind(π57).
iϕj 2 χ here χ(yj) = e , j = 1, 2, where yj are generators of Z , and {πij} is a full system of non-equivalent irreducible representations of stabilizer on the j-th step. The Proposition χ χ 1 implies that Ind(πij) Ind(πkv), ∀i, j, k, v,(i, j) 6= (k, v). If StGf (χ) = StGf (χe) and χ χe χ 6= χe, then Ind(πij) Ind(πij), ∀i, j. Let V (ϕ): ∂F(C2) → M8(C) be the unitary matrix-function such that Ms ∗ χ V (ϕ)Tϕ(h, g)V (ϕ) = πij(h, g). i=1
Proposition.
∗ e C (C2) '{f ∈ C(F(G) → M8(C)) : ∗ V(0,ϕ2)f(0, ϕ2)V(0,ϕ2) ∈ C((0, π) → M4(C) ⊕ M4(C)) ∀ϕ2 ∈ (0, π); ∗ V(ϕ1,π)f(ϕ1, π)V(ϕ1,π) ∈ C((0, π) → M4(C) ⊕ M4(C)) ∀ϕ1 ∈ (0, π); ∗ V(ϕ1,ϕ1)f(ϕ1, ϕ1)V(ϕ1,ϕ1) ∈ C((0, π) × (0, π) → M4(C) ⊕ M4(C)) ∀ϕ1 ∈ (0, π); ∗ V(0,π)f(0, π)V(0,π) ∈ M2(C) ⊕ M2(C) ⊕ M2(C) ⊕ M2(C), ∗ V(0,0)f(0, 0)V(0,0) ∈ M1(C) ⊕ M1(C) ⊕ M1(C) ⊕ M1(C) ⊕ M2(C) ⊗ 12×2, ∗ V(π,π)f(π, π)V(π,π) ∈ M1(C) ⊕ M1(C) ⊕ M1(C) ⊕ M1(C) ⊕ M2(C) ⊗ 12×2}.
21 References [Bou] N. Bourbaki: Groupes et algebres de Lie IV-VI, Hemmann, Paris, (1968). [M] G. W. Mackey: Induced representations of locally compact groups, Ann. Math. 55 (1952), no.1, 101-139.
Kyiv Taras Shevchneko University
On the geometry of the one-parameter families of tame algebras over arbitrary fields
Dirk Kussin
One-parameter families and tame bimodule algebras We are interested in the structure of one-parameter families for tame algebras over arbi- trary fields. Whereas over algebraically closed fields this was explained by Drozd, who showed that the one-parameter families are rational curves, that is, are obtained by the projective line P1, only little is known over arbitrary fields. There is still no satisfactory transformation of the notion of tameness to arbitrary fields. The alternative notion of “generic tameness” is only a very indirect approach. µ ¶ The prototype of a tame algebra (for k = k) is given by the Kronecker algebra k 0 , k2 k where the simple regular modules are parametrized by the projective line P1. Dlab and Ringel studied in the seventies and eightiesµ ¶ a more general concept over arbitrary fields, namely the tame bimodule algebras G 0 , where M = M is a tame bimodule over MF F G k. It is conjectured that each one-parameter family for a tame algebra A can be obtained by a functor from the module category of such a tame bimodule algebra to the module category over A. Therefore the study of the tame bimodule algebras is fundamental.
Throughout this presentation let k be an arbitrary field and Λ be a tame bimodule algebra. As for the Kronecker, the category of finite dimensional modules consists of a “discrete” part (the preprojective and the preinjective component) and a “continuous”
22 part consisting of homogeneous tubes (that is, the regular modules). Whereas the discrete part is not complicated, the problem is to determine the simple regular modules. These are parametrized by a so-called exceptional curve X in the sense of Lenzing [4].
Although there are important partial results by Dlab and Ringel in various papers, by Crawley-Boevey [2] and by Baer, Geigle and Lenzing [1], there is no general description of the explicit structure of these curves. It is the aim of these notes to describe some of our results from [3] which provide a further step in coming closer to an understanding of the general structure.
The generic module and the function field The representation theory of Λ is dominated by a single infinite dimensional module Q, the generic Λ-module. An important invariant of Q, and hence of mod (Λ), is the endomor- phism ring of Q, which is known to be a skew field and of finite dimension over its centre and which we call the function field. Note that over algebraically closed fields k it is given by the field k(t) of rational functions in one variable. In general it is a (non-commutative) algebraic function field of one variable over k. It is a natural question when this func- tion field is commutative. As Ringel pointed out in [5] there is the funny behaviour that the “non-commutative” bimodule RHH (where H are the Hamilton quaternions) yields the 2 2 commutative√ √ function field R(u, v)/(u + v + 1), whereas the “commutative” bimodule √ √ QQ( 2, 3)Q( 2, 3) leads to the non-commutative function field
Qhu, vi/(uv + vu, v2 + 2u2 − 3).
We will explain this phenomenon (see also Example 1 below).
Each simple regular Λ-module determines a multiplicity e(S) (first defined in [5]), which is the End(S)-dimension of the space Ext1(S, P ), where P is a rank one projective module. We always have e(S) ≥ 1. Whereas in case k = k these numbers are always = 1, we have shown that in general the multiplcities may be arbitrarily large.
Theorem 1. The function field is commutative if and only if all the multiplicities e(S) equal 1.
Moreover, assuming chark 6= 2, this happens if and only if there is finite field exten- sion K/k such that either M is the Kronecker bimodule over K, or M = K FF where F is a skew field of quaternions over K. (In these cases the function fields are explicitely known.)
It would be interesting to have a general formula for the dimension of the function field over its centre in terms of the function e.
23 Graded factoriality Similarly as in [1] we describe the geometric structure by projective coordinate algebras. By this we mean a Z-graded algebra R such that the derived category Db(Λ) is the repetitive Z Z category of the quotient category H = mod (R)/ mod 0 (R) (that is, modulo the objects of finite length); the simple regular Λ-modules become the simple objects in this hereditary abelian category H. In [1] such coordinate algebras were obtained by forming orbit algebras M Π(P, σ) = Hom(P, σnP ), n≥0 with σ = τ − the inverse Auslander-Reiten translation. Often the inverse AR translation is not acting “fine enough” on the preprojective component in order to have good ring theoretical properties: for example, for the Kronecker algebra over k we get as orbit algebra k[X2,XY,Y 2] which is not graded factorial; in this case the degree shift by 1 is the better choice, leading to k[X,Y ]. We have shown that it is always possible to find a functor such that the orbit algebra R is graded factorial, which we define in a non-commutative sense: Definition 1. We call a (non-commutative) graded domain R graded factorial if each homogeneous prime ideal of height one is generated by a normal element π (that is, Rπ = πR). Calling these elements π prime elements, it follows that each normal element has a factorization into prime elements which is unique up to permutation and multiplication with units. Theorem 2. Let Λ be a tame bimodule algebra over the field k. There exists an exact autoequivalence σ on Db(Λ) such that the orbit algebra R = Π(P, σ) is a noetherian graded factorial domain. More precisely, we have shown that there is a bijection between the simple regular modules S and the prime elements in Π(P, σ) given by taking the kernel of the S-universal extension of P . Moreover, the multiplicity e(S) is related to the number of irreducible factors of the corresponding prime element. Note also that the function field is the quotient field of Π(P, σ) (of degree zero fractions). Example 1. Let k = Q and √ √ √ √ M = QQ( 2, 3)Q( 2, 3). One shows that for a suitable automorphism σ in the sense of Theorem 2 µ ¶ Π(P, σ) = QhX,Y,Zi/ XY − YX,XZ − ZX,YZ + ZY,Z2 + 2Y 2 − 3X2 .
The element π = x2 − y2 is prime, and from the factorization π = (x − y)(x + y) into irreducibles it follows that for the corresponding simple regular module S we have e(S) = 2. This gives with Theorem 1 an alternative explanation that the function field is not commutative in this case.
24 Ghost automorphisms If k is algebraically closed, then each exact autoequivalence of Db(Λ) which fixes all the objects (up to isomorphism) is isomorphic to the identity functor. This is shown to be wrong in general. Definition 2. An exact autoequivalence γ of Db(Λ) fixing all objects of Db(Λ), but which is not isomorphic to the identity, is called ghost automorphism. Ghost automorphisms occur in the following way: Let R = Π(P, σ) be a graded facto- rial coordinate algebra as above. Then each prime element π ∈ R induces – by normality – a degree preserving automorphism γπ of the algebra R, which induces an exact autoe- ∗ b quivalence γπ of D (Λ). Then ∗ γπ ' 1 ⇐⇒ there is a unit u ∈ R0 such that uπ is central. Thus non-central prime elements (in the preceding sense) induce ghost automorphisms. It is open whether each ghost automorphism arises in this way. Example 2. Let k = R and M = C ⊕ C be the tame bimodule C ⊕ C with canonical left action, and with right action given by (x, y) · z = (xz, yz¯). Then the projective coordinate algebra Π(P, σ) is given by the twisted polynomial algebra C[X, Y ], where X is central and Y z =zY ¯ for all z ∈ C. The prime element Y is not central (even not up to unit), hence induces a ghost auto- morphism (the only one in this case).
The AR translation Whereas the action of the AR translation τ on objects (modules) is studied extensively, little is known about the action of τ on morphisms. In the example above, where Π(P, σ) = C[X, Y ], one could expect that the AR translation τ is given by grading shift by −2 (which is the functor σ−2). This is true on objects, but wrong on morphisms. The reason for this is the occurrence of ghost automorphisms:
Each simple regular module S lying in a tube of index y defines a tubular mutation b σy which is an autoequivalence on D (Λ). The corresponding prime element πy, say of degree d in the graded factorial algebra R = Π(P, σ) (for a suitable, fixed automorphism ∗ σ), defines as described above a ghost automorphism γy (if not ' 1), and with these data one has:
∗ d d ∗ Theorem 3. σy ' γy ◦ σ ' σ ◦ γy .
In particular, tubular mutations σx and σy corresponding to different points (tubes) may be not isomorphic functors, even if the corresponding prime elements have the same degree. Accordingly, the Picard group, defined to be the subgroup of Aut(Db(Λ)) generated by all tubular mutations σx, is not always isomorphic to Z but may have torsion. This effect also has an influence on the AR translation:
25 Example 3. As in Example 2 let M = C ⊕ C. The correct formula for the AR translation is given by −1 −1 −2 ∗ τ ' σx ◦ σy = σ ◦ γy , that is, on morphisms complex conjugation is involved.
It would be very interesting to have a similar formula for the functor τ in general.
References
[1] D. Baer, W. Geigle, and H. Lenzing, The preprojective algebra of a tame hereditary artin algebra, Comm. Algebra 15 (1987), 425–457. MR 88i:16036
[2] W. W. Crawley-Boevey, Regular modules for tame hereditary algebras, Proc. London Math. Soc. (3) 62 (1991), no. 3, 490–508. MR 92b:16024
[3] D. Kussin, Aspects of hereditary representation theory over non-algebraically closed fields, Habilitationsschrift, Paderborn March 2004.
[4] H. Lenzing, Representations of finite dimensional algebras and singularity theory, Trends in ring theory (Miskolc, 1996) (V. Dlab et al., ed.), CMS Conf. Proc., vol. 22, Amer. Math. Soc., Providence, RI, 1998, pp. 71–97. MR 99d:16014
[5] C. M. Ringel, Infinite dimensional representations of finite dimensional hereditary alge- bras, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), vol. 23, Academic Press, London, 1979, pp. 321–412. MR 81i:16032
Institut f¨urMathematik, Universit¨atPaderborn, 33095 Paderborn, Germany e-mail: [email protected]
26 Computer Algebra System Singular 2-2-0 Goes Noncommutative V. Levandovskyy
1 Introduction
Singular 2-2-0 is a further development of both actual 2-0-x release series and 2-1-x pre-release series. The new Singular 2-2-0 offers a wide range of possibilities for modern researcher in the field of both commutative and noncommutative polynomial computation. Singular is a free service to the scientific community. Moreover, it has been proved to be a good development platform, featuring flexibility of different levels of an user/system interaction. The 2004 Richard D. Jenks Memorial Prize for Excellence in Software Engineering for Computer Algebra was awarded to the Singular team at the yearly premier International Symposium on Symbolic and Algebraic Computation (ISSAC) in Santander (Spain).
2 Key Strengths of Singular
• distributed under GPL (GNU Public License) • available for most of hardware and software platforms • the biggest choice of ground fields on the market • the factorization over most of fields is provided • one of the fastest implementations £ of Gr¨obnerbasis algorithm (also noncommutative) £ of standard basis algorithm • primary decomposition for ideals and modules • resolution of singularities • noncommutative subsystem Plural • additional functionality: more than 60 libraries available • exhaustive documentation over 1000 pages long • extensive support via e-mail and online Singular Forum
27 3 Which Rings Can Be Handled with Plural?
Let K be a field and R = K[x1, . . . , xn] be a commutative ring. Suppose there are elements cij ∈ K r {0} and dij ∈ R, ∀ 1 ≤ i < j ≤ n.
Consider an algebra A = Khx1, . . . , xn | ∀ i < j xjxi = cijxixj + diji. It is called a G–algebra (in n variables) if the following conditions hold:
1) ∃ ≺, a well–ordering on R such that ∀ i < j lm(dij) ≺ xixj, 2) Nondegeneracy conditions are fulfilled, that is ∀ 1 ≤ i < j < k ≤ n
cikcjk · dijxk − xkdij + cjk · xjdik − cij · dikxj + djkxi − cijcik · xidjk = 0.
Theorem. Let A be a G–algebra in n variables. Then
α1 α2 αn • A has a PBW basis {x1 x2 . . . xn | αi ∈ N ∪ {0}}, • A is left and right Noetherian,
• A is an integral domain with gldimA ≤ n.
A GR–algebra is a factor of G–algebra by a proper two–sided ideal. There are several comfortable ways to construct GR–algebras.
1. Generic Matrices: creating a G–algebra according to the definition. Input: a ring R, n × n matrices C = (cij) and D = (dij). Shortcuts: if either all cij or all dij are equal, you can input just one value for all of them.
2. Tensor Products: starting from two G–algebras A and B, one can easily create a G–algebra A ⊗K B. 3. Library Procedures: many important algebras are predefined in libraries. One can initialize these algebras over fields of different characteristics. For quantum algebras, specializing the quantum parameter at the given root of unity is possible.
• Weyl and Heisenberg algebras (different realizations) nctools.lib
• U(sln), U(gln), U(g2) ncalg.lib 0 • Uq(sl2), Uq(sl3), Uq(so3) ncalg.lib n •Oq(A ), Oq(Mn(K)) qmatrix.lib • exterior algebras and general fin.–dim. algebras nctools.lib
28 4 New and Revised Functionalities in Plural
Revised and newly–implemented algorithms
£ left Gr¨obnerbasis of a module w.r.t. any well–ordering £ completion of a two–sided generating set to the left Gr¨obnerbasis £ intersection with subalgebras (elimination of variables) £ intersection of a finite set of modules £ saturation and quotient of modules by ideals £ kernel of a module homomorphism £ free resolutions (”normal” and ”minimized” algorithms) I syzygy modules of a given module I Betti numbers for graded modules over graded algebras £ center of an algebra (O. Motsak, center.lib) I centralizer of a finite set of polynomials I works over any field; in any proper factor–algebra
I for example, the center of U(g2) is computable on usual PC £ algebraic dependence of pairwise commutative elements
Newly–developed and implemented algorithms in Plural
♦ central character decomposition of a module (ncdecomp.lib) ¨ central quotient and central saturation of modules by ideals ¨ central character of module ♦ opposite and enveloping algebras, ”opposing” given object ♦ preimage of an ideal/module under a morphism ¨ correctness of a given map of G–algebras ¨ character of module w.r.t. a given (commutative) subalgebra ♦ annihilator of a holonomic (for example, Harish–Chandra) module ¨ annihilator of an element from a module ¨ annihilator of a finitely dimensional module
29 5 Collaborations and Perspectives
Contributed functionality
• V. Gerdt and D. Yanovich (Dubna, Russia) ◦ Involutive (Janet) bases for well–orderings (kernel implementation) • J. G´omez-Torrecillaz, F. Lobillo and C. Rabelo (Granada, Spain) ◦ Gelfand–Kirillov dimension (gkdim.lib) ◦ Quantum matrices and quantum minors (qmatrix.lib) • G. Pfister and W. Decker (Germany) ◦ Tate resolution and cohomology of sheaves (sheafcoh.lib)
Currently under development
♦ New efficient approach to bimodules; basic operations for right modules ♦ Homological algebra for finitely presented modules ¨ for a left module L and a centralizing bimodule N, Hom(L, N), Ext(L, N) and Tor(N,L) are computable modules ¨ Hochschild cohomology Hn(A, M) for (A, A)–bimodule M
6 Contributors Are Welcome!
• Do you want to implement complicated algorithms in an efficient way?
• Have you ever thought of doing this based on Singular?
• Contact us and let us develop together!
Fachbereich Mathematik, TU Kaiserslautern, Germany email: [email protected] http://www.singular.uni-kl.de/plural
30 Cofree Quiver Settings Geert Van de Weyer and Raf Bocklandt
Notations:
• Q is a quiver with vertices Q0, arrows Q1 and source and target maps s, t.
• A quiver setting is a couple of a quiver Q and a dimension vector α : Q0 → N. L • Rep(Q, α) := Mat (C) is the representation space of (Q, α). a∈Q1 αs(a)×αt(a) Q • On Rep(Q, α) we have a base change action of GL := GL(α , C) with categor- α v∈Q0 v ical quotient iss(Q, α). Cofreeness:
Definition 1: Definition 2: Rep(Q, α) is cofree if and only Rep(Q, α) is cofree if and only if if C[Rep(Q, α)] is a free mod- iss(Q, α) is smooth and all fibers ule over the ring of invariants of π : Rep(Q, α) → iss(Q, α) have C[Rep(Q, α)]GLα . the same dimension.
Question: For which (Q, α) is Rep(Q, α) cofree?
Answer: An Algorithm in two steps
1. First reduce the complexity of the quiver by Cutting it into smaller pieces: Applying a reduction step:
...... /.-,()*+i ... '&%$ !"#i CC { 1 D l CC {{ DD zz CC {{ DD zz ! {} D" z} z 1 C '&%$ !"#k C {{ CC D {{ CC a {} { C! ...... 1
↓ RI ↓ ...C ... C { /.-,()*+i1 ... '&%$ !"#il CC {{ 55 CC {{ 5 ! {} 55 C 1 1 CC D 5 {{ C b1 5 bl {{ CC 5 {} { C! 5 Ô ...... 1
31 2. Check whether the remaining bits and pieces are listed below
(a) strongly connected quiver settings (P, ρ) for which
i. There is a vertex v ∈ P0 such that ρ(v) = 1 and through which all cycles run, /o /o / o o/ o/ ii. ∀w 6= v ∈ P0 : ρ(w) ≥ #{ v w } + #{ v w } − 1,
k )'&%$ " !"# 1 \bi m with m ≥ k + l − 1 l
(b) quiver settings (P, ρ) of the form
/.-,()*+=u2 / ··· / /.-,()*+upB {{ BB {{ B /.-,()*+u o o /.-,()*+ 1aB 1 l1 BB }} B }~ } /.-,()*+lq o ··· o /.-,()*+l2
with p, q ≥ 1, such that there is at most one vertex x in the path /.-,()*+u1 /o /o /o //.-,()*+l1 which attains the minimal dimension min{u1, . . . , up, l1, . . . , lq}. The special case where p = 0 and q ≤ 1 (or vice versa)
/.-,()*+l ÔB 2 ÔÔ ( ÔÔ /.-,()*+l . 1 e 1 :\ : . :: : /.-,()*+lq
is always cofree. ˜ (c) quiver settings of extended Dynkin type An with cyclic orientation ˜ ˜ (d) quiver settings (P, ρ) consisting of two cyclic quivers, Ap+s−1 and Aq+s−1, coin- ciding on s subsequent vertices (p, q can be zero)
/.-,()*+;u1 / /.-,()*+u2 / ... / /.-,()*+up D ww DD www D! '&%$ c!"#s Fo ... o 2 o ... o '&%$ c!"#2 o '&%$ =c!"#1 FFF {{ F" {{ /.-,()*+l1 //.-,()*+l2 / ... //.-,()*+lq
with ui, lj ≥ 2 for all 1 ≤ i ≤ p, 1 ≤ j ≤ q and all ck ≥ 4 except for a unique vertex with dimension 2.
University of Antwerp, Belgium
32 33