On the McKay quivers and m-Cartan Matrices.doc
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1、精品论文推荐On the McKay quivers and m-Cartan MatricesJin Yun GuoDepartment of Mathematics Xiangtan Universityemail: Abstract In this paper, we introduce m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram, such as it is positive s
2、emi-definite, can be generalized to m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.Keywords: McKay quiver, $m$-Cartan matrix, linear group, finite abelian group1 IntroductionDynkin diagrams and Euclidean diagrams are very
3、useful in mathematics, they appear in many classification problems. One way to describe them is to use Cartan matrices and relates them to positive definite and positive semidefinite quadratic forms. One of the applications of these diagrams is the represen- tation theory of finite dimensional hered
4、itary algebras. Recently, many peo- ple observed that this theory has some 2-dimensional feature and try to find some higher dimensional generalization9. In fact, Iyama and Yoshino find that McKay quivers are interesting in describing some higher dimensional Auslander- Reiten theory10. In this paper
5、, we provide some further evidence that McKay quivers are the higher dimensional generalization of the Euclidean diagrams, by introducing m-Cartan matrices and proving that McKay quivers share some properties of the Euclidean diagrams. We show that the m-Cartan matrix of an m-dimensional McKay quive
6、r defines a positive semidefinite quadratic form. We also describe the McKay quiver of an abelian group.In 1980, John McKay introduced McKay quiver for a finite subgroup of the general linear group. Let G GL(m, C) = GL(V ) be a finite subgroup, here V is an m-dimensional vector space over C. V is na
7、turally a faithful representation of G. Let Si |i = 1, 2, . . . , n be a complete set of irreducible representations of1 This work is partly supported by Natural Science Foundation of China #10671061, SRFDP#2005050420042 Keywords:McKay quiver, m-Cartan matrix, linear group, finite abelian group10G o
8、ver C. For each Si , decompose the tensor product V Si as a direct sum ofirreducible representations, writeV Si = M ai,j Sj , i = 1, . . . , n,jhere ai,j Sj is a direct sum of ai,j copies of Sj . ai,j is finite since V is finite dimensional. The McKay quiver Q = Q(G) of G is defined as follow. The v
9、ertex set Q0 is the set of the isomorphism classes of irreducible representations of G, and there are ai,j arrows from the vertex i to the vertex j. McKay observes that when G is a subgroup of SL(2, C), then Q = Q(G) is a double quiver of the Euclidean diagram of type A, D, E.McKay quiver has played
10、 an important role in many mathematician fields such as algebraic geometry, mathematics physics and representation theory (See, for example, 15). It also appears in representation theory of algebra, for ex- ample, in the study of the Auslander Reiten quiver of Cohen Macaulay modules 1, 2, preproject
11、ive algebras of tame hereditary algebras 3, 4, 6, 5 and quiver varieties 13, 14, etc. We find that it also plays a critical role in classification of selfinjective Koszul algebras of complexity 2 7.Given a diagram, one associates it with a matrix, its Cartan matrix. Us- ing the Cartan matrices we as
12、sociate this diagram with a quadratic form. It is well known that the quadratic form is positive definite and respectively pos- itive semidefinite definite if and only if the diagram is a Dynkin diagram and respectively an Euclidean diagram. We know that in a Cartan matrix, we have2 as its diagonal
13、entries. One natural question is whether this 2 appears acci- dently? We observe that this 2 can be naturally explained as the dimension. In fact, the Cartan matrix in the case of SL(2, C) is just 2I minus the conjunction matrix of the McKay quiver. In the second section of this paper, we introduce
14、the corresponding m-Cartan matrix and try to generalize partially a property of the Euclidean diagram to a McKay quiver in general, it has a semidefinite quadratic form. This suggests that McKay quiver should be some kind of higher dimensional version of double Euclidean quiver.Though McKay quiver a
15、re well known for SL(2, C). It is difficult to deter- mine the McKay quiver in general. In the last section of this paper, we describe the McKay quiver for the finite abelian subgroup of an arbitrary special linear group, using the skew group algebra construction we introduced early 8. Thencase of a
16、 cyclic group can be regarded as m-dimensional version of double Aquiver.2 The Quadratic Form associated to a McKay quiverQQQLet Q = (Q0 , Q1 ) be a quiver, where Q0 = 1, . . . , n is the vertex set and Q1 is the arrow set. Let MQ = (ai,j ) be the conjunction matrix of Q, that is, an n n matrix with
17、 ai,j the number of arrows from i to j in Q. MQ is an integral matrix with nonnegative entries. Define the m-Cartan matrix of a quiver Q to be the n n matrix C = C (m) = mI MQ . When m = 2, this is exactly the Cartan matrix of the diagram of a double quiver. (Recall that a double quiver can be regar
18、ded as the quiver obtained from a diagram by replacing each edge in the diagram with a pair of arrows pointing at the opposite directions). As for the Cartan matrix, we may defined a bilinear form BQ = B(m) and a quadratic form qQ = q(m) on Qn for Q. This bilinear form and the quadratic form are cal
19、led respectively the m-bilinear form and the m-quadratic form of the quiver x1 y1 Q. They are defined as follow: Let X = . , Y = . Qn , define . xn . ynBQ (X, Y ) = X t C Yand qQ (X ) = X t C X.Q2Here X t denote the transpose of X . For m 2, B(m) is usually not symmetric. Let C 0 = 1 (C + C t ), the
20、n C 0 is symmetric and we have that qQ (X ) = X t C X =X t C 0X .Let G be a finite subgroup of GL(m, C). Let Q = Q(G) be the McKay quiver of G. Denote by MQ = (ai,j ) the conjunction matrix of Q. Let k be an algebraically closed field containing C. Consider mod kG, the category of finite generated l
21、eft kG modules. Denote by k the trivial representation. Thus mk is a direct sum of m copies of the trivial representation k, and its dimension is the same as the natural representation V . For any two modules M, M 0, define an integral function on mod kG as in 12:(M, M 0) = dimk HomkG (M mk, M 0) di
22、mk HomkG (M V, M 0).This is, in fact, a function on the isomorphism classes on mod kG. And we have the following lemma.精品论文推荐Lemma 2.1 1. (M, M 0) = BQ (dim M, dim M 0);2. (M, M ) 0;3. (M, M ) = 0 if M kG;Proof. Since dim(M M 0) = dim M +dim M 0, and we have dimk HomkG (Si , Sj ) =i,j . So, for kG-m
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