1.
Introduction
Path algebras are very important in representation theory and related mathematical fields, it is interesting to study their counterparts in higher representation theory [21,20]. The n-slice algebra is introduced and studied in [15] as the quadratic dual of an n-properly-graded algebra whose trivial extension is an n-translation algebra (see Section 2 for the details). The n-slice algebra was called dual τ-slice algebra in [14,17], for studying algebras related to the higher representation theory introduced and studied by Iyama and his coauthors [21,22,20]. The n-slice algebras bear some interesting features of path algebras in higher dimensional representation theory setting: the τn-closures, higher dimensional counterparts of preprojective and preinjective components, are truncations of Z|n−1Q, a higher version of ZQ [15], and the n-APR tilts and colts can be realized by τ-mutations in Z|n−1Q⊥,op [17], which is obtained by removing a source (resp. sink) and adding a corresponding sink (resp. source).
It is well known that the path algebras of acyclic quivers are classified as finite, tame and wild representation types according to their quivers. This classification has a great influence in representation theory. When Iyama and his coauthors study n-hereditary algebras, they define and study n-representation-finite and n-representation-infinite algebras, they also define and study n-representation-tame algebras inside n-representation-infinite algebras [22,23,20]. Recently, Lee, Li, Mills, Schiffler and Seceleanu give a new characterization of such classification for acyclic quivers in terms of their frieze varieties [24]. The classification of path algebras can also be read out from their preprojective algebras, that is, a path algebra is of finite representation type if and only if its preprojective algebra is finite dimensional, is of tame representation type if and only if its preprojective algebra is of finite Gelfand-Kirilov dimension, and is of wild representation type if and only if its preprojective algebra is of infinite Gelfand-Kirilov dimension [9].
Regarding n-slice algebra as a higher version of path algebra, in this paper, we generalize the classification of path algebras via (n+1)-preprojective algebras and related n-translation algebras. By using the results in [18], we get that for an n-slice algebra, its (n+1)-preprojective algebra is either of finite dimension, of finite Gelfand-Kirilov dimension or of infinite Gelfand-Kirilov dimension. The trivial extension of the quadratic dual of an n-slice algebra is a stable n-translation algebra, so it is either weakly periodic, of finite complexity and of infinite complexity, accordingly. In this way we classify the n-slice algebras as finite type, tame type or wild type according to their (n+1)-preprojective algebras or according to the trivial extensions of their quadratic duals.
McKay quiver for a finite subgroup of a general linear group is introduced in [27], which connects representation theory, singularity theory and many other mathematical fields. McKay quiver is also very interesting in studying higher representation theory of algebras [20,15]. Over the algebraically closed field of characteristic zero, the preprojective algebras of path algebras of tame type are Morita equivalent to the skew group algebras of finite subgroups of SL(2,C) over the polynomial algebra of two variables [29,8]. Inspired by this results, it is observed in [16] that the skew group algebras of a finite subgroup G of GL(n+1,C) over the exterior algebra of an (n+1)-dimensional vector space, and over polynomial algebra of n variables, have the McKay quiver of G as their Gabriel quivers. The former of these algebras is an n-translation algebra and the later is a Noetherian Artin-Schelter regular algebra of Gelfand-Kirilov dimension n+1. In this paper, we show that by constructing universal cover, taking τ-slices, and then taking quadratic duals, we obtain tame n-slice algebras associate to the McKay quiver of a finite subgroup of GL(n+1,C). In this process, we get three pairs of quadratic dual algebras related to a McKay quiver. We present some interesting equivalences of triangulate categories related to these algebras, among them we get a version of Beilinson correspondence and a version of Bernstein-Gelfand-Gelfand correspondences related to McKay quivers.
We also describe the quivers and relations for the tame 2-slice algebras related to McKay quivers. We characterized the relations for 2-slice algebras associated to the McKay quivers of finite Abelian subgroup of SL(3,C) and of finite subgroup of SL(3,C) obtained by embedding SL(2,C) into SL(3,C). The relations of the 2-translation algebras and AS-regular algebras related to the McKay quivers are described first, and the quivers and relations for the associated 2-slice algebras follow immediately.
The paper is organized as follows. In Section 2, concepts and results needed in this paper are recalled. We recall the constructions of the n-translation quivers related to an n-properly-graded quiver, and the algebras related to an n-slice algebra. We also recalled notions and results in [18] needed in Section 3. In Section 3, we discuss the classification of n-slice algebras via the stable n-translation algebras and the (n+1)-preprojective algebras related to them. In Section 4, we discuss the McKay quivers of some finite subgroups of SL(n+1,C). We also discuss the tame n-slice algebras associated to the McKay quivers of the finite subgroups in SL(n+1,C) and present the equivalences of the related triangulate categories. In the last section, we describe the relations for the stable 2-translation algebra, the AS-regular algebras and 2-slice algebras related to the McKay quivers of finite Abelian subgroups of SL(3,C) and finite subgroups of SL(3,C) obtained by embedding SL(2,C) into SL(3,C).
2.
Preliminary
In this paper, we assume that k is a field, and Λ=Λ0+Λ1+⋯ is a graded algebra over k with Λ0 a direct sum of copies of k such that Λ is generated by Λ0 and Λ1. Such algebra is determined by a bound quiver Q=(Q0,Q1,ρ) [13].
Recall that a bound quiver Q=(Q0,Q1,ρ) is a quiver with Q0 the set of vertices, Q1 the set of arrows and ρ a set of relations. The arrow set Q1 is usually defined with two maps s,t from Q1 to Q0 to assign an arrow α its starting vertex s(α) and its ending vertex t(α). Write Qt for the set of paths of length t in the quiver Q, and write kQt for space spanned by Qt. Let kQ=⨁t≥0kQt be the path algebra of the quiver Q over k. We also write s(p) for the starting vertex of a path p and t(p) for the terminating vertex of p. Write s(x)=i if x is a linear combination of paths starting at vertex i, and write t(x)=j if x is a linear combination of paths ending at vertex j. The relation set ρ is a set of linear combinations of paths of length ≥2. We may assume that the paths appearing in each of the element in ρ have the same length, since we deal with graded algebra. Through this paper, we assume that the relations are normalized such that each element in ρ is a linear combination of paths of the same length starting at the same vertex and ending at the same vertex. Conventionally, modules are assumed to be finitely generated left module in this paper.
Let Λ0=⨁i∈Q0ki, with ki≃k as algebras, and let ei be the image of the identity in the ith copy of k under the canonical embedding. Then {ei|i∈Q0} is a complete set of orthogonal primitive idempotents in Λ0 and Λ1=⨁i,j∈Q0ejΛ1ei as Λ0-bimodules. Fix a basis Qij1 of ejΛ1ei for any pair i,j∈Q0, take the elements of Qij1 as arrows from i to j, and let Q1=∪(i,j)∈Q0×Q0Qij1. Thus Λ≃kQ/(ρ) as algebras for some relation set ρ, and Λt≃kQt/((ρ)∩kQt) as Λ0-bimodules. A path whose image is nonzero in Λ is called a bound path.
A bound quiver Q=(Q0,Q1,ρ) is called quadratic if ρ is a set of linear combination of paths of length 2. In this case, Λ=kQ/(ρ) is called an quadratic algebra. Regard the idempotents ei as the linear functions on ⨁i∈Q0kei such that ei(ej)=δi,j, and for i,j∈Q0, arrows from i to j are also regarded as the linear functions on ejkQ1ei such that for any arrows α,β, we have α(β)=δα,β. By defining α1⋯αr(β1,⋯,βr)=α1(β1)⋯αr(βr), ejkQrei is identified with its dual space for each r and each pair i,j of vertices, and the set of paths of length r is regarded as the dual basis to itself in ejkQrei. Take a spanning set ρ⊥i,j of orthogonal subspace of the set ejkρei in the space ejkQ2ei for each pair i,j∈Q0, and set
The quadratic dual quiver of Q is defined as the bound quiver Q⊥=(Q0,Q1,ρ⊥), and the quadratic dual algebra of Λ is the algebra Λ!,op≃kQ/(ρ⊥) defined by the bound quiver Q⊥ (see [14]).
2.1. n-properly-graded quiver and related n-translation quivers
To define and study n-slice and n-slice algebra, we need n-properly-graded quiver and related quivers.
Recall that a bound quiver Q=(Q0,Q1,ρ) is called n-properly-graded if all the maximal bound paths have the same length n. The graded algebra Λ=kQ/(ρ) defined by an n-properly-graded quiver is called an n-properly-graded algebra.
Let Q be an n-properly-graded quiver with n-properly-graded algebra Λ, let M=MQ be a maximal linearly independent set of maximal bound paths in Q. Define the returning arrow quiver ˜Q=(˜Q0,˜Q1) with vertex set ˜Q0=Q0, arrow set ˜Q1=Q1∪Q1,M, with Q1,M={βp:t(p)→s(p)|p∈M}. By proposition 3.1 of [14], ˜Q is the quiver of the trivial extension ΔΛ of Λ. So there is a relation set ˜ρ, such that ΔΛ≃k˜Q/(˜ρ). We use the same notation ˜Q for the bound quiver ˜Q=(˜Q0,˜Q1,˜ρ). By Lemma 3.2 of [14], the relation set ˜ρ can be chosen as a union ρ∪ρM∪ρ0 of three sets, the relation set ρ of Q, the set ρM of all paths formed by two returning arrows from Q1,M, and a set ρ0 of linear combinations of paths containing exactly one returning arrow.
The n-translation quiver and n-translation algebra are introduced to study higher representation theory related to graded self-injective algebras in [13]. The bound quivers of a graded self-injective algebras are exactly the stable n-translation quivers, with the Nakayama permutation τ as its n-translation [13] (called stable bound quiver of Loewy length n+2 in [12]). Since ΔΛ is a graded symmetric algebra, the returning arrow quiver ˜Q is a stable n-translation quiver with trivial n-translation.
With an n-properly-graded quiver Q, we can also define an acyclic stable n-translation quiver Z|n−1Q, via its returning arrow quiver ˜Q, as the bound quiver with vertex set
and arrow set
with
and
The relation set of Z|n−1Q is defined as ρZ|n−1Q=Zρ∪ZρM∪Zρ0, where
and
when the returning arrow quiver ˜Q is quadratic. The quiver Z|n−1Q is a locally finite infinite quiver if Q is locally finite. It is a quiver with infinite copies of Q with successive two neighbours connected by the returning arrows. With the n-translation τ defined by sending each vertex (i,t) to (i,t−1), Z|n−1Q is a stable n-translation quiver.
Recall a complete τ-slice in an acyclic stable n-translation quiver is a convex full bound sub-quiver which intersect each τ-orbit exactly once. It is obvious that each copy of Q in Z|n−1Q is a complete τ-slice.
Given a finite stable n-translation quiver ˜Q, we can construct another infinite acyclic stable n-translation quiver Z⋄˜Q=(Z⋄˜Q0,Z⋄˜Q1,ρZ⋄˜Q) as follows (denoted by ¯Q and called separated directed quiver in [12]) with the vertex set
and the arrow set
If p=αs⋯α1 is a path in ˜Q, define p[m]=(αs,m+s−1)⋯(α1,m), and for an element z=∑tatpt with each pt a path in ˜Q, at∈k, define z[m]=∑tatpt[m] for each m∈Z. Define relations
here ζ[m]=∑tatpt[m] for each ζ=∑tatpt∈ρ. Z⋄˜Q is a locally finite bound quiver if Q is so.
A connected quiver Q is called nicely-graded if there is a map d from Q0 to Z such that d(j)=d(i)+1 for any arrow α:i→j.
Clearly, a nicely graded quiver is acyclic.
Proposition 2.1. 1. Let d be the greatest common divisor of the length of cycles in ˜Q, then Z⋄˜Q has d connected components.
2. All the connected components of Z⋄˜Q are isomorphic.
3. Each connected component of Z⋄˜Q is nicely graded quiver.
Proof. The first and the second assertions follow from Proposition 4.3 and Proposition 4.5 of [12], respectively. The last follows from the definition of Z⋄˜Q.
An n-properly-graded quiver is called n-nicely-graded quiver if it is nicely-graded.
Proposition 2.2. Assume that Q is an n-nicely-graded quiver. Let ˜Q be the returning arrow quiver of Q. Then Z|n−1Q is isomorphic to a connected component of Z⋄˜Q, hence is also nicely graded.
Proof. Let d:Q0→Z be the function such that d(j)=d(i)+1 for an arrow α:i→j in Q. Fix an vertex i0∈Q0, define ϕ:Z|n−1Q0→Z⋄˜Q by
It is easy to see that ϕ is an isomorphism from Z|n−1Q to the connected component of Z⋄˜Q containing the vertex (i0,0).
By definition, an n-translation quiver is an (n+1)-properly graded quiver. The bound quiver Z|n−1Q is an (n+1)-nicely graded quiver and the n-translation sends the ending vertex of a maximal bound path of length n+1 to its starting vertex. So by the definition of complete τ-slice, we immediately have the following.
Proposition 2.3. If ˜Q is a stable n-translation quiver. Then any complete τ-slice in Z⋄˜Q is an n-nicely-graded quiver.
Let m≤l be two integers, write Z⋄Q[m,l] for the full subquiver of Z⋄˜Q with vertex set
Z⋄Q[m,l] is a convex subquiver of Z⋄˜Q, so it is regarded as a bounded subquiver by taking
By Proposition 6.2 of [12], we get the following for any m.
Proposition 2.4. Z⋄Q[m,m+n] is a complete τ-slice of Z⋄˜Q.
The complete τ-slice Z⋄Q[0,n] is called an initial τ-slice in [12]. If the quiver Q is nicely graded, by Lemma 6.5 of [12], we can recover our quiver Q by using τ-mutations on a connected component of Z⋄Q[m,m+n].
Starting with an acyclic n-properly-graded quiver Q, we construct its returning arrow quiver ˜Q, which is a stable n-translation quiver. With ˜Q, we construct a universal cover Z|n−1Q, which is an acyclic stable n-translation quiver, or construct a universal cover Z⋄˜Q, which is a nicely-graded stable n-translation quiver. We can recover Q as a complete τ-slice of Z|n−1Q, and when Q is a nicely-graded, we can recover it as a connected component of complete a τ-slice of Z⋄˜Q.
We have the following picture to illustrate the relationship among these quivers.
2.2. n-slice algebras and related algebras
The quadratic dual quiver Q⊥=(Q0,Q1,ρ⊥) of an acyclic quadratic n-properly-graded quiver Q is called an n-slice.
Recall that a graded algebra ˜Λ=∑t≥0˜Λt is called a (p,q)-Koszul algebra if ˜Λt=0 for t>p and ˜Λ0 has a graded projective resolution
such that Pt is generated by its degree t part for t≤q and kerfq is concentrated in degree q+p [6,13]. A graded algebra defined by an n-translation quiver ˜Q is called an n-translation algebra, if there is a q≥2 or q=∞ such that ˜Λ is an (n+1,q)-Koszul algebra [13]. A stable n-translation algebra ˜Λ is a (n+1,q)-Koszul self-injective algebra for some q≥2 or q=∞, and we call q the Coxeter index of ˜Λ.
Let Λ be the n-properly-graded algebra defined by Q, if the trivial extension ΔΛ is a stable n-translation algebra, the algebra Γ=kQ/(ρ⊥) defined by the n-slice Q⊥ is called an n-slice algebra.
The following Proposition justifies the name n-slice.
Proposition 2.5. Let Γ be an acyclic n-slice algebra with bound quiver Q⊥, then Q is a complete τ-slice of the n-translation quiver Z|n−1Q and Λ=Γ!,op is a τ-slice algebra.
Starting with a quadratic acyclic n-properly-graded quiver Q, let Λ be the n-properly-graded algebra defined by Q and let Γ be the algebra defined by the n-slice Q⊥. If ˜Q is a quadratic quiver, write ˜Q⊥ for its quadratic dual quiver. In this case, Z|n−1Q and Z⋄Q are also quadratic quivers, write Z|n−1Q⊥ and Z⋄Q⊥ for their quadratic dual quivers, respectively.
With the quivers related to Q defined above and their quadratic duals, we have interesting algebras. We associate the trivial extension ˜Λ=ΔΛ of Λ to the returning arrow quiver ˜Q. If Γ is an n-slice algebra, we associate the (n+1)-preprojective algebra ˜Γ=Π(Γ) of Γ to a twist ˜Q⊥,ν of its quadratic dual, by Corollary 5.4 of [14].
We also associate algebras to the bound quivers Z|n−1Q and Z⋄˜Q by constructing smash products over ˜Λ. Note that ˜Λ is generated by the returning arrows over Λ. Taking Λ as degree zero component and taking the returning arrows as degree 1 generators, we get a Z-grading on ˜Λ induced by the n-translation. Construct the smash product ˜Λ#kZ∗ of ˜Λ with respect to this grading. It is shown in Section 5 of [13] that Z|n−1Q is the bound quiver of ˜Λ#kZ∗.
Taking the usual path grading on ˜Λ, that is, taking Λ0 as the degree 0 component and taking the arrows in ˜Q as degree 1 generators, we get a Z-grading on ˜Λ induced by the lengths of the paths of k˜Q. Construct the smash product of ˜Λ#′kZ∗ with respect to this grading. By Theorem 5.12 of [12], Z⋄˜Q is the bound quiver of ˜Λ#′kZ∗.
So we can get criteria for Γ to be n-slice algebra using ˜Λ#kZ∗ and ˜Λ#′kZ∗. Note that ˜Q is always an n-translation quiver, so ˜Λ is n-translation algebra if and only if there is a q≥2 or q=∞, such that ˜Λ is (n+1,q)-Koszul. By Propositions 2.2 and 2.6 of [13], ˜Λ is (n+1,q)-Koszul if and only if ˜Λ!,op is (q,n+1)-Koszul. We also get the following criterion for Γ to be an n-slice algebra.
Proposition 2.6. Let Γ be the algebra defined by an n-slice Q⊥. Then Γ is an n-slice algebra if and only if there is a q≥2 or q=∞, such that the (n+1)-preprojective algebra Π(Γ) is (q,n+1)-Koszul.
Write Z|n−1Q⊥ and Z⋄˜Q⊥ for the quadratic duals of Z|n−1Q and Z⋄˜Q, respectively. Write ˆΛ for the algebra defined by Z|n−1Q or Z⋄˜Q, and write ˆΓ for the algebra defined by Z|n−1Q⊥ or Z⋄˜Q⊥. We have the following picture depicting the relationship of these algebras:
The left triangle is induced by the picture (4) depicting quivers. The left vertically up arrow indicate taking a τ-slice algebra when ˆΛ=˜Λ#kZ∗, and indicate taking a direct summand of τ-slice algebra when ˆΛ=˜Λ#′kZ∗.
2.3. Loewy matrix and Koszul cone of stable n-translation algebra
Now we assume that ˜Λ=∑n+1t=0˜Λt is a stable n-translation algebra, that is, it is an (n+1,q)-Koszul self-injective algebra, with finite bound quiver ˜Q. Let ˜Q0={1,2,…,m}. Then ˜Λ is unital with 1=e1+⋯+em. We need some results in [18] concerning the complexities of ˜Λ and the Gelfand-Kirilov of its quadratic dual ˜Λ!,op.
For 0≤t≤n+1, define
Let E be the m×m identity matrix, then A0(˜Λ)=E.
The Loewy matrix L(˜Λ) for ˜Λ is defined in [18] as a matrix
with size (n+1)m×(n+1)m.
Let M=Mh+⋯+Mh+n be a graded ˜Λ-module of Loewy length ≤n+1 generated in degree h. The level dimensional vector l.dimM is defined as
where dimMt is the dimensional vector of Mt as a ˜Λ0-module. Recall the syzygy ΩM of M is the kernel of the projective cover P(M)→M. Thus ΩM=0 if M is projective, and there is an exact sequence 0→ΩM→P(M)→M→0 if M is not projective. Write Ωt+1M=Ω(ΩtM).
Assume that ˜Λ is graded self-injective, if a finitely generated graded ˜Λ-module M without projective summands has a minimal projective resolution
such that Pt is generated in degree h+t for t≤s, then the level dimensional vector for ΩsM is defined and we have
by Proposition 1.1 of [18].
Write 0 for the zero vector or zero matrix. We call a matrix (vector) nonnegative if all the entries are ≥0, positive if it is nonnegative and at least one entry is >0. We call a matrix (vector) negative if its opposite is positive. For two vectors (matrices) v and v′, write v≤v′ if v′−v is nonnegative and write v<v′ if v′−v is positive. Write
The following Proposition follows from (9) and the definition of the (p,q)-Koszul algebra.
Proposition 2.7. Assume that ˜Λ is a stable n-translation algebra with Coxeter index q. Then q is finite if and only if Lq+1(˜Λ)V0 is negative.
Now assume that q=∞, then ˜Λ is a Koszul self-injective algebra of Loewy length n+2. Write K for the real cone spanned by the level dimensional vectors of finitely generated Koszul ˜Λ-modules, called Koszul cone in [18], and K is contained in the positive cone in the real space R(n+1)m. Let ϱ=ϱL(˜Λ) be the spectral radius of L(˜Λ), that is, the maximal norm of its (complex) eigenvalues, which is called the size of the biggest Jordan block of the eigenvalue λi the algebraic degree of λi.
We can restate the Theorems 2.4 and 2.5 of [18] as follows.
Proposition 2.8. K is a solid cone invariant under the linear transformation defined by L(˜Λ).
Then by Theorem 3.1 of [30], ϱ is an eigenvalue of L(˜Λ) with maximal algebraic degree among the eigenvalues with norm ϱ. By Theorem 2.7 of [18], we have ϱ≥1.
The following is part (a) of Proposition 2.9 of [18].
Proposition 2.9. Let ˜Λ be a Koszul stable n-translation algebra. Let ϱ be the spectral radius of the Loewy matrix L(˜Λ) with algebraic degree d. Then for any vector v∈K, there exists an integer 1≤d′≤d, such that L(v,d′)={Lr(˜Λ)vrd′−1ϱr|r∈N} is a compact set in K and there exists a sequence r1<r2<⋯ of integers and a vector v0 with nonnegative entries such that limj→∞Lrj(˜Λ)vrd′−1jϱrj=v0.
Let M be a ˜Λ-module. Recall that the complexity c˜Λ(M) of M is defined as the least non-negative number d such that there exists λ>0 with dimkΩ(t)M≤λ⋅td−1 for almost all t, or infinite if no such t exist. The complexity c(˜Λ) of an algebra ˜Λ is defined as the supremum of the complexities of finitely generated ˜Λ-modules.
By applying Proposition 2.9, the following is proved as Theorem 2.10 in [18].
Proposition 2.10. If ϱ>1, then c˜Λ(M)=∞ for any non-projective Koszul ˜Λ-module M.
Recall that for a graded algebra Γ=Γ0+Γ1+⋯ generated by Γ0 and Γ1, the Gelfand-Kirilov dimension GKdimΓ is defined as
By using Koszul duality, the Gelfand-Kirilov dimension of the quadratic dual ˜Γ=˜Λ!,op is also discussed in [18].
The following is Theorem 3.2 in [18]. Let ˜Λ be a Koszul stable n-translation algebra, and write ˜Γ=˜Λ!,op.
Theorem 2.11. If ϱ=1 and d is the algebraic degree of the spectral radius, then GKdim˜Γ=d.
If ϱ>1, then GKdim˜Γ=∞.
The following is Theorem 3.1 in [18].
Theorem 2.12. If ˜Γ is Noetherian, then ϱ=1.
3.
Classification of n-slice algebras
By picture (6), we see an n-slice algebra Γ is related to a stable n-translation algebra ˜Λ=(Π(Γ))!,op.
To classify the n-slice algebras, we first discuss a classification of the stable n-translation algebras, by expanding the theory developed in [18]. Let L(˜Λ) be the Loewy matrix of ˜Λ, and ϱ be the spectral radius of L(˜Λ).
We call an algebra ˜Λ weakly periodic if there is an integer l such that Ωl˜Λ0≃˜Λ0 as ˜Λ-module. We remark that in the literature, an algebra ˜Λ is called periodic algebra if there is an integer l such that Ωl˜Λe˜Λ≃˜Λ as ˜Λe-module for the envelop algebra ˜Λe=˜Λop⊗˜Λ.
Lemma 3.1. A stable n-translation algebra ˜Λ is weakly periodic if and only if its Coxeter index q is finite.
Proof. For stable n-translation algebra ˜Λ with finite q, the (q+1)th syzygy Ωq+1˜Λ0ei of a simple ˜Λ-module ˜Λ0ei is a module concentrated in degree n+1+q, by the definition. So Ωq+1˜Λ0ei is semi-simple. Since ˜Λ is self-injective, the syzygies of an indecomposable module are all indecomposable. Thus Ωq+1˜Λ0ei is simple, and action of Ωq+1 permutes the simples. This implies that there is a l>0 such that Ωl(q+1) acts trivially on the simples and ˜Λ is weakly periodic.
Now assume that ˜Λ is weakly periodic with minimal period q′+1, then by the definition Ωq′+1˜Λ0≃˜Λ0 with finite q′. So it is semi-simple and hence concentrated in degree n+1+q′, since all the projective modules have the Loewy length n+1. Since ˜Λ is (n+1,q)-Koszul, so q′≥2 and we see in the projective resolution (5) of ˜Λ0, Pt is generated in degree t for t≤q′ and kerfq=Ωq′+1˜Λ0 is concentrated in degree n+1+q′. So q=q′ is finite.
Clearly, being weakly periodic is a special case of complexity 1.
We have the following characterization of the periodicity and of the complexities for stable n-translation algebras using Loewy matrices.
Theorem 3.2. Let ˜Λ be a stable n-translation algebra with Coxeter index q and Loewy matrix L(˜Λ). Then
1. The algebra ˜Λ is weakly periodic if and only if there is an positive integer h, such that Lh(˜Λ)V0 is negative;
2. The algebra ˜Λ is of finite complexity if and only if q is infinite and the spectral radius of L(˜Λ) is 1;
3. The algebra ˜Λ is of infinite complexity if and only if q is infinite and the spectral radius of L(˜Λ) is larger than 1.
Proof. Assume that ˜Λ is (n+1,q)-Koszul algebra. By Lemma 3.1, ˜Λ is weakly periodic if and only if q is finite. By Proposition 2.7, q is finite if and only if Lh(˜Λ)v0 is negative for h=q+1. This proves the first assertion.
Now assume that q is infinite, then ˜Λ is Koszul. By proposition 2.8, K is a solid cone invariant under the action of Loewy matrix L(˜Λ). By Theorem 3.1 of [30], the spectral radius ϱ is an eigenvalue of L(˜Λ) with maximal algebraic degree among the eigenvalues with norm ϱ. Note that the constant term of the characteristic polynomial of L(˜Λ) is 1 (see also Theorem 2.7 of [18]). So ϱ≥1.
If ϱ>1, by Proposition 2.10, ˜Λ is of infinite complexity.
Now assume that ϱ=1, then by Proposition 2.9, there exists an integer 1≤d′≤d, such that {Lr(˜Λ)l.dimMrd′−1|r∈N} is bounded set for any finitely generated Koszul ˜Λ-module M. So l.dimΩrM=Lr(˜Λ)l.dimM≤rd−1v for some positive vector v. This implies that for sufficient large r, dimkΩrM≤λrd−1 for some positive λ. So the complexity of M is ≤d. This proves that ˜Λ has finite complexity.
Let ˜Γ be the quadratic dual of a stable n-translation algebra ˜Λ, then ˜Γ is a (q,n+1)-algebra. We can characterize the growth of ˜Γ as follows.
Theorem 3.3. Let ˜Λ be a stable n-translation algebra with Coxeter index q and spectral radius ϱ and let ˜Γ=˜Λ!,op be its quadratic dual. Then
1. ˜Γ is finite dimensional if and only if q is finite;
2. ˜Γ is of finite Gelfand-Kirilov dimension if and only if q is infinite and ϱ=1;
3. ˜Γ is of infinite Gelfand-Kirilov dimension if and only if q is infinite and ϱ>1.
Proof. If q is finite, then ˜Λ is an almost Koszul algebra of type (n+1,q), by the definition. So ˜Γ is almost Koszul of type (q,n+1) by Propositions 3.11 and 3.4 of [6]. Thus it is of finite Loewy length, and hence is finite dimensional. This proves the first assertion.
If q is infinite, then ˜Λ is a Koszul self-injective algebra, and ˜Γ is an infinite dimensional AS-regular algebra. The rest of the theorem follows from Theorem 2.11.
Combine Theorems 3.2 and 3.3, we get the following.
Theorem 3.4. Let ˜Λ be a stable n-translation algebra and let ˜Γ=˜Λ!,op be its quadratic dual. Then
1. ˜Λ is weakly periodic if and only if ˜Γ is finite dimensional;
2. ˜Λ is of finite complexity if and only if ˜Γ is of finite Gelfand-Kirilov dimension;
3. ˜Λ is of infinite complexity if and only if ˜Γ is of infinite Gelfand-Kirilov dimension.
By Theorem 2.12, we also have the following.
Theorem 3.5. If ˜Γ is Noetherian, then ˜Γ is of finite Gelfand-Kirilov dimension.
Assume that Γ is an acyclic n-slice algebra with quiver Q⊥, let Λ=Γ!,op be its quadratic dual. Let ν be the automorphism of Λ defined by sending each arrow α in Q to (−1)nα. By Theorem 6.6 of [14], we have that
where Π(Γ) be the (n+1)-preprojective algebra, and ΔνΛ be a twisted trivial extension of Λ with respect ν. Let ˜Γ=Π(Γ) and ˜Λ=ΔνΛ. ˜Λ is a stable n-translation algebra with trivial n-translation, and ˜Γ is an AS-regular algebra. We have classification for ˜Λ and ˜Γ, using numerical invariants such as complexity and Gelfand-Kirilov dimension, respectively.
When n=1, Γ is an hereditary algebra, then Γ=kQ is the path algebra of a quiver Q. It is well known that the path algebras are classified, according to their representation type, as finite, tame and wild ones. This classification can be done using the underlying graphs of the quivers. That is, a path algebra kQ is of finite type if the underlying graph of Q is a Dynkin diagram, of tame type if the underlying graph of Q is an Euclidean diagram and of wild type otherwise. This classification can also be read out from the preprojective algebra Π(Q) of the path algebra kQ. The path algebra kQ is of finite representation type if Π(Q) is finite dimensional, of tame type if Π(Q) is of Gelfand-Kirilov dimension 2 and of wild type if Π(Q) is of infinite Gelfand-Kirilov dimension [9].
Now we define that an n-slice algebra Γ is of finite type if ˜Γ is finite dimensional, of tame type if ˜Γ is of finite Gelfand-Kirilov dimension (not zero), and of wild type if ˜Γ is of infinite Gelfand-Kirilov dimension.
As an immediate consequence of Theorem 3.4, we get a classification of n-slice algebras.
Theorem 3.6. An n-slice algebra is of finite type, or of tame type, or of wild type.
For an n-slice algebra Γ, we have the following characterizations of the classification.
Theorem 3.7. 1. Γ is of finite type if and only if ˜Λ is weakly periodic, if and only if Lh(˜Λ)v0 is negative for some h, if and only if q is finite.
2. Γ is of tame type if and only if ˜Λ is of finite complexity, if and only if q is infinite and the spectral radius of L(˜Λ) is 1.
3. Γ is of wild type if and only if ˜Λ is of infinite complexity, if and only if q is infinite and the spectral radius of L(˜Λ) is larger than 1.
Proof. This follows from the above Theorems 3.3 and 3.4.
When n=1, ˜Λ is the algebra with vanishing radical cube and q is related to the Coxeter number of a Dynkin diagram [6]. When n=1, we also have that, Γ is tame if and only if ˜Γ is Noetherian, by [2]. For the general case, we have that the following Proposition, as a consequence of Theorem 2.12.
Proposition 3.8. If ˜Γ is Noetherian of infinite dimension, then Γ is tame.
It is natural to ask if the converse of Proposition 3.8 is true?
4.
McKay quivers and n-slice algebras
By Proposition 3.8, if the (n+1)-preprojective algebra ˜Γ of an n-slice algebra is Noetherian, then Γ is tame. In this section, we discuss n-slice algebras related to McKay quivers, which are always tame.
In this section, we assume that k is an algebraically closed field of characteristic 0. It is well known that the quiver ˜Q of the preprojective algebra of a path algebra kQ is the double quiver of Q. The double quivers of Euclidean quivers are exactly the McKay quivers of finite subgroup of SL(2,C) [27].
McKay quiver was introduced in [27]. Let V be an n-dimensional vector space over C and let G⊂GL(n,C)=GL(V) be a finite subgroup. V is naturally a faithful representation of G. Let {Si|i=1,2,…,l} be a complete set of irreducible representations of G over C. For each Si, decompose the tensor product V⊗Si as a direct sum of irreducible representations, i.e.,
here Sai,jj denotes a direct sum of ai,j copies of Sj. The McKay quiver ˜Q=˜Q(G) of G is defined as follows. The vertex set ˜Q0 is the set of (indices of) the isomorphism classes of irreducible representations of G, and there are ai,j arrows from the vertex i to the vertex j.
We recall some results on the McKay quivers of Abelian groups and on the relationship between McKay quivers of same group in GL(n,C) and in SL(n+1,C) [10,11].
Let G be an Abelian group, then
is a direct product of m cyclic groups of order r1,…,rm. Write diag(x1,⋯,xm) for the diagonal matrices with diagonal entries x1,⋯,xm. Let ξr be the rth root of the unit, embed G into SL(m+1,C) by sending
for i=(i1,⋯,im)∈Z/r1Z×⋯×Z/rmZ. Let et=(0,⋯,0,1,0⋯,0)∈Z/r1Z×⋯×Z/rmZ be the element with 1 of Z/rtZ at the tth position, and 0 otherwise, and e=∑mt=1et. The following Proposition can be proved by using Proposition 3.2 of [11] and Section 3 of [10]. We give an inductive proof to show how such quiver is constructed.
Proposition 4.1. The McKay quiver ˜Q(r1,⋯,rm) of the group G(r1,⋯,rm) in SL(m+1,C) is the quiver with the vertex set
and the arrow set
Proof. We prove by induction on m.
The Abelian subgroup in SL(1,C) is trivial and its McKay quiver is a loop. So Proposition holds for m=0.
Assume Proposition holds for m=h.
Embed the group G(r1,⋯,rh,rh+1) in GL(h+1,C) by sending
then group G(r1,⋯,rh) is a subgroup of G(r1,⋯,rh,rh+1) with
and we have
where C′rh+1=(diag(1,⋯,1,ξrh+1)) is the cyclic group generated by the element diag(1,⋯,1,ξrh+1). So
By Theorem 1.2 of [11], the McKay quiver of G(r1,⋯,rh,rh+1) in GL(h+1,C) is a regular covering of the McKay quiver of G(r1,⋯,rh) in SL(h+1,C) with the automorphism group the cyclic group (ξrh+1) of order rh+1, whose generator is induced by the Nakayama permutation. So the vertex set of the McKay quiver ˜Q′(r1,⋯,rh,rh+1) for G(r1,⋯,rh,rh+1) in GL(h+1,C) is rh+1 copies of ˜Q(r1,⋯,rh), indexed by Z/rh+1Z. Write v(h) for a vector v to indicate that its dimension is h. Since the arrows given by the vector e(h) represents the Nakayama transformation, it induces arrows
from one copy ˜Q(r1,⋯,rh) to the next in ˜Q′(r1,⋯,rh,rh+1). The other arrows are determined by the representations of G(r1,⋯,rh), so the arrows given by the vectors e(h)t induce arrows inside each copy. Changing the indices by subtracting (e(h),0) from the indices of all the vertices, we may take
as the arrow set.
Now embed G(r1,⋯,rh,rh+1) to SL(h+2,C) by sending
Since Nakayama permutation σ of ˜Q′(r1,⋯,rh,rh+1) sends each vertex i to the ending vertex of a path of length h+1 with arrows defined by different et's, that is, sending i to i−e. Then by Theorem 3.1 of [11], the McKay quiver ˜Q(r1,⋯,rh,rh+1) of G(r1,⋯,rh,rh+1) in SL(h+2,C) is obtained from the quiver ˜Q′(r1,⋯,rh,rh+1) by adding an arrow αi,m+2 from i to i−e for each vertex i in ˜Q0(r1,⋯,rh,rh+1).
This shows that Proposition holds for h+1 and finishes the proof.
Note that the Nakayama permutation for the subgroup of a special linear group is trivial. As a direct consequence of Proposition 3.1 of [11], we also have the following Proposition to describe the McKay quiver of finite group G in SL(m+1,C) obtained by embed SL(m,C) into SL(m+1,C).
Proposition 4.2. Let G be a finite subgroup of SL(m,C) with McKay quiver ˜Qm(G). Then there is an embedding of G into SL(m+1,C) such that the McKay quiver ˜Qm+1(G) of G, as a subgroup of SL(m+1,C), is obtained from ˜Qm(G) by adding a loop at each vertex of ˜Qm(G).
Let V be an (n+1) dimensional vector space over k. let k[V]=k[x0,x1,⋯,xn] be the polynomial algebra of n+1 variables over k and ∧V be exterior algebra of V. In fact, both k[V] and ∧V are quotient algebras of the tensor algebra Tk(V) of V over quadratic ideals, and they are quadratic dual of each other. Let G be a finite subgroup of GL(V)≃GL(n+1,C). The following is the Theorem 1.8 of [16].
Theorem 4.3. Let ˜Q be the McKay quiver of G in GL(V), the skew group algebra Tk(V)∗G is Morita equivalent to k˜Q.
Let k[V]∗G and ∧V∗G be the skew group algebras of G over k[V] and over ∧V, respectively. Let ˜Γ=˜Γ(G) be the basic algebra Morita equivalent to k[V]∗G and let ˜Λ=˜Λ(G) be the basic algebra Morita equivalent to ∧V∗G. By Theorem 4.3, the quivers of both ˜Γ and ˜Λ are the McKay quiver ˜Q=˜Q(G) of G in GL(n+1,C).
Note that ∧V is the Yoneda algebra of k[V], and both are Koszul algebra. By Theorem 10 of [26], ∧V∗G is the Yoneda algebra of k[V]∗G, and by Theorem 14 of [26], they are both Koszul. This shows that ˜Γ and ˜Λ are Yoneda algebras from each other, so they are quadratic dual of each other. By Lemma 13 of [26], k[V]∗G is an AS-regular algebra(called generalized Auslander algebra there) and ∧V∗G is a self-injective algebra. By choosing a quadratic relation ˜ρ=˜ρ(G) of ˜Q(G) such that ˜Λ≃k˜Q/(˜ρ). The following Proposition follows from the definition of n-translation algebra.
Proposition 4.4. Let G be the finite subgroup of GL(n+1,C), then its McKay quiver ˜Q(G)=(˜Q0,˜Q1,˜ρ) is an n-translation quiver and ˜Λ(G) is a Koszul stable n-translation algebra.
Starting with a finite group G⊆GL(n+1,C), we have a pair of bound quivers ˜Q(G) and ˜Q⊥(G) associated to the McKay quiver of G. The bound quiver ˜Q(G) defines a stable n-translation algebra ˜Λ(G) and ˜Q⊥(G) defines an AS-regular algebra ˜Γ(G)=k˜Q/(ρ⊥). We call ˜Λ(G) a stable n-translation algebra associated to ˜Q(G) and ˜Γ(G) an AS-regular algebra associated to ˜Q(G).
Let QN(G)=Z⋄˜Q[0,n], then by Theorem 5.1 of [12], the algebra ΛN defined by the bound quiver QN is the Beilinson algebra of ˜Λ(G). Let ˜Λ(G)#′kZ∗ be the smash product of ˜Λ(G) with respect to the usual gradation (see [13]). Write ˆΛ for the repetitive algebra of an algebra Λ of finite global dimension, then by Theorem 5.12 of [12], we have ˜Λ(G)#′kZ∗≃^ΛN.
Let Q=Q(G) be a connected component of a complete τ-slice of Z⋄˜Q, then Q is a quadratic n-nicely-graded quiver. Let Q⊥ be its quadratic dual quiver, and Q⊥ is a nicely-graded n-slice. We have that Z|n−1Q(G) is a connected component of Z⋄˜Q(G), and use the notations of the vertices and arrows in Z⋄˜Q(G) for the vertices and arrows in Z|n−1Q(G). Let Q′ be the full subquiver of Z|n−1Q(G) defined by the vertex set {(i,m)∈(Z|n−1Q(G))0|0≤m≤n}. Then Q′ is a complete τ-slice in Z|n−1Q(G) and Q is obtained by a sequence of τ-mutations from Q′. Let Λ′ be the τ-slice algebra defined by Q′, then by Theorem 6.8, Δ(Λ′)≃Δ(Λ)=Δ(G). So the bound quiver of Δ(G) is the returning arrow quiver ˜Q′ of Q′ whose vertex set is indexed as
For a path p=αl⋯αl∈˜Ql(G) and ¯m∈Z/(n+1)Z, write p[¯m] for the path (αl,¯m+l−1)⋯(α1,¯m) of length l in Δ(G). Then we have that ∑p,¯map,¯mp[¯m]=0 if and only if ∑p,¯m=¯m′ap,¯mp=0 for all ¯m′∈Z/(n+1)Z.
Proposition 4.5. Γ(G)=kQ⊥/(ρ⊥) is a tame n-slice algebra.
Proof. For any i∈˜Q(G), we have a minimal projective resolution
for the simple ˜Λ(G)-module ˜Λ0(G)ei, such ˜Pt is generated in degree t. We may assume that ˜Pt=⨁hts=1˜Λ(G)eit,s for a sequence It=(it,1,…,it,ht) of vertices in ˜Q0(G). Then ft is presented as a matrix Ct=(c(t)u,v) of size ht×ht−1. We have that elements c(t)u,v of Ct are all in ∑s≥1˜Λs(G) for 1≤u≤ht−1,1≤v≤ht.
Clearly (13) is exact if and only if CtCt+1=0 and if (d1,⋯,dht)Ct=0 for some (d1,⋯,dht)∈⨁hts=1˜Λ(G)eit,s, then there is (d′1,⋯,d′ht+1)∈⨁ht+1s=1˜Λ(G)eit+1,s such that
Now for any (i,¯m)∈Δ(G), define ˜Pt[¯m]=⨁hts=1Δ(G)eit,s,¯m+t, and let ~ft[¯m] be the map defined by the ht×ht−1 matrix Ct[¯m]=(c(t)u,v[¯m+t]). Then we get immediately a sequence
By comparing the matrices defining the sequences (13) and (14), we see that (13) is exact if and only if (14) is exact. That is, (13) is a projective resolution of simple ˜Λ(G)-module ˜Λ0(G)ei if and only if (14) is a projective resolution of simple Δ(G)-module Δ0(G)ei,¯m for any ¯m∈Z/(n+1)Z.
Since ˜Λ(G) is Koszul, Pt is generated in degree t, so c(t)u,v∈˜Λ1(G) and hence c(t)u,v[¯m+t]∈Δ1(G) and ˜Pt[¯m] is generated in degree t, for all (i,¯m). This shows that Δ(G) is Koszul and ˜Γ(G) is an n-slice algebra.
By comparing (13) and (14), the simple modules Δ0(G)ei,¯m has the same complexity as ˜Λ0(G)ei. By Theorem 2.12 and Theorem 3.2, ˜Λ(G) is of finite complexity, so Δ(G) has finite complexity. So by Theorem 3.7, Γ(G) is a tame n-slice algebra.
We call Γ(G) an n-slice algebra associated to the McKay quiver ˜Q(G).
It is interesting to know if the converse of Proposition 4.5 is true, that is, for an indecomposable nicely graded tame n-slice algebra Γ, if there is a finite subgroup G⊂GL(n+1,C), such that Γ≃Γ(G) for some connected component Q of a complete τ-slice in Z⋄˜Q(G)? The quiver Q(G) is not unique by the definition, but they can be related by a sequences of τ-mutations. Thus n-slice algebra Γ(G) associated to the McKay quiver of G is not unique, and such algebras are related by a sequence of n-APR tilts (see [17]). Let ΓN=ΛN,!,op be the quadratic dual of ΛN. We remark that Γ(G) can be obtained from a direct summand of ΓN by a sequence of n-APR tilts.
For the three pairs of quadratic duals of algebras: (˜Λ(G),˜Γ(G)), (^ΛN,^ΓN) and (ΛN,ΓN). We can construct ^ΛN from ˜Λ(G) by smash product, and ΛN from ^ΛN by taking τ-slice algebra, we don't know how to get back ˜Λ(G) from Λ(G).
For an AS-regular algebra ˜Γ, denote by qgr˜Γ the non-commutative projective scheme of ˜Γ (see [1]). We have the following equivalences of triangulate categories related to McKay quiver.
Theorem 4.6. The following categories are equivalent as triangulate categories:
1. the bounded derived category Db(qgr˜Γ(G)) of the non-commutative projective scheme of ˜Γ(G);
2. the bounded derived category Db(ΓN) of the finitely generated ΓN-modules;
3. the bounded derived category Db(ΛN) of the finitely generated ΛN-modules;
4. the stable category grmod_˜Λ(G) of finitely generated graded ˜Λ(G)-modules;
5. the stable category mod_^ΛN of finitely generated ^ΛN-modules;
6. the stable category mod_^ΓN of finitely generated ^ΓN-modules.
If the lengths of the oriented circles in ˜Q(G) is coprime, they are also equivalent to the following triangulate categories:
(7) the bounded derived category Db(qgrΠ(G)) of the non-commutative projective scheme of the (n+1)-preprojective algebra Π(G) of Γ(G);
(8) the stable category grmod_Δν(G) of finitely generated graded modules over the (twisted) trivial extension Δν(G) of Λ(G).
Proof. By Theorem 4.14 of [28], we have that Db(qgr˜Γ(G)) is equivalent to Db(ΓN) as triangulate categories, since ˜Γ(G) is AS-regular.
By Theorem 1.1 of [7], we have that grmod˜Λ(G) and grmodΔ(ΛN) are equivalent, thus grmod_˜Λ(G) and grmod_Δ(ΛN) are equivalent as triangulated categories.
By Corollary 1.2 of [7], we have that grmod_˜Λ(G) and Db(modΛN) are equivalent as triangulated categories.
On the other hand, we have ΓN≃ΛN,!,op is Koszul by Propositions 2.6 and 6, 5 of [14]. By Proposition 2.5, and Corollary 2.4 of [14], we have that Db(modΓN) and Db(modΛN) are equivalent to the orbit categories of Db(grΓN) and of Db(grΛN), respectively, using the proof Theorem 2.12.1 of [3] (see the arguments before Theorem 6.7 of [14]). So by Theorem 2.12.6 of [3], Db(modΓN) and Db(modΛN) are equivalent as triangulated categories. Similar to the proof of Theorem 6.7 of [14], we also get that Db(qgr˜Γ(G)) and grmod_˜Λ(G) are equivalent as triangulated categories.
By Lemma Ⅱ.2.4 of [19], mod^ΛN and grmodΔ(ΛN) are equivalent, so mod_^ΛN and grmod_Δ(ΛN) are equivalent as triangulated categories. By Theorem Ⅱ.4.9 of [19], Db(modΛN) and mod_^ΛN are equivalent as triangulated categories.
Similarly, Db(modΓN) and mod_^ΓN are equivalent as triangulated categories.
When the lengths of the oriented circles in ˜Q(G) is coprime, then Z⋄˜Q(G) has only one connected component by Proposition 2.1. So ΛN=Λ(G) and equivalences for (2), (3), (7) and (8) follow from Theorem 6.7 of [14].
We remark that the equivalence of (1) and (2) can be regarded as a McKay quiver version of Beilinson correspondence and the equivalence of (1) and (4) can be regarded as a McKay quiver version of Berstein-Gelfand-Gelfand correspondence [5,4]. So we have the following analog to (6), for the equivalences of the triangulate categories in Theorem 4.6:
5.
2-slice algebras associated to McKay quivers
In the classical representation theory, we take a slice from the translation quiver and view the path algebra as 1-slice algebra. For n≥2, we need to know both quiver and the relations for a n-slice algebra, especially for the tame ones. For any n, we can construct tame n-slice algebras using the McKay quivers.
Now we consider the case of n=2, by writing down their quivers and relations. We first determine the relations which produce stable 2-translation algebra of finite complexity for a given McKay quiver ˜Q. By constructing the bound quiver Z⋄˜Q and taking a complete τ-slice, then taking the quadratic dual, we get the related 2-slice quiver and 2-slice algebra.
Assume that ˜Q=˜Q(G)=(˜Q0,˜Q1) is a McKay quiver of finite subgroup G of SL(3,C). Let ˜Λ=k˜Q/I be a stable 2-translation algebra of finite complexity. Note that ˜Λ=∑3t=1˜Λt is a stable 2-translation algebra with trivial 2-translation. Now we want to determine the relation ˜ρ such that I=(ρ).
Since ˜Λ is Koszul, ˜ρ⊂k˜Q2, the k-space spanned by the paths of length 2. By Lemma 2.1 of [18], we have dimkei˜Λ1ej=dimkej˜Λ2ei and ˜Λ3ei≃˜Λ0ei as ˜Λ0 module. Let M(˜Q) be the adjacent matrix of ˜Q, that is, an |˜Q0|×|˜Q0| matrix with the (i,j) entry the number of arrows from i to j, then the Loewy matrix of ˜Λ is
This is exactly the Loewy matrix of ˜Λ(G), the basic algebra Morita equivalent to ∧[x0,x1,x2]∗G. Therefore we have the following Proposition.
Proposition 5.1. Let i,j∈˜Q0. Then
1. If there is an arrow from i to j, then there is a bound path of length 2 from j to i in ˜Λ;
2. If there is only one arrow from i to j, any two paths of length 2 from j to i are linearly dependent in ˜Λ.
Proof. The proposition follows directly from dimkei˜Λ1ej=dimkej˜Λ2ei.
Therefore the number of arrows from i to j is exactly the order of the set of maximal linearly independent sets of bound paths of length 2 from j to i.
We also have the following Proposition.
Proposition 5.2. If α:i→j,β:j→h are arrows in ˜Q with i,j,h different vertices in ˜Q and there is no arrow from h to i, then βα=0 in ˜Λ.
Proof. Due to that there is no arrow from h to i, dimkeh˜Λ1ei=0. Since ˜Q is a 2-translation quiver with τi=i, thus by definition, dimkei˜Λ2eh=dimkeh˜Λ1ei=0. Thus for any arrows α:i→j,β:j→h, we have βα=0 in ˜Λ.
Now we determine the relations for the McKay quiver of finite Abelian subgroup of SL(3,C) and of the subgroup which is a finite subgroup of SL(2,C) embedded in SL(3,C). Due to Proposition 5.2, we excluded the McKay quivers of Abelian group which has a direct summand of Z/3Z.
Let s≥4,r≥4 and let G(s,r) be the finite subgroup of SL(3,C) obtained from the Abelian groups Z/sZ×Z/rZ, using the embedding in Proposition 4.1. Write Ξ for Al,Dl,E6,E7 and E8 with l≥5. Let G(Ξ) be the finite subgroup of SL(3,C) obtained from finite subgroup G(Ξ) of SL(2,C) with McKay quiver of type Ξ, using the embedding in Proposition 4.2. Let ˜Q(G) be the McKay quiver of subgroup G, now we determine the relations for the stable 2-translation algebras with McKay quiver ˜Q(G) as their quivers and for the 2-slice algebras defined by the quiver ˜Q(G). Denote by k∗ the set of nonzero elements in k. Write the McKay quiver for G(s,r) as ˜Q(s,r) and the one for G(Ξ) as ˜Q(Ξ).
5.1. Relations for ˜Q(s,r)
Let ˜Q=˜Q(s,r) be the McKay quiver of an Abelian group which is a direct sum of cyclic group of order s and r. By Proposition 4.1, its vertex set ˜Q0=Z/sZ×Z/rZ, and the arrows are described as follows: for each vertex i∈˜Q0, there is an arrow αi from i to i+e1, an arrow βi from i to i+e2 and an arrow γi from i to i−e. So the McKay quiver ˜Q(s,r) is as follows:
For each vertex i∈˜Q0, write
and
for ai,bi,ci∈k∗. Let C(a,b,c)={ai,bi,ci|i∈˜Q0} be a subset of k∗ and let
and let
Proposition 5.3. If the quotient algebra ˜Λ(s,r)=k˜Q(s,r)/I is a 2-translation algebra, then there is {ai,bi,ci|i∈˜Q0}⊆k∗, such that I is generated by the set ˜ρ(s,r,C(a,b,c)) in (17).
Proof. Assume that ˜Λ(s,r)=k˜Q/I is a 2-translation algebra.
Consider the square with vertices i,i+e1,i+e2,i+e. Since there is an arrow γi from i to i−e, one gets that there are ci,c′i such that
by (2) of Proposition 5.1. Since ˜Q(G) is 2-translation quiver with trivial 2-translation and ei˜Λ1(G)ei+e×ei+e˜Λ2(G)ei→ei˜Λ3ei defines an non-degenerated bilinear form, we have that ci≠0≠c′i. And we may take c′i=1, so we get
Similarly, there are ai,bi∈k∗ such that
and
For each i∈˜Q0, since i≠i+2et for 1≤t≤3, we have that
by Proposition 5.2.
So ˜ρ(s,r,C(a,b,c))⊂I.
Let ˜Λ=k˜Q/(˜ρ(s,r,C(a,b,c))), then ˜Λ(s,r) is a factor algebra of Λ. Then we have that dimkej˜Λ1ei=dimkej˜Λ1(G(s,r))ei for all i,j since ˜Λ and ˜Λ(s,r) have the same quiver. In fact, by using the same notation for a path in ˜Q and its image in ˜Λ, for each i∈˜Q0=Z/sZ×Z/rZ, we have that
and
by computing directly using the relations in ˜ρ(s,r,C(a,b,c)). So we have
and
This implies that
for any i,i′, by comparing the Loewy matrix of ˜Λ(s,r).
So
for all i,i′ and ˜Λ=˜Λ(s,r).
This proves I=(˜ρ(s,r,C(a,b,c))) is generated by ˜ρ(s,r,C(a,b,c)).
For the quadratic dual quiver ˜Q⊥(s,r), we have the following Proposition describe its relations, by an easy calculation.
Proposition 5.4. If the quotient algebra ˜Γ(s,r)=k˜Q(s,r)/I′ is Koszul AS-regular algebra of finite Gelfand-Kirilov dimension, then the relation set can be chosen as
Proof. Let ˜Λ(s,r)=k˜Q(s,r)/(˜ρ(s,r)) for some as defined in (17). It is immediate that and are both linear independent sets of elements in the subspace spanned by paths of length , and by identifying the space of the paths of length in with its dual space, i.e., for and . For each pair , it is immediate that so and span orthogonal subspaces in . By Proposition 5.3, we have that if is a Koszul -translation algebra then is generated by . Thus if is Koszul AS-regular algebra of finite Gelfand-Kirilov dimension then is generated by for some , by Theorem 5.1 of [25].
Now construct the quiver from , this is an infinite stable acyclic -translation quiver. It has connected components when are both multiples of , and has only one connected component otherwise.
Recall by (2), the relation set for is defined as . Write
and
for . So we have for each ,
and let
By taking a connected component in the complete -slice of , one obtains a -nicely-graded quiver as follows:
Here we denote by the vertex in , so is the arrow from to , is the arrow from to and is the arrow from to .
Let , the relation set
Since any sequence of relations in such that the successive pair share an arrow does not form a circle, by changing the representative for the arrows, the coefficient in the commutative relations in can all be taken as , and
Proposition 5.5. If the quotient algebra is a -properly-graded algebra, then the ideal is generated by .
The quadratic dual quiver is a -slice quiver associated to the McKay quiver . It is easy to check that
Proposition 5.6. If the quotient algebra is a -slice algebra then is generated by .
So the relations for the -slice algebra are independent of the parameter set.
5.2. Relations for
Let be a subgroup of with McKay quiver . Let be the McKay quiver obtained by embedding in as in Proposition 4.2. Then is one of the following quivers:
The vertices for are indexed by . Write for the loop at the vertex , for the arrow from to with ( in the case ) and for the arrow from to with ( in the case ). For the sake of writing down the arrows properly, we do not follow the convention to index the vertices.
We have immediate the following on the arrows of these McKay quivers.
Lemma 5.7. Let be the McKay quiver defined above for , . Then
1. There is a loop at each vertex of ;
2. There is at most one arrow from to for any two vertices in ;
3. There is an arrow from to if and only if there is an arrow from to ;
4. There are at most arrows leaving a vertex, and at most arrows heading to a vertex.
Let be a -translation algebra for the ideal , we determine relations for .
Lemma 5.8. If , then
if such paths exist.
Proof. If , one sees from the quivers that is the only path of length from to , and there is no arrow from to . By using Proposition 5.2, we have that
Similarly we have
if such paths exist.
Write
Take
As a corollary of Lemma 5.8, we get the following.
Proposition 5.9. .
Let , that is, choose a nonzero number for each arrow , and set
Proposition 5.10. By choosing the representatives of the arrows suitably, we have a set such that
Proof. By Lemma 5.7, there is an arrow from to if and only if there is an arrow from to in . By Proposition 5.1, we see that for each arrow , we have that are bound paths and
for some , similar to the proof of Proposition 5.3. Now apply Proposition 5.1 for the arrow , we have that are bound paths and
for some . Take .
Starting from in cases and from in cases , by changing the representatives of arrows by one by one, we get for all . That is,
for all arrows in .
This proves that by choosing the representatives of the arrows suitably, we have
Write for the set of paths of length from to in . The set is a orthogonal basis of the subspace spanned by the paths of length in . We have immediately the following.
Lemma 5.11. For each arrow and nonzero , is a basis of the orthogonal subspace of in the space spanned by paths of length of .
Write
Now consider . By Lemma 5.7, for each vertex in , there is a loop at and the number of arrows heading to and the number of arrows leaving are the same, and there are at most arrows leaving .
Fix , write
Then is an arrow starting at and is an arrow heading to .
Consider the following cases.
Lemma 5.12. Assume that there is only one arrow leaves .
1. If , then there is a , such that ;
2. We have and for any
is a orthogonal basis for .
Proof. Apply Proposition 5.1 for the arrow , the image of and are linearly dependent. So there is a such that .
The second assertion follows from direct computation.
Lemma 5.13. Assume that there are exactly two arrows leave .
1. There is a , such that
2. If , then there is a such that .
3. We have and for any
are orthogonal bases for .
Proof. By Proposition 5.1, we have that the images of and of are bound paths and they are linearly dependent. So there is a such that .
If , then its image is linearly dependent on , so there is a such that .
The rest follows from direct computations.
Lemma 5.14. Assume that there are three arrows leave with .
1. There are , such that
2. If , then there is a such that .
3. We have and for any ,
are orthogonal bases of .
Proof. The lemma follows from Proposition 5.1, similar to above two lemmas.
Denote by the set of vertices in from which only one arrow leaves, the set of vertices in from which exact two arrows leave and the set of vertices in from which three arrows leave. Set
and set
Let
and let
Lemma 5.15. A basis of the orthogonal subspace of and of are respectively:
and let
Proof. This follow immediately from (2) of Lemma 5.12, (3) of Lemma 5.13 and (3) of Lemma 5.14.
Fix , for a set
for and as defined in (29) and (30), set
By Lemma 5.11 and Lemma 5.15, we have the following.
Proposition 5.16.
Proposition 5.17. Let be , , with , and . If is -translation algebra then there is a subset of and a set as in (29) of nonzero elements in , such that by choosing the representatives of the arrows properly, we have is generated by .
Proof. Take . Then by choosing the representatives of the arrows properly, we have that there is a set as in (29), such that , by Propositions 5.9, 5.10 and Lemmas 5.12, 5.13 and 5.14.
Let . By Lemma 5.8, we have that if there is no arrow from to , by Lemma 5.11, we have that if there is an arrow from to and by (2) of Lemma 5.12, (3) of Lemma 5.13 and (3) of Lemma 5.14, and for all . Since is -translation and the Loewy matrix of is (16) with and , this implies that is a generating set of .
It is also immediate that spans the orthogonal spaces of in . So we have the following characterization of relations for .
By Lemma 5.16, a quadratic dual relation of is defined in (37). So we get the following, similar to the proof of Proposition 5.4.
Proposition 5.18. Let be , , with , and . If is Koszul AS-regular algebra of finite Gelfand-Kirilov dimension then there is a subset of and a set as in (29) of nonzero elements in , such that .
Now construct the nicely-graded quiver over , it has only one connected component since has loops. With the relation set
for some parameter set , forms a -translation quiver with -translation algebra
if is an -translation algebra.
By taking the complete -slices , we obtain the following quivers:
They are all nicely-graded quivers. We get -properly-graded quivers by taking the relation
For any parameter set , we can replace the representatives for the arrows and such that all the parameters become .
Write
Let
and let
and let
Take
and
Write
They are subsets of the space spanned by the paths of length , and and span orthogonal subspaces in the subspace spanned by .
Take a subset . Let
and
We have the following descriptions of the relation sets for the -slice algebras and for the -slice algebras associated to the McKay quiver .
Proposition 5.19. Let be , , with , and . If is -slice algebra with a -translation algebra then there is a subset of such that .
Proposition 5.20. Let be , , with , and . If is -slice algebra, then there is a subset of such that .
5.3. Remarks
1. By constructing for given , and taking a complete -slice in one of its connected components, we get an indecomposable -slice algebra , and a -slice algebra . For a given complete -slice, we can change the representatives of the arrows so the relation set has the same expressions, so the -slice algebras and the -slice algebras are isomorphic, respectively (for a fixed subset of the vertices in the case ). The algebra dependents only on the quiver and the set , but is independent of the parameters in .
2. The complete -slices in are usually non-isomorphic as quivers. But they all can be obtained from any one of them by a sequence of -mutations by [12]. By [17], the corresponding -slice algebras are obtained by a sequence of -APR tilts from the one described in Propositions 5.6 and 5.20.
3. Though we need the field containing for defining the McKay quiver, our results in this section are valid for any field, we only need the McKay quiver and the combinatory data in the Loewy matrix related to the McKay quiver, at least in case of .
Acknowledgments
We would like to thank the referees for reading the manuscript carefully and for suggestions and comments on revising and improving the paper. They also thank the referee for bring [24] to their attention.