Auslander's Theorem for dihedral actions on preprojective algebras of type A

Given an algebra $R$ and $G$ a finite group of automorphisms of $R$, there is a natural map $\eta_{R,G}:R\#G \to \mathrm{End}_{R^G} R$, called the Auslander map. A theorem of Auslander shows that $\eta_{R,G}$ is an isomorphism when $R=\mathbb{C}[V]$ and $G$ is a finite group acting linearly and without reflections on the finite-dimensional vector space $V$. The work of Mori and Bao-He-Zhang has encouraged study of this theorem in the context of Artin-Schelter regular algebras. We initiate a study of Auslander's result in the setting of non-connected graded Calabi-Yau algebras. When $R$ is a preprojective algebra of type $A$ and $G$ is a finite subgroup of $D_n$ acting on $R$ by automorphism, our main result shows that $\eta_{R,G}$ is an isomorphism if and only if $G$ does not contain all of the reflections through a vertex.


Introduction
In [18], Qin, Wang, and Zhang initiated a study of the McKay correspondence for non-connected N-graded algebras in (global) dimension two. An important component to this study is Auslander's Theorem. This project is an attempt to study this result in the context of preprojective algebras of type A.
Let V be a finite-dimensional vector space and G a finite group acting linearly on R = C[V ]. The Auslander map γ R,G : R#G → End R G R is defined as Define a multiplication on R#G by (r 1 #g 1 )(r 2 #g 2 ) = r 1 g 1 (r 2 )#g 1 g 2 , r i #g i ∈ R#G, extended linearly. We call R#G the skew group ring R#G. Note that, under this definition, the group ring kG is k#G with trivial G action.
A quiver Q is a tuple (Q 0 , Q 1 , s, t) consisting of a set of vertices Q 0 = {e 0 , . . . , e n }, a set of arrows Q 1 , and source and target functions s, t : Q 1 → Q 0 . For any arrow α, we call the vertex s(α) the source of α and the vertex t(α) the target of α. We say Q is finite if Q 0 and Q 1 are finite, and schurian if given any two vertices i and j there is at most one arrow with source i and target j. The adjacency matrix of a quiver Q is the matrix M Q in which (M Q ) ij denotes the number of arrows i → j.
We denote the set of paths of length ℓ by Q ℓ , and treat the vertices e i as trivial paths of length 0 with s(e i ) = t(e i ) = e i .
Let Q be a quiver. The path algebra of a quiver Q over the field k, denoted kQ, is the algebra with k-basis the set of paths ∞ ℓ=0 Q ℓ and multiplication defined by concatenation. That is, given paths p = α 1 · · · α ℓ and q = β 1 · · · β k , pq = α 1 · · · α ℓ β 1 · · · β k if t(p) = s(q) and pq = 0 otherwise. For any e ∈ Q 0 and any path p, we also have the following: Note that a path algebra kQ on a finite quiver is graded by {(kQ) ℓ } ∞ ℓ=0 where (kQ) ℓ = Span k (Q ℓ ), and there are only finitely many paths of any given length, so kQ is locally finite.
The double of a quiver Q, denoted Q, is defined by setting Q 0 = Q 0 and for every arrow α ∈ Q 1 with s(α) = e i and t(α) = e j , we add an arrow α * with s(α * ) = e j and t(α * ) = e i . We call Q 1 the set of nonstar arrows and Q * 1 := Q 1 \ Q 1 the set of star arrows.
The type A extended Dynkin quivers A n−1 , n ≥ 3, are given by e 0 e 1 e 2 · · · e n−2 e n−1 α0 α1 α2 αn−3 αn−2 αn−1 We focus on preprojective algebras corresponding to this type. If Q = A n−1 , then Q can be characterized as follows. The vertex set Q 0 of Q is {e 0 , . . . , e n−1 }, and there is exactly one nonstar arrow, α i , from e i to e i+1 and one star arrow, α * i , from e i+1 to e i for each i = 0, . . . , n − 1 where the index is taken mod n. That is, Q is schurian. For example, the double of A 2 is presented below: We note that A 1 is also defined. However, as its double is not schurian, it does not fit into the theory we have developed.
The preprojective algebra Π An−1 has nice ring-theoretic properties, as discussed in the next proposition. Before this, we review some of the definitions. For others, such as graded injectively smooth and the generalized Gorenstein condition, we refer to [20] and [22], respectively.
Let R = n∈N R n be a locally finite graded algebra. The Gelfand-Kirillov (GK) dimension of R is defined as Now let δ be a dimension function on R. For example, we may have δ = GKdim or δ = Kdim, the Krull dimension. The ring R is δ-Cohen-Macaulay (δ-CM) if δ(R) is finite, and for every nontrivial finitely The (total) Hilbert series of R is the formal power series If R 0 = k n with primitive idempotents {e 0 , . . . , e n−1 }, then the matrix-valued Hilbert series is the matrix with entries  [20,Lemma 4.3]. The statement on the Hilbert series follows from [6] and [19].
Proof. Given any idempotent e i ∈ Q 0 , e i Ωe i ∈ (Ω) and so the following relations hold in Π An−1 : where the indices are taken mod n.
Given any star arrow α * i , the only nonstar arrow β such that α * i β = 0 is α i . Consequently, whenever a star arrow is followed by a nonstar arrow, we can use (2.4) to obtain a nonstar arrow followed by a star arrow instead. By repeated use of (2.4) we have By induction on the number of star arrows, it follows that we can push all star arrows to the right.
The invariant theory of preprojective algebras was studied by Weispfenning, with particular interest towards a version of the Shephard-Todd-Chevalley theorem [21,22]. Our interest is in a version of Auslander's Theorem for group actions on the projective algebra R = Π An−1 . Particularly relevant to the present investigation is the following theorem due to Bao, He, and Zhang. Let R ′ be the image of R in the composition We call R ′ the identity component of (R#G)/(f G ), which we can associate with R. By  Since R is locally finite, k ∪ ℓ<m R ℓ is finite dimensional for all m ∈ N. Consequently, if there exists m ∈ N such that every path of length at least m is in (f G ), then R ′ has a complete set of coset representatives We conclude this section with a discussion of graded automorphisms of Π An−1 .
Conversely, if σ is a graded automorphism of Π An−1 , then σ 0 permutes the set Q 0 . Since Q is schurian, then for any α ∈ Q 1 , σ(α) is necessarily a nonzero scalar multiple of the unique arrow from σ 0 (s(α)) to σ 0 (t(α)). First, we will consider automorphisms which fix the vertices of A n−1 . In Section 3, we study automorphisms corresponding to dihedral automorphisms on A n−1 .
For the remainder of this section, let Q = A n−1 and R = Π An−1 . Let Fixed subrings of R under automorphisms in F were studied by Weispfenning [21,22].
must be a root of unity. On the other hand, σ(Ω) ∈ Span k (Ω) and so σ(Ω) = ωΩ for some ω ∈ k × . It is not difficult to show using the preprojective relation that ξ i ξ * i = ω for all i. The value ω in this case is the homological determinant of the σ-action on R [15,22]. We will consider cases in which ω = 1.
Our primary tool for studying the Auslander map for cyclic subgroups of F is the following result of He and Zhang, which we have rephrased for our purpose.
We now apply Lemma 2.7 to establish an isomorphism of the Auslander map for certain scalar automorphisms.
In each of the following cases, η R,G is an isomorphism.
Proof. First, suppose there exists a pure nonstar path q of length ℓ such that q#1 ∈ (f G ). Then let p be a path containing at least 2ℓ nonstar arrows. Pushing all nonstar arrows to the left using the Structure Lemma (Lemma 2.3), it follows that p contains q, so p#1 ∈ (f G ). The same argument applies if q, p are pure star paths. Hence, if (f G ) contains both a pure nonstar path and a pure star path of length ℓ, then dim k (R ′ ) < ∞ and η R,G is an isomorphism by Theorem 2.6. Thus, in each case we will attempt to produce such paths.
(1) Assuming ξ i = ζ as in the hypothesis, take a i = α i and apply Lemma 2.7. Taking indices mod n, it follows that a 0 · · · a m−1 ∈ (f G ) is a pure nonstar path. One similarly obtains a pure star path. Hence, η R,G is an isomorphism by the above argument.
(2) Let p be any pure nonstar path of length n, and q any pure star path of length n. Since p contains each nonstar arrow exactly once, σ(p) = ζp. No power of p is zero and so we apply Lemma 2.7 with a i = p Similarly we obtain q m #1 ∈ (f G ). Hence, η R,G is an isomorphism.
(3) The order of σ is determined by its image on R 1 which in turn is determined by its image on Since the orders of the scalars ξ i are relatively prime, then we have |ζ| = |ξ 0 |·|ξ 1 | · · · |ξ n−1 | = |σ|.
The result now follows from (2).

Dihedral actions on Π An−1
In this section we establish our main theorem regarding the Auslander map for dihedral actions on Π An−1 .
Let σ be a quiver automorphism of a schurian quiver Q. As discussed above, if α ∈ Q 1 , then σ(α) is a scalar multiple of the unique arrow from σ 0 (s(α)) to σ 0 (t(α)). Throughout this section, we assume that scalar is 1.
For the remainder of this section, let Q = A n−1 and R = Π An−1 . We will show that there is a group of quiver automorphisms of Q that is isomorphic to the dihedral group on n vertices. We first identify two quiver automorphisms of Q which extend to automorphisms of R.
By Proposition 3.1, G = ρ, r extends to a subgroup of Aut gr (R) where R = Π Q . It is clear that G ∼ = D n , and so we identify D n with the group G acting on R by graded automorphisms.
Theorem 3.2. Let G be a subgroup of D n . If there exists a reflection τ ∈ D n that fixes a vertex and τ / ∈ G, Proof. Let τ be the reflection that fixes e i and suppose τ / ∈ G. Since τ is the only nontrivial element of D n that fixes e i , we have e i g(e i ) = 0 for all g ∈ G \ {1}. Consequently e i (f G )e i = e i #1. Let p be a path of length at least 2n + 1, so p contains at least n + 1 nonstar arrows or at least n + 1 star arrows. Without loss of generality, suppose p has at least n + 1 nonstar arrows. By the Structure Lemma (Lemma 2.3), we may push all star arrows to the right, so that p = α j α j+1 · · · α j+n−1 α j+n p ′ for some path p ′ and some j = 0, . . . , n − 1 where the indices are taken mod n. Then for some 0 ≤ k ≤ n − 1, i = j + k + 1 mod n, so p = (α j · · · α j+k )e i (α j+k+1 · · · α j p ′ ) Hence p#1 ∈ (f G ) and so q#1 ∈ (f G ) for all paths q of length at least 2m + 1. Thus, dim k (R ′ ) < ∞.
Theorem 3.2 shows that the Auslander map is an isomorphism for the pair (R, G) so long as G is missing a reflection which fixes some vertex. In case n is odd, this includes all proper subgroups of D n . However, in the case that n is even, there is one additional subgroup, W n . It remains to show that the Auslander map fails to be an isomorphism in the case of W n and the full dihedral group D n .
3.1. The D n case. For x ∈ R, we denote by the O(x) the orbit of x under D n . We begin by describing the orbits of R under the D n action so as to find a k-basis of R Dn .
Recall that for k ≥ 0, we let Q k (resp. Q * k ) denote the set of paths of length k containing only nonstar (resp. star) arrows, and Q 0 = Q * 0 is the set of trivial paths. Further, let Q ℓ Q * k denote the set of paths containing exactly ℓ nonstar arrows followed by k star arrows. Then in the double quiver, we have Clearly, Q ℓ is a generating set for the graded piece R ℓ of R. Finally, for ℓ ≥ k ≥ 0, set B ℓ,k = Q ℓ Q * k ∪ Q k Q * ℓ .

Lemma 3.3. For any
Proof. By Structure Lemma (Lemma 2.3), a path p is uniquely determined by its source along with the number of nonstar and star arrows it contains. Consequently, each p ∈ Q ℓ Q * k is uniquely determined by its source, as is each q ∈ Q k Q * ℓ . Thus for each i = 0, . . . , n − 1, let p i (resp. q i ) denote the unique path in Q ℓ Q * k (resp. Q k Q * ℓ ) with source e i . Then B ℓ,k = {p 0 , . . . , p n−1 , q 0 , . . . , q n−1 }. Let x ∈ B ℓ,k and y ∈ O(x), so y = g(x) for some g ∈ D n . If g is a rotation, then g bijectively maps Q 1 to Q 1 and Q * 1 to Q * 1 . Consequently g(x) has the same number of nonstar arrows as x, and the same number of star arrows as x. That is, if x ∈ Q ℓ Q * k , then y ∈ Q ℓ Q * k . Thus y ∈ B ℓ,k . If g is a reflection, then g bijectively maps Q 1 to Q * 1 and Q * 1 to Q 1 . Hence g(x) has the same number of nonstar arrows as x has star arrows, and the same number of star arrows as x has nonstar arrows. That is, if We have |O(x)| = |D n |/| stab(x)|, and g ∈ stab(x) only if g fixes the source of x. Hence g is the identity or the unique reflection r fixing s(x). Now, if ℓ = k, then Q ℓ Q * k = Q k Q * ℓ so |B ℓ,k | = 2n, and r inverts the number of star and nonstar arrows, so r( By Lemma 3.3, these are exactly the orbit sums of homogeneous elements in R, and hence form a k-basis for R Dn . This shows that R Dn has Hilbert series .  Proof. To prove (3.5), we suppose that ℓ > k and then On the other hand, if ℓ = k, then For (3.6), the result is obvious if m = 1. Suppose it holds for some m, then Proof. We recall first that for every arrow α there is exactly one nonstar arrow β and one star arrow γ such that αβ = 0 and αγ = 0. Using this fact and the preprojective relation we have, Proof. We already have Then By another induction, we have O(ℓ, k) ∈ k[s 1 , s 2 ] with ℓ + k = d + 1.

Now assume d is odd. Then d + 1 is even and since
. Now the argument proceeds as in the even case.
We now proceed to our main result for this section. is a basis for R over R Dn . That is, R is a rank 2n free module over R Dn . Then we have as R Dn -modules. Since α n−1 R Dn ∼ = e 0 R Dn (−1), then End R Dn R contains a map of negative degree and so the Auslander map is not an isomorphism for (R, D n ).
First we show that the set S generates R as a R Dn -module. Clearly R 0 ⊂ Span R Dn S. Moreover, for all . Hence, R 1 ⊂ Span R Dn S. Note that there are exactly three paths of degree 2 for each vertex. Consider the degree 2 paths based at vertex 0. We have Hence, {α 0 α * 0 , α 0 α 1 , α * n−1 α * n−2 } ⊂ Span R Dn S. A similar argument for the remaining vertices shows that R 2 ⊂ Span R Dn S.
In particular, the above argument shows that for all m. Thus, R is generated as a right R Dn -module by R 0 and R 1 . It follows that R ⊂ Span R Dn S. That is, S is a generating set for R as an R Dn module.
For independence, we note that for every element of S there is exactly one other element in S with the same source. Hence, it suffices to prove that e i R Dn ∩ α i R Dn = {0}. We do this computation for i = 0 and the other vertices follow similarly.
Suppose a ∈ e i R Dn ∩ α i R Dn . We may assume without loss of generality that a is homogeneous of degree d. Suppose first that d is even. Then there exist scalars k i , k ′ i ∈ k such that From the second expression, we note that every path summand of a must contain at least one non-starred arrow. Hence, k 0 = 0. But then from the first expression we note that every path summand of a must contain at least one starred arrow, so k ′ 0 = 0. Continuing in this way, we see that a = 0.
3.2. The W n case. In case n is odd, η R,G is an isomorphism if and only if G is a proper subgroup of D n .
In case n is even, there is one additional instance when η R,G fails to be an isomorphism, namely for the subgroup W n defined as: W n = τ ∈ D n : τ (e i ) = e i for some i = 0, . . . , n − 1 .
That is, W n is generated by the reflections in D n that pass through a vertex. If n is odd, then every reflection fixes a vertex, so W n contains every reflection and W n = D n . If n is even, only half of the reflections fix a vertex, so W n is a proper subgroup of D n . Since W n is of index 2 in D n , W n is a maximal subgroup of D n .
Throughout this section we assume n is even. Our strategy will be similar to the previous section. The key difference is that the invariant ring is no longer connected graded. In particular, there are exactly twice as many orbits in each graded piece as in the D n case.
As in the previous section, for ℓ ≥ k ≥ 0, set B ℓ,k = Q ℓ Q * k ∪ Q k Q * ℓ . Then define B even ℓ,k = e i B ℓ,k for i even and B odd ℓ,k = e i B ℓ,k for i odd.
Proof. This is similar to the proof of Lemma 3.3. In particular, the B ℓ,k partition the paths of Q. However, since g ∈ W n preserves the parity of the idempotents e i , it follows that g(B even ℓ,k ) ⊂ B even ℓ,k . Because g is bijective then in fact we have equality.
It remains to show that we have an equivalence with the orbits. If p ∈ B even ℓ,k , then clearly O(p) ⊂ B even ℓ,k . Since |B even ℓ,k | = 1 2 B ℓ,k and |W n | = 1 2 D n , then it follows from the argument in Lemma 3.3 that |B even ℓ,k | = |O(p)|. A similar argument applies to |B odd ℓ,k |.

Set
O(ℓ, k) even = p∈B even These form a k-basis for R Wn . Thus, R Wn has total Hilbert series .
However, since (R Wn ) 0 = k 2 , then we can also record the matrix-valued Hilbert series. Let M be the 2 × 2 matrix defined by M 0,0 = #{p ∈ B even ℓ,k with target e i , i even}, M 0,1 = #{p ∈ B even ℓ,k with target e i , i odd}, Note that for p ∈ B ℓ,k , the parity of the target depends on the source and the parity of ℓ + k. Hence, it follows that the matrix-valued Hilbert series of R Wn is Proofs of the relations in the next lemma are similar to the corresponding proofs in Lemma 3.4. Let C denote the subalgebra of R Wn generated by these elements. Let Q be the following quiver: and let kQ denote its path algebra. We assign degree 1 to the arrows u 1 , u 2 and degree 2 to v 1 , v 2 . We will show that C = R Wn and that Remark 3.14. The algebra kQ/(v 1 u 1 − u 1 v 2 , v 2 u 2 − u 2 v 1 ) is a quotient-derivation algebra appearing in the classification of graded twisted Calabi-Yau algebras of global dimension 2 [19]. In particular, the matrix corresponding to the Nakayama automorphism µ is ( 0 1 1 0 ) and the µ-twisted superpotential is Proof. We prove the first relation. The second is similar.
Thus, O(1, 1) even ∈ C. A similar proof with s ′ 1 s 1 shows that O(1, 1) odd ∈ C so that (R Wn ) 2 ⊂ C. The remainder of the proof follows similarly to Lemma 3.8 with proper respect shown towards parity. In particular, we use (3.12)-(3.13).
Theorem 3.17. The Auslander map is not an isomorphism for the pair (R, W n ).
Proof. Denote the trivial paths of Q by f 0 , f 1 . There is a map φ : kQ → R Wn defined by setting It is easy to verify that this determines a well-defined surjective map kQ → R Wn and belongs to ker φ. By comparing the matrix-valued Hilbert series, it is clear that kQ/K ∼ = R Wn .