aa r X i v : . [ m a t h . R T ] J u l The natural quiver of an artinian algebra ∗ Fang Li † and Lili Chen ‡ Department of Mathematics, Zhejiang UniversityHangzhou, Zhejiang 310027, ChinaJuly 6, 2018
Abstract
The motivation of this paper is to study the natural quiver of an artinian algebra,a new kind of quivers, as a tool independing upon the associated basic algebra.In [5], the notion of the natural quiver of an artinian algebra was introduced andthen was used to generalize the Gabriel theorem for non-basic artinian algebras split-ting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicalsvia pseudo path algebras and generalized path algebras respectively.In this paper, firstly we consider the relationship between the natural quiver andthe ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabrieltheorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type alge-bras are introduced to outline those artinian algebras satisfying the generalized Gabrieltheorem here and in [5]. For such algebras, the uniqueness of the related generalizedpath algebra and quiver holds up to isomorphism in the case when the ideal is admis-sible. For an artinian algebra, there are two basic algebras, the first is that associatedto the algebra itself; the second is that associated to the correspondent generalizedpath algebra. In the final part, it is shown that for a Gabriel-type artinian algebra,the first basic algebra is a quotient of the second basic algebra.In the end, we give an example of a skew group algebra in which the relationbetween the natural quiver and the ordinary quiver is discussed.
Suppose that A is a left artinian algebra over a field k , and r = r ( A ) is the radical of A .In this paper left artinian algebras are written briefly as “artinian algebras”. ∗ Project supported by the Program for New Century Excellent Talents in University (No.04-0522) andthe National Natural Science Foundation of China (No.10571153) † [email protected] ‡ [email protected] NATURAL QUIVER AND THE RELATION WITH ORDINARY QUIVER { S , S , · · · , S n } be the complete set of non-isomorphic simple A -modules of A .One can define a finite quiver Γ A , called the ordinary quiver of A as follows: Γ = { , , · · · , n } , and the number m ij of arrows from i to j equals to the dimensional numberdim k Ext A ( S i , S j ). By [2], when A is a finite-dimensional basic algebra over an algebraicallyclosed field k and 1 A = ε + · · · + ε n a decomposition of 1 A into a sum of primitive or-thogonal idempotents. Then, we can re-index { S , S , · · · , S n } such that S i ∼ = Aε i /rε i ,and moreover, dim k Ext A ( S i , S j ) =dim k ( ε j r/r ε i ). Clearly, if Q is a finite quiver withoutoriented cycles, the ordinary quiver of the path algebra kQ is just Q .Now, we introduce the so-called natural quivers from artinian algebras.Write A/r = L si =1 A i where A i is a simple ideal of A/r for each i . Then, the algebra r/r is an A/r -bimodule by ¯ a · ( r/r ) · ¯ b = arb/r for any ¯ a = a + r, ¯ b = b + r ∈ A/r . Let i M j = A i · r/r · A j , then i M j is finitely generated as A i - A j -bimodule for each pair ( i, j ).For two artinian algebras A and B , the rank of a finitely generated A - B -bimodule M is defined as the least cardinal number of the sets of generators. Clearly, for any finitelygenerated A - B -bimodule, such rank always exists uniquely.Now we can associate with A a quiver ∆ A = (∆ , ∆ ), which is called the naturalquiver of A , in the following way. Let ∆ = { , · · · , s } as the set of vertices. For i, j ∈ ∆ ,let the number t ij of arrows from i to j in ∆ A be the rank of the finitely generated A j - A i -bimodules j M i . Obviously, if j M i = 0, there are no arrows from i to j .The notion of natural quiver was firstly introduced in [5], where the aim of the authoris to use the generalized path algebra from the natural quiver of an artinian algebra A tocharacterize A through the generalized Gabriel theorem. In the further research, one ismotivated to study the representation of an artinian algebra via the associated generalizedpath algebra or pseudo path algebra but not the basic algebra of the artinian algebra.In order to clean the relation between the ordinary quiver and the natural quiver ofan artinian algebra, it is necessary to note that the natural quiver defined here is indeedopposite to the quiver defined in [5]. Now, we consider the relation between the ordinaryquiver and the natural quiver of an artinian k -algebra over an algebraically closed field k .Clearly the number of the vertices in two quivers are equal since A and A/r have thesame simple modules, that is, we have n = s as above.When A is a finite-dimensional basic algebra over an algebraically closed field k , A/r ∼ = Q k where the number of copies of k equals the number of primitive orthogo-nal idempotents. As mentioned above, dim k Ext A ( S i , S j ) = dim k ( ε j r/r ε i ) where S i , S j are the simple modules of A corresponding to the primitive orthogonal idempotents ε i , ε j respectively, which means that the number of arrows from i to j in the ordinary quiverΓ A of A is equal to that in the natural quiver ∆ A of A . Thus, we have: Lemma 1.1.
For a finite-dimensional basic algebra A over an algebraically closed field k , the ordinary quiver and the natural quiver of A coincide. In particular, if Q is a GENERALIZED GABRIEL THEOREM IN THE RADICAL-GRADED CASE finite quiver without oriented cycles, the ordinary quiver and the natural quiver of the pathalgebra kQ are both Q . In order to discuss similarly for non-basic algebras, we introduce the following notion:Let Q be a quiver and Q ′ a sub-quiver of Q . If ( Q ′ ) = Q and for any vertices i, j ,there exist arrows from i to j in Q ′ if and only if there exist arrows from i to j in Q , thenwe call this Q ′ a dense sub-quiver of Q .When A is over an algebraically closed field k , by Proposition 7.4.4 in [6], the relation: t ij ≤ m ij ≤ n i n j t ij holds where n i and n j are integers such that A i ∼ = M n i ( k ) and A j ∼ = M n j ( k ). Trivially,if each n i = 1, then t ij = m ij , thus the ordinary quiver ∆ A and Γ A the natural quiver of A are coincided. But, when some n i = 1, it is possible that t ij < m ij , and t ij = 0 if andonly if m ij = 0, which means usually, ∆ A is a dense sub-quiver of Γ A .As well-known, for an artinian algebra A , there is the correspondent basic algebra B and they are Morita-equivalent, i.e. the module categories Mod A and Mod B are equiv-alent, which follows that there is an equivalent functor F such that Hom A ( S i , S j ) F ∼ = Hom B ( F ( S i ) , F ( S j )) for any simple modules S i and S j in Mod A . Moreover, Ext A ( S i , S j ) ∼ = Ext B ( F ( S i ) , F ( S j )). It means the ordinary quiver of A is the same with that of B . If A isof finite dimension, its basic algebra is also of finite dimension. In the summary, we have: Proposition 1.2.
Let A be a finite dimensional algebra over a field k with r its radicaland A/r = A ⊕ · · · ⊕ A n the direct sum of simple ideals, and B is the corresponding basicalgebra of A . Let Γ A and Γ B be the ordinary quivers of A and B respectively, meanwhile ∆ A and ∆ B the natural quivers of A and B respectively. Then,(i) Γ A = Γ B ;(ii) Γ B = ∆ B if k is algebraically closed;(iii) ∆ A is a dense sub-quiver of Γ A , also of Γ B and ∆ B , if k is algebraically closed. The concepts of generalized path algebras were introduced early in [3] in order to find ageneralization of path algebras so as to obtain a generalized type of the Gabriel Theoremfor arbitrary finite dimensional algebras which would admit this algebra to be isomorphicto a quotient algebra of such a generalized path algebra. It is natural to ask how welook for a generalized path algebra via the natural quiver to cover the artinian algebra.Unfortunately, in general, as shown by the counter-example in [5], an artinian algebrawith lifted quotient may not be a homomorphic image of its correspondent A -path-typetensor algebra. In this reason, the concepts of pseudo path algebras were introduced in[5] and it was shown that when the quotient algebra of an artinian algebra can be lifted, GENERALIZED GABRIEL THEOREM IN THE RADICAL-GRADED CASE A -path-type tensor algebras and equivalently by the generalizedpath algebras. This point can be seen in [5] from the generalized Gabriel theorem for afinite dimensional algebra with 2-nilpotent radical in the case it is splitting over its radical.In this section, we will give another class of artinian algebras which can be coveredby the generalized path algebra via the natural quiver, that is, the generalized Gabrieltheorem for this class of artinian algebras is true, too. This class of artinian algebras arejust the so-called radical-graded artinian algebra as follows.For an artinian algebra A , let r = rad A be the radical of A and the Loewy length rl ( A ) = s . Define gr A = A/r ⊕ r/r ⊕ · · · ⊕ r s − /r s − ⊕ r s − as a graded-algebra withmultiplication ( x + r i +1 )( y + r j +1 ) = xy + r i + j +1 for x ∈ r i , y ∈ r j . Trivially, this gradedalgebra is strict.An artinian algebra A is said to be radical-graded if A = L i ≥ A i is strictly gradedwith A semisimple. In this case, there is a minimal positive integer t such that A i = 0for all i ≥ t since A is artinian. By this definition, it is easy to see that for any artinianalgebra A , gr A is always radical-graded. We have the following characterization: Proposition 2.1.
An artinian algebra A is radical-graded if and only if A ∼ = gr A . In thissituation, A ∼ = A/r , r ∼ = r/r ⊕ · · · ⊕ r s − /r s − ⊕ r s − as algebras.Proof. “ ⇐ =” is trivial since gr A is radical-graded.“= ⇒ ”: Suppose that A = L i ≥ A i is strictly graded with A semisimple. Thus, thereis a minimal positive integer t such that A i = 0 for i ≥ t . Write r ′ = L i ≥ A i , clearly itis an ideal of A , A/r ′ = A semisimple and r ′ j = L i ≥ j A i which is zero when j ≥ t andhence r ′ is nilpotent. So r ′ = r the radical of A , and clearly r ′ i /r ′ i +1 ∼ = A i for i ≥ i ≥ t , and r ′ l = 0 if and only if A l = 0 for any l . Therefore t = s = rl ( A ),gr A = A/r ⊕ r/r ⊕ · · · ⊕ r s − ∼ = A/r ′ ⊕ r ′ /r ′ ⊕ · · · ⊕ r ′ s − = L i ≥ A i = A . (cid:3) From the above proposition, we get
Corollary 2.2.
For any artinian algebra A , gr(gr A ) ∼ = gr A . Now, we introduce briefly some notions about generalized path algebras.Let Q = ( Q , Q ) be a quiver and A = { A i : i ∈ Q } a family of k -algebras A i withidentity e i , indexed by the vertices of Q . The elements a j = 0 of S i ∈ Q A i are called A -paths of length zero , with starting vertex s ( a j ) and the ending vertex e ( a j ) are both j . For each n ≥
1, an A -path P of length n is given by a β a β · · · a n β n a n +1 , where( s ( β ) | β · · · β n | e ( β n )) is a path in Q of length n , for each i = 1 , ..., n , 0 = a i ∈ A s ( β i ) and0 = a n +1 ∈ A e ( β n ) . s ( β ) and e ( β n ) are also called respectively the starting vertex and theending vertex of P . Write s ( P ) = s ( α ) and e ( P ) = e ( α n ). Now, consider the quotient GENERALIZED GABRIEL THEOREM IN THE RADICAL-GRADED CASE R of the k -linear space with basis the set of all A -paths by the subspace generated by allthe elements of the form a β · · · β j − ( a j + · · · + a mj ) β j a j +1 · · · β n a n +1 − m X l =1 a β · · · β j − a lj β j a j +1 · · · β n a n +1 where ( s ( β ) | β · · · β n | e ( β n )) is a path in Q of length n , for each i = 1 , ..., n , a i ∈ A s ( β i ) , a n +1 ∈ A e ( β n ) and a lj ∈ A s ( β j ) for l = 1 , ..., m . In R , given two elements [ a β a β · · · a n β n a n +1 ] and [ b γ b γ · · · b n γ n b n +1 ], define the multiplication as follows:[ a β a β · · · a n β n a n +1 ] · [ b γ b γ · · · b n γ n b n +1 ]= ( [ a β a β · · · a n β n ( a n +1 b ) γ b γ · · · b n γ n b n +1 ] , if a n +1 , b ∈ A i for the same i , otherwiseIt is easy to check that the above multiplication is well-defined and makes R to becomea k -algebra. This algebra R defined above is called an A -path algebra of Q respecting to A , or generally generalized path algebras . Denote it by R = k ( Q, A ). Clearly, R is an A -bimodule, where A = ⊕ i ∈ Q A i .A generalized path algebra k ( Q, A ) is said to be normal if all algebras A i ( i ∈ Q ) aresimple algebras for A = { A i : i ∈ Q } .Associated with the pair ( A, A M A ) for a k -algebra A and an A -bimodule M , wewrite the n -fold A -tensor product M ⊗ A M ⊗ · · · ⊗ A M as M n . Writing M = A , then T ( A, M ) = A ⊕ M ⊕ M ⊕ · · · ⊕ M n ⊕ · · · becomes a k -algebra with multiplication inducedby the natural A -bilinear maps M i × M j → M i + j for i ≥ j ≥ T ( A, M ) is calledthe tensor algebra of M over A .Define a special class of tensor algebras so as to characterize generalized path algebras.An A -path-type tensor algebra is defined to be the tensor algebra T ( A, M ) satisfying that(i) A = L i ∈ I A i for a family of k -algebras A = { A i : i ∈ I } , (ii) M = L i,j ∈ I i M j where i M j are finitely generated A i - A j -bimodules for all i and j in I and A k · i M j = 0 if k = i and i M j · A k = 0 if k = j . A free A -path-type tensor algebra is the A -path-type tensoralgebra T ( A, M ) whose each finitely generated A i - A j -bimodule i M j for i and j in I is afree bimodule with a basis and the cardinality of this basis is equal to the rank of i M j asa finitely generated A i - A j -bimodule.In an A -path algebra k ( Q, A ), let A = L i ∈ Q A i . For any i , j , let i M Fj be the free A i - A j -bimodule with basis given by the arrows from j to i . Then the number of free generatorsin the basis is the rank of i M Fj as a finitely generated bimodule. Define A k · i M Fj = 0 if k = i and i M Fj · A k = 0 if k = j . Then M F = L j → ii M Fj is an A -bimodule. We get theunique free A -path-type tensor algebras T ( A, M F ).Conversely, given an A -path-type tensor algebra T ( A, M ) with A = { A i : i ∈ I } and finitely generated A i - A j -bimodules i M j for i, j ∈ I such that A = L i ∈ I A i , M = L i,j ∈ I i M j , A k · i M j = 0 if k = i and i M j · A k = 0 if k = j . Trivially, i M j = A i M A j . Let GENERALIZED GABRIEL THEOREM IN THE RADICAL-GRADED CASE r ij be the rank of j M i . One can associate with T ( A, M ) a quiver Q = ( Q , Q ), called thequiver of T ( A, M ), via Q = I as the set of vertices and for i, j ∈ I , r ij as the numberof arrows from i to j in Q . Its A -path algebra k ( Q, A ) is called the corresponding A -pathalgebra of T ( A, M ). By definition, the quiver of T ( A/r, r/r ) is just ∆ A .From the above discussion, every A -path-type tensor algebra T ( A, M ) can be usedto construct its corresponding A -path algebra k ( Q, A ); but, from this A -path algebra k ( Q, A ), we can get uniquely the free A -path-type tensor algebra T ( A, M F ). In summary,we have the following in [5]: Lemma 2.3. (i) Every A -path-type tensor algebra T ( A, M ) can be used to constructuniquely the free A -path-type tensor algebra T ( A, M F ) . There is a surjective k -algebramorphism π : T ( A, M F ) → T ( A, M ) such that π ( i M Fj ) = i M j for any i, j ∈ I ;(ii) Let T ( A, M F ) be the free A -path-type tensor algebra built by a A -path algebra k ( Q, A ) . Then there is a k -algebra isomorphism e φ : T ( A, M F ) → k ( Q, A ) such that forany t ≥ , e φ ( L j ≥ t ( M F ) j ) = J t ;(iii) Let T ( A, M ) be an A -path-type tensor algebra with the corresponding A -path alge-bra k ( Q, A ) . Then there is a surjective k -algebra homomorphism e ϕ : k ( Q, A ) → T ( A, M ) such that for any t ≥ , e ϕ ( J t ) = L j ≥ t M j .Here J denotes the ideal generated by all A -paths of length in k ( Q, A ) . In the sequel, we always denote by J the ideal generated by all generalized paths oflength one in the discussed generalized path algebras. When Q is admissible, i.e. is acyclic, J is just the radical of a normal generalized path algebra k ( Q, A ) (see [3]).A relation σ on an A -path algebra k (∆ , A ) is a k -linear combination of some A -paths P i with the same starting vertex and the same ending vertex, that is, σ = k P + ··· + k n P n with k i ∈ k and s ( P ) = · · · = s ( P n ) and e ( P ) = · · · = e ( P n ). If ρ = { σ t } t ∈ T is a setof relations on k (∆ , A ), the pair ( k (∆ , A ) , ρ ) is called an A -path algebra with relations .Associated with ( k (∆ , A ) , ρ ) is the quotient k -algebra k (∆ , A , ρ ) def = k (∆ , A ) / h ρ i , where h ρ i denotes the ideal in k (∆ , A ) generated by the set of relations ρ . When the length l ( P i )of each P i is at least j , it holds h ρ i ⊂ J j .Now, let M = r/r as A/r -bimodule, i M j = A i · r/r · A j , then i M j is finitelygenerated as A i - A j -bimodule for each pair ( i, j ) and M = L i,j i M j . Thus, we get thetensor algebra T ( A/r, r/r ) and the corresponding generalized path algebra k (∆ A , A ) fromthe natural quiver ∆ A of A .A set of some A -paths or their linear combinations in k ( Q, A ) is said to be A -finite ifall A -paths in this set are constructed from a finite number of paths in Q with elementsof S i ∈ Q A i . A quotient or an ideal of k ( Q, A ) is said to be A -finitely generated if it isgenerated by an A -finite set.The following is the main result in this section: GENERALIZED GABRIEL THEOREM IN THE RADICAL-GRADED CASE Theorem 2.4. (Generalized Gabriel Theorem in radical-graded case)
Assume that A is aradical-graded artinian k -algebra. Then, there is an A -finite set ρ of relations of k (∆ A , A ) such that A ∼ = k (∆ A , A ) / h ρ i with J s ⊂ h ρ i ⊂ J for some positive integer s .Proof : Let r be the radical of A with the Loewy length rl ( A ) = s + 1. Since A is radical-graded, we have A ∼ = A/r ⊕ r/r ⊕ r /r ⊕ · · · ⊕ r s − /r s ⊕ r s . Thus, r ∼ = r/r ⊕ r /r ⊕ · · · ⊕ r s − /r s ⊕ r s and A ∼ = A/r ⊕ r as algebras.Write A/r = ⊕ si =1 A i with simple ideals A i for all i . Then, we have the A -pathtype tensor algebra T ( A/r, r/r ) with A = { A i : i = 1 · · · s } . Firstly, we can find asurjective morphism of algebras from T ( A/r, r/r ) to A . In fact, for r m /r m +1 , define f m : r/r ⊗ A/r · · · ⊗
A/r r/r (with m copies of r/r ) −→ r m /r m +1 satisfying that f m ( x ⊗· · · ⊗ x m ) = x · · · x m where x i ∈ r/r for x i ∈ r and x · · · x m ∈ r m /r m +1 . It is easyto see that f m is well-defined as a morphism of A/r - A/r -bimodules and trivially, f m issurjective. Then, f = id A/r ⊕ f ⊕ · · · ⊕ f m ⊕ · · · is a surjective algebra morphism from T ( A/r, r/r ) to A , where f m = 0 when m ≥ s + 1.Moreover, by Lemma 2.3, there is a surjective k -algebra homomorphism e ϕ : k (∆ A , A ) → T ( A/r, r/r ) such that for any t ≥ e ϕ ( J t ) = L j ≥ t ( r/r ) ⊗ j , where ( r/r ) ⊗ j denotes r/r ⊗ A/r r/r ⊗ A/r ···⊗
A/r r/r with j copies of r/r . Then, f e ϕ : k (∆ A , A ) → A is a surjec-tive algebra morphism. Therefore, for the kernel I = ker ( f e ϕ ), we obtain k (∆ A , A ) /I ∼ = A .Now, we prove that ⊕ j ≥ rl ( A ) ( r/r ) ⊗ j ⊂ Kerf ⊂ ⊕ j ≥ ( r/r ) ⊗ j . In fact, by the def-inition, f = id r/r , so f | A/r ⊕ r/r = id A/r ⊕ f : A/r ⊕ r/r −→ A is a monomor-phism with image intersecting r trivially. It follows that Kerf ⊂ ⊕ j ≥ ( r/r ) ⊗ j . On theother hand, f (( r/r ) ⊗ j ) = 0 for j ≥ rl ( A ) since r j = 0 in this case. Therefore we get ⊕ j ≥ rl ( A ) ( r/r ) ⊗ j ⊂ Kerf .But, by Lemma 2.3, for t = rl ( A ) and t = 2 respectively, e ϕ ( J rl ( A ) ) = ⊕ j ≥ rl ( A ) ( r/r ) ⊗ j and e ϕ ( J ) = ⊕ j ≥ ( r/r ) ⊗ j . So, e ϕ ( J rl ( A ) ) ⊂ Kerf ⊂ e ϕ ( J ).Then, we prove J t ⊂ e ϕ − e ϕ ( J t ) ⊂ J t + e φ ( ⊕ j ≤ t − (( r/r ) F ) ⊗ j ) ∩ e φ ( Kerπ ) for t ≥ e ϕ is that in Lemma 2.3. Trivially, J t ⊂ e ϕ − e ϕ ( J t ). On the other hand, e ϕ = π e φ − and then e ϕ − = e φπ − . By Lemma 2.3(iii), e ϕ ( J t ) = ⊕ j ≥ t ( r/r ) ⊗ j . From the defini-tion of π in Lemma 2.3, it can be seen that π − ( ⊕ j ≥ t ( r/r ) ⊗ j ) ⊂ ⊕ j ≥ t (( r/r ) F ) ⊗ j +( ⊕ j ≤ t − (( r/r ) F ) ⊗ j ) ∩ Ker π . Thus, by Lemma 2.3, we have e ϕ − e ϕ ( J t ) = e φπ − ( ⊕ j ≥ t ( r/r ) j ) ⊂ e φ ( ⊕ j ≥ t (( r/r ) F ) ⊗ j )+ e φ ( ⊕ j ≤ t − (( r/r ) F ) ⊗ j ) ∩ e φ ( Kerπ )= J t + e φ ( ⊕ j ≤ t − (( r/r ) F ) ⊗ j ) ∩ e φ ( Kerπ ).Hence, J rl ( A ) ⊂ e ϕ − e ϕ ( J rl ( A ) ) ⊂ e ϕ − ( Kerf ) = I ⊂ e ϕ − e ϕ ( J ) ⊂ J + e φ ( ⊕ j ≤ (( r/r ) F ) ⊗ j ) ∩ e φ ( Kerπ ) ⊂ J + J ∩ e φ ( Kerπ ).But, e φ ( Kerπ ) = e φ ( π − (0)) = e ϕ − (0) = Ker e ϕ . Then, J rl ( A ) ⊂ I ⊂ J + J ∩ Ker e ϕ ⊂ J. Lastly, we present I through an A -finite set of relations on k (∆ A , A ). J rl ( A ) is theideal A -finitely generated in k (∆ A , A ) by all A -paths of length rl ( A ). k (∆ A , A ) /J rl ( A ) TWO BASIC ALGEBRAS FROM AN ARTINIAN ALGEBRA A -finitely, under the meaning of isomorphism, by all A -paths of length lessthan rl ( A ), so as well as I/J rl ( A ) as a k -subspace. Then, I is an A -finitely generatedideal in k (∆ A , A ). Assume { σ l } l ∈ Λ is a set of A -finite generators for the ideal I . For theidentity 1 of A/r , we have the decomposition of orthogonal idempotents 1 = e + · · · + e s ,where e i is the identity of A i . Then σ l = 1 σ l P ≤ i,j ≤ s e i σ l e j . Obviously, e i σ l e j can beexpanded as a k -linear combination of some such A -paths which have the same startingvertex j and the same ending vertex i . So, σ ilj = e i σ l e j is a relation on the A -pathalgebra k (∆ A , A ). Moreover, I is generated by all σ ilj due to σ l = P i,j σ ilj . Therefore,for ρ = { σ ilj : 1 ≤ i, j ≤ s, l ∈ Λ } , we get I = h ρ i . Hence k (∆ A , A , ρ ) = k (∆ A , A ) / h ρ i ∼ = A with h ρ i = Ker ( f e ϕ ) and J rl ( A ) ⊂ h ρ i ⊂ J + J ∩ Ker e ϕ ⊂ J . (cid:3) The uniqueness of the correspondent generalized path algebra and natural quiver of aradical-graded artinian algebra holds up to isomorphism if the ideal h ρ i is restricted into J . That is, if there exists another quiver and its related generalized path algebra such thatthe same isomorphism relation as in Theorem 2.4 is satisfied, then this quiver and relatedgeneralized path algebra are just respectively the natural quiver and the corresponding oneof the radical-graded artinian algebra. This can be seen as a special case of the uniquenessof the so-called Gabriel-type algebras, see Theorem 3.3 in the next section. For an artinian algebra A , write A/r = L si =1 A i with simple ideals A i , we get k (∆ A , A )where ∆ A is the natural quiver of A and A = { A i : i = 1 , , · · · , s } .It is known that the associated basic algebra B which is Morita-equivalent to A isimportant for representations of A . In order to realize our approach, it is valid to considerthe associated basic algebra C of the generalized path algebra k (∆ A , A ) of the naturalquiver ∆ A of A and moreover, the relationship between B and C .However, in general, the generalized path algebra k (∆ A , A ) is not an artinian algebra,e.g. when the natural quiver ∆ A contains an oriented cycle. So, k (∆ A , A ) has not theso-called related basic algebra under the meaning of “artinian” such that they are Morita-equivalent each other. In this reason, C is different from that for artinian algebras.A complete set of non-isomorphic primitive orthogonal idempotents of A is a set ofprimitive orthogonal idempotents { ε i : i ∈ I ⊂ (∆ A ) } such that Aε i = Aε j as left A -modules for any i = j in I and for each primitive idempotent ε s the module Aε s isisomorphic to one of the modules Aε i ( i ∈ I ).Every indecomposable projective module P is decided by a primitive idempotent e i ,that is, P ∼ = Aε i for some i . And, there exists a bijective correspondence between theiso-classes of indecomposable projective modules and the iso-classes of simple modules.The set of the latter is equal to the vertex set (Γ A ) of the ordinary quiver Γ A of A , andthen to the vertex set (∆ A ) of the natural quiver ∆ A of A . Hence, I = (∆ A ) . Let each TWO BASIC ALGEBRAS FROM AN ARTINIAN ALGEBRA P i be chosen as a representative from the iso-class of indecomposable projective module Aε i and let i run over the vertex set (∆ A ) . Then the basic algebra B of A is given by B = End ( ` i ∈ (∆ A ) P i ) ∼ = L i,j ∈ (∆ A ) Hom A ( P i , P j ) ∼ = L i,j ∈ (∆ A ) ε i Aε j . Lemma 3.1.
Let A be an artinian algebra. Then the complete set of non-isomorphicprimitive orthogonal idempotents of A , A/r ( r is the radical of A ) and k (∆ A , A ) are thesame, whose cardinality is equal to that of the vertex set of the natural quiver of A .Proof : Let ε i be the image of ε i under the canonical homomorphism from A to A/r .Since A and A/r have the same simple modules, { ε i : i ∈ (∆ A ) } is a complete set of non-isomorphic primitive orthogonal idempotents of A/r . But the idempotents of k (∆ A , A )must have length zero, hence { ε i : i ∈ (∆ A ) } is also a complete set of non-isomorphicprimitive orthogonal idempotents of k (∆ A , A ). (cid:3) As discussed before Lemma 3.1, the basic algebra C satisfies C = End ( a i ∈ (∆ A ) k (∆ A , A ) ε i ) ∼ = M i, j ∈ (∆ A ) ε i k (∆ A , A ) ε j . Then, we get the following:
Proposition 3.2.
For an artinian algebra A over a field k with the natural quiver ∆ A , let { ε i : i ∈ (∆ A ) } be the complete set of non-isomorphic primitive orthogonal idempotentsof A . Denote by ε i the image of ε i under the canonical morphism from A to A/r . Then,(i) the basic algebra B of A is isomorphic to L i ∈ (∆ A ) ε i Aε j ;(ii) the basic algebra C of the associated generalized path algebra k (∆ A , A ) of A isisomorphic to L i ∈ (∆ A ) ε i k (∆ A , A ) ε j . As we have said, k (∆ A , A ) is not artinian when ∆ A has an oriented cycle. Hence,we cannot affirm whether C is Morita equivalent to k (∆ A , A ) in general. But, C is stilldecided uniquely by k (∆ A , A ) and then by A .For an arbitrary artinian algebra A , we still cannot obtain the explicit relation betweentwo basic algebras B and C depending upon Proposition 3.2. However, for the followingspecial case, that is, for the so-called Gabriel-type algebras , we will give an exact conclusionfor the two basic algebras.
Definition 3.1.
Let A be an artinian algebra over a field k and k (∆ A , A ) its associatednormal generalized path algebra. If there exists an ideal I of k (∆ A , A ) such that A ∼ = k (∆ A , A ) /I , then we say A to be of Gabriel-type . Since in [5], we have the Generalized Gabriel Theorem for a finite dimensional algebra A with 2-nilpotent radical r = r ( A ) in the case A is splitting over r , that is, A ∼ = k (∆ A , A ) / h ρ i with J ⊂ h ρ i ⊂ J + J ∩ Ker e ϕ where h ρ i is an ideal generated by the set of relations ρ of k (∆ , A ) and e ϕ is that in Lemma 2.3. It means that any such finite dimensional algebra isalways of Gabriel-Type. TWO BASIC ALGEBRAS FROM AN ARTINIAN ALGEBRA I , then this quiver and the related generalized path algebra are just re-spectively the natural quiver and the corresponding one of this algebra. Exactly, we havethe following statement on the uniqueness: Theorem 3.3.
Assume A is an artinian algebra, r = r ( A ) is the radical of A . Let A/r = L pi =1 A i with simple ideals A i . If there is a quiver Q and a normal generalizedpath algebra k ( Q, B ) with a set of simple algebras B = { B , · · · , B q } and an admissibleideal I of k ( Q, B ) (i.e. for some s , J s ⊂ I ⊂ J ) such that A ∼ = k ( Q, B ) /I where J theideal of k ( Q, B ) generated by all B -paths of length one, then Q is just the natural quiver ∆ A of A and p = q such that A i ∼ = B i for i = 1 , ..., p after reindexed. It follows that A isa Gabriel-type algebra.Proof: Since (
J/I ) s ⊆ J s /I = 0 and k ( Q, B ) /I/J/I ∼ = k ( Q, B ) /J = B ⊕ · · · ⊕ B q semisimple, then rad( k ( Q, B ) /I ) = J/I . From the isomorphism A ∼ = k ( Q, B ) /I , we have A/ rad A ∼ = k ( Q, B ) /I/J/I , i.e. A ⊕ · · · ⊕ A p ∼ = B ⊕ · · · ⊕ B q . Thus p = q and A i ∼ = B i for i = 1 , ..., p after reindexed.By the isomorphism, the two algebras A and k ( Q, B ) /I have the same natural quivers,i.e. ∆ A = ∆. Then we only need to show that the natural quiver ∆ of k ( Q, B ) /I isjust Q . Firstly since p = q , ∆ = Q . And the number of arrows from i to j in ∆ is rank ( B j ( J/I/J /I ) B i ) = rank ( B j ( J/J ) B i ), which is just the number of arrows from i to j in Q . Therefore Q = ∆ = ∆ A . (cid:3) Lemma 3.4.
Let A be a Gabriel-type artinian algebra with A π ∼ = k (∆ A , A ) /I for an ideal I of k (∆ A , A ) satisfying I ⊂ J . Assume that { ε i : i ∈ (∆ A ) } is the complete setof non-isomorphic primitive orthogonal idempotents of A . Then, there is a complete setof non-isomorphic primitive orthogonal idempotents { d i : i ∈ (∆ A ) } of A/r such that π ( ε i ) = d i + I for any i ∈ (∆ A ) .Proof : Let π ( ε i ) = e ε i + I , then { e ε i + I : i ∈ (∆ A ) } is a complete set of non-isomorphicprimitive orthogonal idempotents of k (∆ A , A ) /I since π is an isomorphism.Since ( e ε i + I ) = e ε i + I , we get e ε i − e ε i ∈ I . Note that I lies in the ideal of k (∆ A , A )generated by all A -paths of length one. Because the square of any non-cyclic path is zero,either e ε i + I = E i c i + I or e ε i + I = d i + I where E i are circles in ∆ A , c i and d i are primitiveidempotents in k (∆ A , A ), or equivalently in A/r . TWO BASIC ALGEBRAS FROM AN ARTINIAN ALGEBRA w = 1 k (∆ A , A ) /I − P l ∈ (∆ A ) ( e ε i + I ), then w is an idempotent and can be decomposedinto a sum of some primitive orthogonal idempotents e f j + I , write w = ( e f + I )+ · · · +( e f t + I ).Thus, 1 k (∆ A , A ) /I = P i ∈ (∆ A ) ( e ε i + I ) + P tj =1 ( e f j + I ) . Let X + I and Y + I denote the sums of those idempotents in { e ε i + I : i ∈ (∆ A ) } S { e f j + I : j = 1 + · · · + t } respectively in the forms E p c p + I and d q + I , where c p , d q ∈ A/r . Thus,1 k (∆ A , A ) + I = 1 k (∆ A , A ) /I = ( X + I ) + ( Y + I ), it follows that X + I = (1 k (∆ A , A ) − Y ) + I .Suppose there are some i such that e ε i + I = E i c i + I = 0. Then X + I = 0. Hence1 k (∆ A , A ) − Y = 0, then 1 k (∆ A , A ) − Y ∈ X + I ⊂ J , which is impossible due to 1 k (∆ A , A ) − Y ∈ k ((∆ A ) , A ).The above contradiction means that each e ε i + I = d i + I where each d i is primitiveidempotent in A/r .Clearly { d i : i ∈ (∆ A ) } is a set of non-isomorphic primitive orthogonal idempotents of A/r , by Lemma 3.1 it is a complete set of non-isomorphic primitive orthogonal idempotentsof
A/r . (cid:3) Theorem 3.5.
Let A be a Gabriel-type artinian algebra over a field k with A π ∼ = k (∆ A , A ) /I for an ideal I of k (∆ A , A ) satisfying I ⊂ J . Then for the basic algebra B of A and thebasic algebra C of k (∆ A , A ) , it holds that B ∼ = ( C + I ) /I .Proof: Let { ε i : i ∈ (∆ A ) } be a complete set of non-isomorphic primitive orthogonalidempotents of A . Then, by Lemma 3.4, there is a complete set of non-isomorphic primitiveorthogonal idempotents { d i : i ∈ (∆ A ) } of A/r such that π ( ε i ) = d i + I for each i ∈ (∆ A ) .By Lemma 3.1, { d i : i ∈ (∆ A ) } is also a complete set of non-isomorphic prim-itive orthogonal idempotents of k (∆ A , A ). Thus, by Proposition 3.2, we have C ∼ = L i,j ∈ (∆ A ) d i k (∆ A , A ) d j . Moreover, under the isomorphism π , B ∼ = M i,j ∈ (∆ A ) ε i Aε j ∼ = M i,j ∈ (∆ A ) ( d i + I )( k (∆ A , A ) /I )( d j + I ) ∼ = M i,j ∈ (∆ A ) ( d i k (∆ A , A ) d j + I ) /I ∼ = ( C + I ) /I. (cid:3) This theorem mentions the relation between the two basic algebras B and C which areboth decided by the same artinian algebra A .In general, for a Gabriel-type artinian A whose the ideal I is admissible (even onlywith I ⊂ J ), the two natural quivers ∆ B and ∆ C of the associated basic algebras B of A and C of k (∆ A , A ) are not equal. In fact, although B ∼ = ( C + I ) /I , rad B ∼ = (rad C + I ) /I ,in general B/ rad B = C/ rad C and (rad B ) / (rad B ) = (rad C ) / (rad C ) . Proposition 3.6.
For a Gabriel-type artinian algebra A with A π ∼ = k (∆ A , A ) /I , if I is anadmissible ideal, the natural quivers of A and k (∆ A , A ) are the same, i.e. ∆ A = ∆ k (∆ A , A ) . TWO BASIC ALGEBRAS FROM AN ARTINIAN ALGEBRA Proof : Since I is admissible, there is a positive integer s such that J s ⊂ I ⊂ J .rad A ∼ = J/I as proved in Theorem 3.3. And A ∼ = k (∆ A , A ) /I , then A/ rad A ∼ = k (∆ A , A ) /J .Moreover, rad A/ (rad A ) ∼ = J/J . Thus, by the definition, ∆ A = ∆ k (∆ A , A ) . (cid:3) In the other case, for an artinian algebra A , when ∆ A is admissible (i.e. is acyclic), itis true that ∆ A = ∆ k (∆ A , A ) , since J is just the radical of k (∆ A , A ).To sum up, for a finite dimensional algebras A over algebraically closed field k , wheneither ∆ A is admissible or A is of Gabriel-type satisfying A ∼ = k (∆ A , A ) /I with admissible I , we have the following diagram:Γ A ⊃ ∆ A = ∆ k (∆ A , A ) ⊂ Γ k (∆ A , A ) k ∩ ∩ k Γ B = ∆ B ∆ C = Γ C where Γ A is the ordinary quiver of A , etc.; ⊂ , ⊃ and ∩ mean the embeddings of the densesub-quivers.We feel the relations in this diagram would still hold for any artinian algebras. Thispoint of view will be discussed in the subsequent work.As we say above, the ordinary quiver and the natural quiver of a finite dimensionalbasic algebra coincide each other. In the end of this section, we give an example whichmeans the coincidence is also possible to happen for some non-basic algebras. Meanwhile,in this example, we show a method of computing the number of arrows of the naturalquiver of an artinian algebra. Example
Let k be an algebraically closed field of characteristic different from 2 andlet Q be the quiver: • e ✲✛ • e ′ ✲ • e ′ • e ✛ β α α ′ β ′ • e Denote the path algebra kQ by Λ and let G = h σ i be the group of order 2. For theelements e , e , e , α, β in Λ, let σe = e , σe = e ′ , σe = e ′ , σα = α ′ , σβ = β ′ . Then,there is only one way of extending σ to a k -algebra automorphism of Q and this is theway we will consider G as a group of automorphisms of Q . Now, we consider the ordinaryquiver and the natural quiver of the skew group algebra Λ G (see [2]).Let r be the radical of Λ. By Proposition 4.11 in [2], r Λ G = rad (Λ G ). It is easyto see that (Λ G ) / ( r Λ G ) ∼ = (Λ /r ) G . In the page 84 of [2], it was given that (Λ /r ) G ∼ = A × A × A × A = k × k × k kk k ! × k kk k ! as algebras and the associated basicalgebra B is obtained in the reduced form from Λ G , which is Mortia-equivalent to Λ G ,and moreover, it was proved in [2] that B is isomorphic to the path algebra of the followingquiver:. INTERPRETATIONS • e (1) e (2) ✲❄ • νe (3) ✲ • e (4) • λ µ This quiver is just the ordinary quiver Γ Λ G of Λ G . Therefore, all m ij = 0, or 1.For i = 1 , , , , dim k A i = n i where n = n = 1, n = n = 2. By definition, for i, j = 1 , , , t ij is the rank of j M i = A j ( r Λ G ) / ( r Λ G ) A i as A j - A i -bimodule, equiva-lently, as a right A i ⊗ A opj -module. A i ⊗ A opj ∼ = M n i n j ( k ) is a simple algebra with dimension n i n j . Thus, j M i is semisimple over this simple algebra. Let j M i = L ⊕ · · · ⊕ L s whereall L v are simple A i ⊗ A opj -modules for v = 1 , · · · , s . L v can be considered as a simpleright ideal of A i ⊗ A opj , therefore, L v ∼ = ( k k · · · k ) the whole set of all 1 × n i n j matricesover k . For any 0 = x v ∈ L v , L v = x v ( A i ⊗ A opj ). Then, j M i = L sv =1 x v ( A i ⊗ A opj ) as( A i ⊗ A opj )-modules.First, we prove s = m ij the number of the arrows from the vertex i to the other vertex j in the ordinary quiver Γ Λ G of Λ G .Let { S , S , · · · , S n } be the complete set of non-isomorphic simple Λ G -modules. Then m ij = dim k Ext A ( S i , S j ). By [6][2], dim k Ext A ( S i , S j ) = dim k ( kε j ( r Λ G ) / ( r Λ G ) kε i )where { ε , · · · , ε n } is the complete set of primitive orthogonal idempotents of (Λ G ) /rad (Λ G )with ε i ∈ A i . And, kε j ( r Λ G ) / ( r Λ G ) kε i = ε j A j ( r Λ G ) / ( r Λ G ) A i ε i = ε jj M i ε i ∼ = j M i ( ε i ⊗ ε j ) = ( L ⊕· · ·⊕ L s )( e i ⊗ e j ) = ( x ⊕· · ·⊕ x s )( A i ⊗ A opj )( e i ⊗ e j ) ∼ = ( x ⊕· · ·⊕ x s ) M n i n j ( k ) E ll where ε i ⊗ ε j is a primitive idempotent of A i ⊗ A opj so let E ll be the correspondent elementof ε i ⊗ ε j in M n i n j ( k ) under the isomorphism.Obviously, dim k x v M n i n j ( k ) E ll = 1 for all v = 1 , · · · , s . Then, dim k ( x ⊕ · · · ⊕ x s ) M n i n j ( k ) E ll = s . It follows that m ij = s .For each pair ( i, j ), when m ij = 0, we have kε j ( r Λ G ) / ( r Λ G ) kε i = 0. Then j M i = 0.Thus, the rank t ij of j M i equals 0. When m ij = 1, then s = m ij = 1, that is, j M i = L isa simple ( A i ⊗ A opj )-module. Hence, the rank t ij of j M i is 1 in this case.According to the above discussion, for each pair ( i, j ), we have t ij = m ij = 0 or 1.Therefore, the natural quiver ∆ Λ G is equal to the ordinary quiver Γ Λ G . In [2][1], given a finite dimensional algebra A , the ordinary quiver Γ A can be constructedby the indecomposable projective modules and the irreducible morphisms between them.So the ordinary quiver of A provides a convenient way to study its projective (or injective)modules and morphisms between them, even when A is not a basic algebra. By theGabriel theorem, the ordinary quiver of a finite-dimensional algebra A is used as a tool tocharacterize the structure of its associated basic algebra but not of A . In this reason, theordinary quiver is not effective enough to characterize a non-basic algebra. The generalized EFERENCES proj A with irreducible morphisms isisomorphic to the opposite of the ordinary quiver of A .Through [5] and here, we think the method of natural quiver may offset some shortageof ordinary quiver and AR-quiver. In certain sense, the natural quiver of an artinianalgebra A will also be available for the theory of representations of an artinian algebra.Under certain condition, the representation category Rep A of A can be decided whollyby the ordinary quiver and the AR-quiver. The category of representations of A may bepartially induced from the category of representations of Γ B through the basic algebra B .For example, when A is Gabriel-type, that is, A is isomorphic to some quotient of thegeneralized path algebra of ∆ A = ∆ B , any representations of A can be induced directlyfrom some of representations of the generalized path algebra of ∆ A . In the classical theoryof representations of artin algebras (see [2][1][4] etc.), one wants to characterize Rep A through representations of ∆ B with B . However, the difficulty is that in general, it is noteasy to construct concretely the basic algebra B from A . By comparison, the method ofnatural quivers is more straightforward through representations of the generalized pathalgebra of ∆ A . Therefore, we hope to set up this new approach to representations of anartinian algebra via representations of the generalized path algebra of its natural quiver. References [1] I. Assem, D. Simson and A. Skowro ´ N ski, Elements of the Representation Theoryof Associative Algebras, Volume 1: techniques of representation theory , LMSST 65,Cambridge University Press, 2006[2] M.Auslander, I.Reiten and S.O.Smalø, Representation Theory of Artin Algebra,
Cam-bridge University Press , Cambridge, 1995[3] F.U.Coelho and S.X.Liu, Generalized path algebras, In:
Interactions between ringtheory and repersentations of algebras (Murcia), Lecture Notes in Pure and Appl.Math,
Marcel-Dekker, New York, , 2000, pp.53-66[4] V.Dlab, Representations of Valued graph, Seminaire de mathematiques superieures,
Les presses de luniversite de montreal , Montreal, Canada, 1980[5] F.Li, Characterization of left Artinian algebras through pseudo path algebras, toappear in