DG Algebra structures on the quantum affine n -space O −1 ( k n )
aa r X i v : . [ m a t h . R A ] S e p DG ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) X.-F. MAO, X.-T. WANG, AND M.-Y.ZHANG
Abstract.
Let A be a connected cochain DG algebra, whose underlyinggraded algebra A is the quantum affine n -space O − ( k n ). We computeall possible differential structures of A and show that there exists a one-to-onecorrespondence between { cochain DG algebra A | A = O − ( k n ) } and the n × n matrices M n ( k ). We also study the isomorphism problems ofthese non-commutative DG algebras. For the cases n ≤
3, we further showthat such DG algebras are all Koszul Calabi-Yau DG algebras. introduction
Along this paper, k will denote an algebraically closed field of characteristiczero. Recall that a cochain DG k -algebra is a graded k -algebra together with adifferential of degree 1, which satisfies the Leibniz rule. Algebras with additionaldifferential structures provide convenient models for intrinsic and homological infor-mation from diverse array of areas ranging from representation theory to symplecticand algebraic geometry. For example, a Gorenstein topological space X in algebraictopology is characterized by the Gorensteinness of the cochain algebra C ∗ ( X ; k ) ofnormalized singular cochains on X (cf. [FHT1, Gam]). And it is well known thatthe rational homotopy type of a simply connected space of finite type is encodedin its Sullivan model.In the derived algebraic geometry, a fundamental fact discovered by A. Bondaland M. Van den Bergh is that any quasi-compact and quasi-separated scheme X is affine in the derived sense, i.e. D Qcoh ( X ) is equivalent to D( A ) for a suitableDG algebra A (cf. [BV]). For a quasi-compact and quasi-separated scheme X ,the property of smoothness is equivalent to the homologically smoothness of thecorresponding A . In the smooth case, the triviality of the canonical bundle for thescheme is equivalent to the Calabi-Yau properties of the DG algebra A . Calabi-Yau DG algebras are introduced by Ginzburg in [Gin], and have a multitude ofconnections to representation theory, mathematical physics and non-commutativealgebraic geometry. Therefore the constructions and studies of Calabi-Yau DGalgebras have become tremendous helpful to people working in different areas ofmathematics.In [HM], the first author and J.-W. He give a criterion for a connected cochainDG algebra to be 0-Calabi-Yau, and prove that a locally finite connected cochainDG algebra is 0-Calabi-Yau if and only if it is defined by a potential. For a n -Calabi-Yau connected cochain DG algebra A , one sees that the full triangulatedsubcategory D blf ( A ) of D( A ) containing DG A -modules with finite dimensional totalcohomology is a n -Calabi-Yau triangulated category (cf. [CV]). The notion ofCalabi-Yau triangulated category was introduced by Kontsevich [Kon] in the late Mathematics Subject Classification.
Primary 16E45, 16E65, 16W20,16W50.
Key words and phrases.
Calabi-Yau DG algebra, cochain DG algebra, Ext-algebra, Koszul DGalgebra, quantum affine space. A is a 0-Calabi-Yau DG algebra if A = k h x , x i with | x | = | x | = 1. The proofthere relies on the classification of the differential of A . Note that A in this caseis just the quantum plane O − ( k ). This motivates us to consider more generalcase. For the quantum affine n -space O − ( k n ) , n ≥
3, we want to see what kindcochain DG algebras can be constructed over it. We describe all possible cochainDG algebra structures over O − ( k n ) by the following theorem (see Theorem 2.1). Theorem A.
Let A be a connected cochain DG algebra such that A is the k -algebra with degree one generators x , · · · , x n and relations x i x j = − x j x i , for all1 ≤ i < j ≤ n . Then ∂ A is determined by a matrix M = ( m ij ) n × n such that ∂ A ( x ) ∂ A ( x )... ∂ A ( x n ) = M x x ... x n . For any M = ( m ij ) ∈ M n ( k ), it is reasonable to define a connected cochain DGalgebra A O − ( k n ) ( M ) such that[ A O − ( k n ) ( M )] = O − ( k n )and its differential ∂ A is defined by ∂ A ( x ) ∂ A ( x )... ∂ A ( x n ) = M x x ... x n . To consider the homological properties of A O − ( k n ) ( M ), it is necessary to study theisomorphism problem. We have the following theorem (see Theorem 3.6). Theorem B.
Let M and M ′ be two matrixes in M n ( k ). Then A O − ( k n ) ( M ) ∼ = A O − ( k n ) ( M ′ )if and only if there exists C = ( c ij ) n × n ∈ QPL n ( k ) such that M ′ = C − M ( c ij ) n × n . Here QPL n ( k ) is the set of quasi-permutation matrixes in GL n ( k ). One sees thatQPL n ( k ) is a subgroup of GL n ( k ) (see Proposition 3.4). By Theorem B, one seesthat any DG algebra automorphism group of A O − ( k n ) ( M ) is QPL n ( k )’s subgroup { C = ( c ij ) n × n ∈ QPL n ( k ) | M = C − M ( c ij ) n × n } G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 3 for any M ∈ M n ( k ). Theorem B also indicates that one can define a right groupaction χ : M n ( k ) × QPL n ( k ) → M n ( k )of QPL n ( k ) on M n ( k ) such that χ [( M, C = ( c ij ) n × n )] = C − M (( c ij ) ) n × n . Theset of all orbits of this group action is one to one correspondence with the set ofisomorphism classes of DG algebras in {A O − ( k n ) ( M ) | M ∈ M n ( k ) } .It is natural for one to ask whether each A O − ( k n ) ( M ) is a Calabi-Yau DGalgebra. For the case n = 2, we know it is right by [MH]. It is worth noting thatas n grows large, the classifications, cohomologies and homological properties of A O − ( k n ) ( M ) become increasingly difficult to compute and study. This increasedcomplexity is, in large part, due to the irregular increase of the number of casesone need to study separately. In this paper, we focus our attentions on the case of n = 3. It involves further classifications and complicated matrix analysis.In general, the cohomology graded algebra H ( A ) of a cochain DG algebra A usually contains some homological information. One sees that A is a Calabi-YauDG algebra if the trivial DG algebra ( H ( A ) ,
0) is Calabi-Yau by [MYY], and it isproved in [MH] that a connected cochain DG algebra A is a Kozul Calabi-Yau DGalgebra if H ( A ) belongs to one of the following cases:( a ) H ( A ) ∼ = k ; ( b ) H ( A ) = k [ ⌈ z ⌉ ] , z ∈ ker( ∂ A );( c ) H ( A ) = k h⌈ z ⌉ , ⌈ z ⌉i ( ⌈ z ⌉⌈ z ⌉ + ⌈ z ⌉⌈ z ⌉ ) , z , z ∈ ker( ∂ A ) . Recently, it is proved in [MHLX, Proposition 6.5] that a connected cochain DGalgebra A is Calabi-Yau if H ( A ) = k [ ⌈ z ⌉ , ⌈ z ⌉ ] where z ∈ ker( ∂ A ) and z ∈ ker( ∂ A ). In this paper, we show the following proposition (see Proposition 4.3). Proposition A.
A connected cochain DG algebra A is a Koszul Calabi-Yau DGalgebra if H ( A ) = k h⌈ y ⌉ , ⌈ y ⌉i / ( t ⌈ y ⌉ + t ⌈ y ⌉ + t ( ⌈ y ⌉⌈ y ⌉ + ⌈ y ⌉⌈ y ⌉ )) , where ( t , t , t ) ∈ P k − { (1 , , , (0 , , } .With the help of the computational results of H [ A O − ( k ) ( M )] in [MWYZ], wepartially prove the Calabi-Yau properties of A O − ( k ) ( M ). There are some cases ofDG algebras whose Calabi-Yau properties one can’t judge from their cohomologies.For such kind of A O − ( k ) ( M ), we construct the minimal semi-free resolution of k in each case and compute the corresponding Ext-algebras. It involves furtherclassifications and complicated matrix analysis. In our proof, we rely heavily on aresult proved in [HM] that a Koszul connected cochain DG algebra A is Calabi-Yauif and only if its Ext-algebra is a symmetric Frobenius algebra.In DG homological algebra, it is reasonable to ask the following question onrealizability. Question 0.1.
Let A and A ′ be two connected cochain DG algebra with A = A ′ .Assume that H ( A ) ∼ = H ( A ′ ) as graded algebras. Can we conclude that A is quasi-isomorphic to A ′ ? From the classifications in Section 6, we can see many counter-examples forQuestion 0.1 (See Remark 6.14).1. preliminaries
We assume that the reader is familiar with basic definitions concerning DGhomologically algebra. If this is not the case, we refer to [AFH, FHT2, MW1, MW2]for more details on them. We begin by fixing some notations and terminology.There are some overlaps here in [MHLX, MGYC].
X.-F. MAO, X.-T. WANG, AND M.-Y.ZHANG
Some conventions.
For any k -vector space V , we write V ∗ = Hom k ( V, k ).Let { e i | i ∈ I } be a basis of a finite dimensional k -vector space V . We denote thedual basis of V by { e ∗ i | i ∈ I } , i.e., { e ∗ i | i ∈ I } is a basis of V ∗ such that e ∗ i ( e j ) = δ i,j .For any graded vector space W and j ∈ Z , the j -th suspension Σ j W of W is agraded vector space defined by (Σ j W ) i = W i + j .1.2. Notations on DG algebras.
For any cochain DG algebra A , we denote A op as its opposite DG algebra, whose multiplication is defined as a · b = ( − | a |·| b | ba for all graded elements a and b in A . A cochain DG algebra A is called non-trivialif ∂ A = 0, and A is said to be connected if its underlying graded algebra A is aconnected graded algebra.Given a cochain DG algebra A , we denote by A i its i -th homogeneous component.The differential ∂ A is a family of linear maps ∂ i A : A i → A i +1 with ∂ i +1 A ◦ ∂ i A = 0,for all i ∈ Z . The cohomology graded algebra of A is the graded algebra H ( A ) = M i ∈ Z ker( ∂ i A )im( ∂ i − A ) . For any cocycle element z ∈ ker( ∂ i A ), we write ⌈ z ⌉ as the cohomology class in H ( A )represented by z . One sees that H ( A ) is a connected graded algebra if A is aconnected cochain DG algebra.For any connected cochain DG algebra A , we denote by m A its maximal DGideal · · · → → A ∂ A → A ∂ A → · · · ∂ n − A → A n ∂ n A → · · · . Clearly, k has a structure of DG A -module via the augmentation map ε : A → A / m A = k. One sees that the enveloping DG algebra A e = A ⊗ A op of A is also a connectedcochain DG algebra with H ( A e ) ∼ = H ( A ) e , and m A e = m A ⊗ A op + A ⊗ m A op . Derived categories.
A morphism f : A → A ′ of DG algebras is a chain mapof complexes which respects multiplication and unit; f is said to be a DG algebraisomorphism (resp. quasi-isomorphism) if f (resp. H ( f )) is an isomorphism. ADG algebra isomorphism f is called a DG automorphism when A ′ = A . The set ofall DG algebra automorphisms of A is a group, denoted by Aut dg ( A ).For any cochain DG algebra A , we write D( A ) as the derived category of left DGmodules over A (DG A -modules for short). A DG A -module M is compact if thefunctor Hom D( A ) ( M, − ) preserves all coproducts in D( A ). It is worth noticing thata DG A -module is compact if and only if it admits a minimal semi-free resolutionwith a finite semi-basis (see [MW1, Proposition 3.3]). The full subcategory of D( A )consisting of compact DG A -modules is denoted by D c ( A ).We write D b ( A ) as the full subcategories of D( A ), whose objects are cohomolog-ically bounded. We say a graded vector space M = ⊕ i ∈ Z M i is locally finite, if each M i is finite dimensional. The full subcategory of D( A ) consisting of DG moduleswith locally finite cohomology is denoted by D lf ( A ).1.4. Definitions of some homological properties.
Let A be a connected cochainDG algebra.(1) If dim k H ( R Hom A ( k, A )) = 1, then A is called Gorenstein (cf. [FHT1]);(2) If A k , or equivalently A e A , has a minimal semi-free resolution with a semi-basis concentrated in degree 0, then A is called Koszul (cf. [HW]);(3) If A k , or equivalently the DG A e -module A is compact, then A is calledhomologically smooth (cf. [MW3, Corollary 2.7]); G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 5 (4) If A is homologically smooth and R Hom A e ( A , A e ) ∼ = Σ − n A in the derivedcategory D(( A e ) op ) of right DG A e -modules, then A is called an n -Calabi-Yau DG algebra (cf. [Gin, VdB]).2. dg algebra structures In this section, we study all possible differential structures of a connected cochainDG algebra, whose underlying graded algebra is the quantum affine n -space O − ( k n ). Theorem 2.1.
Let A be a connected cochain DG algebra such that A is the k -algebra with degree one generators x , · · · , x n and relations x i x j = − x j x i , for all ≤ i < j ≤ n . Then ∂ A is determined by a matrix M = ( m ij ) n × n such that ∂ A ( x ) ∂ A ( x ) ... ∂ A ( x n ) = M x x ... x n . Proof.
Since the differential ∂ A of A is a k -linear map of degree 1, we may let ∂ A ( x i ) = n P s =1 n P t = s m ist x s x t , where m is,t ∈ k , for any 1 ≤ s ≤ t ≤ n and 1 ≤ i ≤ n .We have0 = ∂ A ( x i x j + x j x i ) = ∂ A ( x i ) x j − x i ∂ A ( x j ) + ∂ A ( x j ) x i − x j ∂ A ( x i )= ∂ A ( x i ) x j − x j ∂ A ( x i ) + ∂ A ( x j ) x i − x i ∂ A ( x j )= [ n X s =1 n X t = s m ist x s x t ] x j − x j [ n X s =1 n X t = s m ist x s x t ]+ [ n X s =1 n X t = s m jst x s x t ] x i − x i [ n X s =1 n X t = s m jst x s x t ]= [ n X s =1 n X t = s +1 m ist x s x t ] x j − x j [ n X s =1 n X t = s +1 m ist x s x t ]+ [ n X s =1 n X t = s +1 m jst x s x t ] x i − x i [ n X s =1 n X t = s +1 m jst x s x t ] X.-F. MAO, X.-T. WANG, AND M.-Y.ZHANG = [ j − X s =1 n X t = s +1 m ist x s x t ] x j + n X t = j +1 m ijt x j x t x j + [ n X s = j +1 n X t = s +1 m ist x s x t ] x j − x j [ j − X s =1 n X t = s m ist x s x t ] − n X t = j +1 m ijt x j x t − x j [ n X s = j +1 n X t = s m ist x s x t ]+ [ i − X s =1 n X t = s m jst x s x t ] x i + n X t = i +1 m jit x i x t x i + [ n X s = i +1 n X t = s m jst x s x t ] x i − x i [ i − X s =1 n X t = s +1 m jst x s x t ] − n X t = i +1 m jit x i x t − x i [ n X s = i +1 n X t = s +1 m jst x s x t ]= [ j − X s =1 n X t = s +1 m ist x s x t ] x j + [ n X s = j +1 n X t = s +1 m ist x s x t ] x j − x j [ j − X s =1 n X t = s m ist x s x t ] − x j [ n X s = j +1 n X t = s m ist x s x t ] − n X t = j +1 m ijt x j x t + [ i − X s =1 n X t = s m jst x s x t ] x i + [ n X s = i +1 n X t = s m jst x s x t ] x i − x i [ i − X s =1 n X t = s +1 m js,t x s x t ] − x i [ n X s = i +1 n X t = s +1 m jst x s x t ] − n X t = i +1 m jit x i x t = [ n X s =1 s = j n X t = s +1 t = j m ist x s x t ] x j + j − X s =1 m isj x s x j − x j [ n X s =1 s = j n X t = s +1 t = j m ist x s x t ] − x j j − X s =1 m isj x s x j − n X t = j +1 m ijt x j x t + [ n X s =1 s = i n X t = s +1 t = i m jst x s x t ] x i + i − X s =1 m jsi x s x i − x i [ n X s =1 s = i n X t = s +1 t = i m js,t x s x t ] − i − X s =1 m jsi x i x s x i − n X t = i +1 m jit x i x t = 2 j − X s =1 m is,j x s x j − n X t = j +1 m ijt x j x t + 2 i − X s =1 m jsi x s x i − n X t = i +1 m jit x i x t , for any 1 ≤ i < j ≤ n . This implies that m is,j = 0 , for all s ∈ { , , · · · , j − } m ij,t = 0 , for all t ∈ { j + 1 , j + 2 , · · · , n } m js,i = 0 , for all s ∈ { , , · · · , i − } m ji,t = 0 , for all t ∈ { i + 1 , i + 2 , · · · , n } . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 7 Hence ∂ A ( x i ) = n P s =1 m is,s x s . One sees that ∂ A ( x i ) = ∂ A ( n X s =1 m is,s x s )= n X s =1 m is,s [ ∂ A ( x i ) x i − x i ∂ A ( x i )]= n X s =1 m is,s [ n X t =1 m it,t x t x i − x i n X r =1 m ir,r x r ]= n X s =1 m is,s n X q =1 m iq,q ( x q x i − x i x q ) = 0 , for any i ∈ Z . Therefore, ∂ A is uniquely determined by the n × n matrix M = ( m ij ),where m ij = m ij,j , i, j ∈ { , , · · · , n } . (cid:3) By Theorem 2.1, one sees that the following definition is reasonable.
Definition 2.2.
For any M = ( m ij ) ∈ M n ( k ) , we define a connected cochain DGalgebra A O − ( k n ) ( M ) such that [ A O − ( k n ) ( M )] = O − ( k n ) and its differential ∂ A is defined by ∂ A ( x ) ∂ A ( x ) ... ∂ A ( x n ) = M x x ... x n . Lemma 2.3.
For any M = ( m ij ) ∈ M n ( k ) and t ∈ N , each x ti is a cocycle centralelement of A O − ( k n ) ( M ) .Proof. For the sake of convenience, we let A = A O − ( k n ) ( M ). Since x i x j = x i x i x j = − x i x j x i = x j x i when i = j , one sees that x i is a central element of A . This implies that each x ti is a central element of A . By Proposition 2.1, we have ∂ A ( x i ) = ∂ A ( x i ) x i − x i ∂ A ( x i )= n X j =1 m ij x j x i − x i n X j =1 m ij x j = n X j =1 m ij ( x j x i − x i x j ) = 0 . Using this, we can inductively prove ∂ A ( x ti ) = 0. (cid:3) By Lemma 2.3, one sees that the graded ideal I = ( x , x , · · · , x n ) is a DG idealof A O − ( k n ) ( M ), and the quotient DG algebra A O − ( k n ) ( M ) /I = ^ ( x , x , · · · , x n )is the exterior algebra with zero differential. We have the following short exactsequence 0 → I ֒ → A O − ( k n ) ( M ) → ^ ( x , x , · · · , x n ) → . X.-F. MAO, X.-T. WANG, AND M.-Y.ZHANG Isomorphism problem
It is well known in linear algebra that a square matrix is called a permutationmatrix if its each row and each column have only one non-zero element 1. In [AJL],a more general notion is introduced. This is the following definition.
Definition 3.1.
A square matrix is called a quasi-permutation matrix if each rowand each column has at most one non-zero element.
Remark 3.2.
By the definition above, a quasi-permutation matrix is possible to besingular. A quasi-permutation matrix is non-singular if and only if each row andeach column have only one non-zero element.
Lemma 3.3.
Let M = ( m ij ) n × n be a matrix in GL n ( k ) such that m ir m jr = 0 , forany ≤ i < j ≤ n and r ∈ { , , · · · , n } . Then each row and each column of M has only one non-zero element. In brief, M is a quasi-permutation matrix.Proof. Since M ∈ GL n ( k ), each column of M has at least one non-zero element. Ifthe r -th column of M has two non-zero elements m i r and m i r , then m i r m i r = 0,which contradicts with the assumption. Thus each column of M has only onenon-zero element. Then we conclude that M has n non-zero elements. By the non-singularity of M , we show that each row of M has exactly one non-zero element. (cid:3) Proposition 3.4.
The set of quasi-permutation matrixes in GL n ( k ) is a subgroupof the general linear group GL n ( k ) .Proof. For any quasi-permutation matrixes B = ( b ij ) n × n and D = ( d ij ) n × n inGL n ( k ), there exist σ, τ ∈ S n such that b iσ ( i ) = 0 , b ij = 0 , if j = σ ( i ) d iτ ( i ) = 0 , d ij = 0 , if j = τ ( i ) , for any i ∈ { , , · · · , n } . One sees that B and D can be written by B = ( E r E r − · · · E )diag( b σ (1) , b σ (2) , · · · , b nσ ( n ) ) ,D = ( E ′ s E ′ s − · · · E ′ )diag( d τ (1) , d τ (2) · · · , d nτ ( n ) ) , where E i and E ′ j are elementary matrixes obtained by swapping two rows of theidentity matrix I n . Hence BD − = ( E r · · · E )diag( b σ (1) , · · · , b nσ ( n ) )diag( 1 d τ (1) , · · · , d nτ ( n ) )( E ′ · · · E ′ s )= ( E r · · · E )diag( b σ (1) d τ (1) , · · · , b nσ ( n ) d nτ ( n ) )( E ′ · · · E ′ s ) . So BD − is obtained from the diagonal matrix diag( b σ (1) d τ (1) , · · · , b nσ ( n ) d nτ ( n ) ) by swappingtwo rows or two columns several times. Then each row and each column of BD − has only one non-zero element. Hence the set of quasi-permutation matrixes inGL n ( k ) is a subgroup of the general linear group GL n ( k ). (cid:3) By Proposition 3.4, we can introduce the following definition.
Definition 3.5.
We use
QPL n ( k ) to denote the set of non-singular quasi-permutation n × n matrixes. By Proposition 3.4, QPL n ( k ) is a subgroup of GL n ( k ) . Theorem 3.6.
Let M and M ′ be two matrixes in M n ( k ) . Then A O − ( k n ) ( M ) ∼ = A O − ( k n ) ( M ′ ) if and only if there exists C = ( c ij ) n × n ∈ QPL n ( k ) such that M ′ = C − M ( c ij ) n × n . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 9 Proof.
We write A = A O − ( k n ) ( M ) and A ′ = A O − ( k n ) ( M ′ ) for simplicity. In orderto distinguish, we assume that A ′ is the k -algebra with degree one generators x ′ , · · · , x ′ n and relations x ′ i x ′ j = − x ′ j x ′ i for all 1 ≤ i < j ≤ n .If the DG algebras A ∼ = A ′ , then there exists an isomorphism f : A → A ′ of DGalgebras. Since f : A → A ′ is a k -linear isomorphism, we may let f ( x ) f ( x )... f ( x n ) = C x ′ x ′ ... x ′ n for some C = ( c ij ) n × n ∈ GL n ( k ). We have0 = f ( x i x j + x j x i )= f ( x i ) f ( x j ) + f ( x j ) f ( x i )= ( n X s =1 c is x ′ s )( n X t =1 c jt x ′ t ) + ( n X t =1 c jt x ′ t )( n X s =1 c is x ′ s )= 2 n X r =1 c ir c jr ( x ′ r ) , for any 1 ≤ i < j ≤ n . Since char k = 2, we get c ir c jr = 0 for any 1 ≤ i < j ≤ n and r ∈ { , , · · · , n } . By Lemma 3.3, each row and each column of C have onlyone non-zero element. Hence C is a quasi-permutation non-singular matrix. Since f is a chain map, we have f ◦ ∂ A = ∂ A ′ ◦ f . For any i ∈ { , , · · · , n } , we have ∂ A ′ ◦ f ( x i ) = ∂ A ′ ( n X j =1 c ij x ′ j )(Eq1) = n X j =1 c ij ( n X l =1 m ′ jl ( x ′ l ) )= n X l =1 [ n X j =1 c ij m ′ jl ]( x ′ l ) and f ◦ ∂ A ( x i ) = f ( n X j =1 m ij ( x j ) )(Eq2) = n X j =1 m ij [ f ( x j )] = n X j =1 m ij [ n X l =1 c jl x ′ l ] = n X j =1 m ij n X l =1 ( c jl ) ( x ′ l ) = n X l =1 [ n X j =1 m ij ( c jl ) ]( x ′ l ) . Hence n P j =1 c ij m ′ jl = n P j =1 m ij ( c jl ) for any i, l ∈ { , , · · · , n } . Then we get CM ′ = M (( c ij ) ) n × n . Since C ∈ GL n ( k ), we have M ′ = C − M (( c ij ) ) n × n .Conversely, if there exists a quasi-permutation matrix C = ( c ij ) n × n ∈ GL n ( k )such that M ′ = C − M (( c ij ) ) n × n . Then we have CM ′ = M (( c ij ) ) n × n , which implies that n P j =1 c ij m ′ jl = n P j =1 m ij ( c jl ) for any i, l ∈ { , , · · · , n } . Define alinear map f : A → A ′ by f ( x ) f ( x )... f ( x n ) = C x ′ x ′ ... x ′ n . Obviously, f is invertible since C ∈ GL n ( k ). Since C is a quasi-permutation matrix,we have f ( x i ) f ( x j ) + f ( x j ) f ( x i ) = ( n X s =1 c is x ′ s )( n X t =1 c jt x ′ t ) + ( n X t =1 c jt x ′ t )( n X s =1 c is x ′ s )= 2 n X r =1 c ir c jr ( x ′ r ) = 0 , for any 1 ≤ i < j ≤ n . Hence f : A → A ′ can be extended to a morphismof graded algebras between A and A ′ . We still denote it by f . For any i ∈{ , , · · · , n } , we still have (Eq1) and (Eq2). Since CM ′ = M ( c ij ) n × n , we have n P j =1 c ij m ′ jl = n P j =1 m ij ( c jl ) for any i, l ∈ { , , · · · , n } . This implies ∂ A ′ ◦ f ( x i ) = f ◦ ∂ A ( x i ) , for any i ∈ { , , · · · , n } . Hence, f : A → A ′ is an isomorphism of DG algebras. (cid:3) Corollary 3.7.
For any M ∈ M n ( k ) , we have Aut dg A O − ( k n ) ( M ) = { C = ( c ij ) n × n ∈ QPL n ( k ) | M = C − M ( c ij ) n × n } . Proof.
This is immediate from Theorem 3.6. (cid:3)
Definition 3.8.
Theorem 3.6 indicates that we can define a map χ : M n ( k ) × QPL n ( k ) → M n ( k )such that χ [( M, C = ( c ij ) n × n )] = C − M (( c ij ) ) n × n .Let X be a set and G a group. A right action of G on X is a map X × G → X ,denoted ( x, g ) x · g such that x · G = x for all x ∈ X and ( x · g ) · g ′ = x · ( gg ′ )for all x ∈ X and g ′ , g ∈ G . A left action of G on X is defined similarly. We write X/G for the set of G -orbits. Proposition 3.9.
The map χ defined in Definition 3.8 is a right group action of QPL n ( k ) on M n ( k ) . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 11 Proof.
Obviously, I n is the identity element in QPL n ( k ). For any M in M n ( k ),we have χ [( M, I n )] = I − n M I n = M . For any C = ( c ij ) n × n and C ′ = ( c ′ ij ) n × n inQPL n ( k ), we have χ { ( χ [( M, C )] , C ′ ) } = χ [( C − M (( c ij ) ) n × n , C ′ )]= ( C ′ ) − C − M (( c ij ) ) n × n (( c ′ ij ) ) n × n and χ [( M, CC ′ )] = ( C ′ ) − C − M (( n X l =1 c il c ′ lj ) ) n × n . It remains to show that (( c ij ) ) n × n (( c ′ ij ) ) n × n = (( n P l =1 c il c ′ lj ) ) n × n . Since C , C ′ ∈ QPL n ( k ), there exist σ, τ ∈ S n such that c iσ ( i ) = 0 , c ij = 0 , if j = σ ( i ) c ′ iτ ( i ) = 0 , c ′ ij = 0 , if j = τ ( i ) , for any i ∈ { , , · · · , n } . One sees that C and C ′ can be written by C = ( E r E r − · · · E )diag( c σ (1) , c σ (2) , · · · , c nσ ( n ) ) ,C ′ = diag( c ′ τ (1) , c ′ τ (2) · · · , c ′ nτ ( n ) )( E ′ E ′ · · · E ′ s ) , where E i and E ′ j are elementary matrixes obtained by swapping two rows of theidentity matrix I n . Then CC ′ = E s E s − · · · E diag( c σ (1) c ′ τ (1) , c σ (2) c ′ τ (2) , · · · , c nσ ( n ) c ′ nτ ( n ) ) E ′ E ′ · · · E ′ t , and hence(( n X l =1 c il c ′ lj ) ) n × n = E r · · · E diag(( c σ (1) ) ( c ′ τ (1) ) , · · · , ( c nσ ( n ) ) ( c ′ nτ ( n ) ) ) E ′ · · · E ′ s = E r · · · E diag(( c σ (1) ) , · · · , ( c nσ ( n ) ) )diag(( c ′ τ (1) ) , · · · , ( c ′ nτ ( n ) ) ) E ′ · · · E ′ s = (( c ij ) ) n × n (( c ′ ij ) ) n × n , which implies χ [( M, CC ′ )] = χ { ( χ [( M, C )] , C ′ ) } . Therefore, the map χ defined inDefinition 3.8 is a right group action of QPL n ( k ) on M n ( k ). (cid:3) Corollary 3.10. In M n ( k ), there is a natural equivalence relation ∼ R defined by M ∼ R M ′ ⇔ ∃ C = ( c ij ) n × n ∈ QPL n ( k ) such that M ′ = χ ( M, C ) . Hence the set of isomorphism classes of DG algebras in {A O − ( k n ) ( M ) | M ∈ M n ( k ) } is the quotient set {A O − ( k n ) ( M ) | M ∈ M n ( k ) } / QPL n ( k ) . some useful lemmas The cohomology graded algebra H ( A ) of a cochain DG algebra A usually con-tains a lot of homological informations. In some cases, it is possible to detect theCalabi-Yau properties of A from H ( A ). For example, It is proved in [MYY], that A is a Calabi-Yau DG algebra if the trivial DG algebra ( H ( A ) ,
0) is Calabi-Yau.And we have the following lemma.
Lemma 4.1. [MH, Theorem A]
Let A be a connected cochain DG algebra. Then A is a Koszul Calabi-Yau DG algebra if H ( A ) belongs to one of the following cases: ( a ) H ( A ) ∼ = k ; ( b ) H ( A ) = k [ ⌈ z ⌉ ] , z ∈ ker( ∂ A );( c ) H ( A ) = k h⌈ z ⌉ , ⌈ z ⌉i ( ⌈ z ⌉⌈ z ⌉ + ⌈ z ⌉⌈ z ⌉ ) , z , z ∈ ker( ∂ A ) . In the rest of section, we will give another useful criterion to detect the Calabi-Yau properties of A . For this, we need the following interesting lemma first. Lemma 4.2.
For any α, β, γ , E λ,µ,ν = ( a b a c a d λb + νc νb + µc a | a, b, c, d ∈ k ) is a commutative symmetric k -algebra of -dimension under the usual multiplicationof matrices.Proof. We claim that E λ,µ,ν is closed under the usual multiplication of matrices andthe multiplication in E λ,µ,ν is commutative. Indeed, we have a b a c a d λb + νc νb + µc a a ′ b ′ a ′ c ′ a ′ d ′ λb ′ + νc ′ νb ′ + µc ′ a ′ = aa ′ ab ′ + a ′ b aa ′ ac ′ + a ′ c aa ′ d ′′ λ ( a ′ b + ab ′ ) + ν ( a ′ c + ac ′ ) ν ( a ′ b + ab ′ ) + µ ( a ′ c + ac ′ ) aa ′ and a ′ b ′ a ′ c ′ a ′ d ′ λb ′ + νc ′ νb ′ + µc ′ a ′ a b a c a d λb + νc νb + µc a = aa ′ ab ′ + a ′ b aa ′ ac ′ + a ′ c aa ′ d ′′ λ ( a ′ b + ab ′ ) + ν ( a ′ c + ac ′ ) ν ( a ′ b + ab ′ ) + µ ( a ′ c + ac ′ ) aa ′ , where d ′′ = a ′ d + ad ′ + ( λb + νc ) b ′ + c ′ ( νb + µc ).Clearly, 1 E = P i =1 E ii . Let e = E + νE + λE , e = E + νE + µE and e = E . Then we have E λ,µ,ν = k E ⊕ ke ⊕ ke ⊕ ke . One sees that the map θ : E λ,µ,ν → Hom k ( E λ,µ,ν , k )defined by e e e −→−→−→−→ e ∗ e ∗ e ∗ ∗ is an isomorphism of left E λ,µ,ν -modules. Thus E λ,µ,ν is a commutative Frobe-nius algebra. Since any commutative Frobenius algebra is symmetric, E λ,µ,ν is acommutative symmetric algebra. (cid:3) Lemma 4.3.
Let A be a connected cochain DG algebra such that H ( A ) = k h⌈ y ⌉ , ⌈ y ⌉i / ( t ⌈ y ⌉ + t ⌈ y ⌉ + t ( ⌈ y ⌉⌈ y ⌉ + ⌈ y ⌉⌈ y ⌉ )) , where ( t , t , t ) ∈ P k −{ (1 , , , (0 , , } . Then the Ext -algebra of A is isomorphicto E t ,t ,t and A is a Koszul Calabi-Yau DG algebra. G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 13 Proof.
The graded module H ( A ) k has the following minimal graded free resolution:0 → H ( A ) e r d → H ( A ) ⊗ ( M i =1 ke y i ) d → H ( A ) ε → k → , where ε, d and d are defined by ε | H ≥ ( A ) = 0, ε | H ( A ) = id k , d ( e y i ) = y i and d ( e r ) = t y e + t y e + t y e + t y e . Applying the constructing procedure ofEilenberg-Moore resolution, we can construct a minimal semi-free resolution F ofthe DG A -module k . We have F = A ⊕ [ A ⊗ ( M i =1 k Σ e y i )] ⊕ A Σ e r and ∂ F is defined by ∂ F (1) ∂ F (Σ e ) ∂ F (Σ e ) ∂ F (Σ e r ) = y y t y + t y t y + t y e Σ e Σ e r . Hence A is a Koszul and homologically smooth DG algebra. By the minimality of F , we have Hom A ( F, k ) = { k ∗ ⊕ [ n M i =1 k (Σ e y i ) ∗ ] ⊕ k (Σ e r ) ∗ } . So the Ext-algebra E = H (Hom A ( F, F )) is concentrated in degree 0. On the otherhand, Hom A ( F, F ) ∼ = { k ∗ ⊕ [ n M i =1 k (Σ e y i ) ∗ ] ⊕ k (Σ e r ) ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F, F )). Since F is a free graded A -module with a basis { , Σ e y , Σ e y , Σ e r } concentratedin degree 0, the elements in Hom A ( F, F ) is one to one correspondence with thematrixes in M ( k ). Indeed, any f ∈ Hom A ( F, F ) is uniquely determined by amatrix A f = ( a ij ) × ∈ M ( k ) with f (1) f (Σ e ) f (Σ e ) f (Σ e r ) = A f · e Σ e Σ e r . And f ∈ Z [Hom A ( F, F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f y y t y + t y t y + t y = y y t y + t y t y + t y A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = 0 a = a t + a t a = a t + a t
24 X.-F. MAO, X.-T. WANG, AND M.-Y.ZHANG by direct computations. Hence the the Ext-algebra E ∼ = a b a c a d t b + t c t b + t c a | a, b, c, d ∈ k = E t ,t ,t . By Lemma 4.2, E is a symmetric algebra and hence Tor A ( k A , A k ) ∼ = E ∗ is asymmetric coalgebra. By [HM, Theorem A], A is a Koszul Calabi-Yau DG algebra. (cid:3) cohomology and Calabi-Yau properties From this section, we will do research on homological properties of A O − ( k n ) ( M ).For the case n = 2, we have the following proposition. Proposition 5.1. [MH, Proposition 3.3]
For M = ( m ij ) × ∈ M ( k ) , we have H [ A O − ( k ) ( M )] = k, if | M | 6 = 0 k [ ⌈ x ⌉ ] , if m = 0 and m = m = m = 0 k [ ⌈ x ⌉ , ⌈ x ⌉ ] / ( ⌈ x ⌉ ) , if m = 0 , m = m = m = 0 k [ ⌈ x ⌉ ] , if m = 0 , m = 0 and m = m = 0 k [ ⌈ m x − m x ⌉ ] , if m = 0 , m = 0 and m = m = 0 k [ ⌈ m x − m x ⌉ ] , if m ij = 0 , ∀ i, j, m = m m , | M | = 0 k [ ⌈ m x − m x ⌉ , ⌈ x ⌉ ] / ( ⌈ m x − m x ⌉ ) , if m ij = 0 , ∀ i, j,m = m m , | M | = 0 . Remark 5.2.
By Proposition 5.1 and Lemma 4.1, one sees that A O − ( k ) ( M ) is aKoszul Calabi-Yau DG algebra in the following cases: (1) | M | 6 = 0 ; (2) m = 0 and m = m = m = 0 ; (3) m = 0 , m = 0 and m = m = 0 ; (4) m = 0 , m = 0 and m = m = 0 ; (5) m ij = 0 , ∀ i, j, m = m m , | M | = 0 .For the other cases, the statement that A O − ( k ) ( M ) is a Koszul Calabi-Yau DGalgebra also holds by [MH, Theorem C] . It is natural for us to ask whether each A O − ( k ) ( M ) , M ∈ M ( k ) is also a KoszulCalabi-Yau DG algebra. By [MYY, Proposition 3 . A O − ( k ) ( M ) isa Calabi-Yau DG algebra when M = 0. Note that each A O − ( k ) ( M ) is actually a3-dimensional DG Sklyanin algebra in [MWYZ]. When M is not a zero matrix, wehave the following proposition on H [ A O − ( k ) ( M )]. We refer the reader to [MWYZ,Section 6] for detailed computations. Proposition 5.3. [MWYZ]
Assume that M is a matrix in M ( k ) . Then we havethe following statements on H [ A O − ( k ) ( M )] . (1) H ( A O − ( k ) ( M )) = k , when r ( M ) = 3 ; (2) H [ A O − ( k ) ( M )] = k [ ⌈ t x + t y + t z ⌉ ] if r ( M ) = 2 , s t + s t + s t = 0 ,where k ( s , s , s ) T and k ( t , t , t ) T be the solution spaces of homogeneouslinear equations M X = 0 and M T X = 0 , respectively; (3) H ( A O − ( k ) ( M )) is k [ ⌈ t x + t x + t x ⌉ , ⌈ s x + s x + s x ⌉ ] / ( ⌈ t x + t x + t x ⌉ ) , if r ( M ) = 2 , s t + s t + s t = 0 , where k ( s , s , s ) T and k ( t , t , t ) T arethe solution spaces of homogeneous linear equations M X = 0 and M T X =0 , respectively; G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 15 (4) H [ A O − ( k ) ( M )] is k h⌈ l x − x ⌉ , ⌈ l x − x ⌉i ( m ⌈ l x − x ⌉ + m ⌈ l x − x ⌉ − ⌈ l x − x ⌉⌈ l x − x ⌉ + ⌈ l x − x ⌉⌈ l x − x ⌉ l l m l m l − m ) when M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 with l l = 0 and m l + m l = m ; (5) H [ A O − ( k ) ( M )] = k h⌈ l x − x ⌉ , ⌈ l x − x ⌉i ( ⌈ l x − x ⌉⌈ l x − x ⌉ + ⌈ l x − x ⌉⌈ l x − x ⌉ ) , when M = m m m l m l m l m l m l m l m with ( m , m , m ) = 0 , m l + m l = m and l l = 0 ; (6) H [ A O − ( k ) ( M )] = k h⌈ l x − x ⌉ , ⌈ l x − x ⌉i ( m ⌈ l x − x ⌉ + m ⌈ l x − x ⌉ ) , when M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 with m l + m l = m and l l = 0 ; (7) H [ A O − ( k ) ( M )] = k h⌈ l x − x ⌉ , ⌈ x ⌉ , ⌈ x ⌉i m ⌈ l x − x ⌉ + m ⌈ x ⌉ ⌈ x ⌉⌈ l x − x ⌉ − ⌈ l x − x ⌉⌈ x ⌉⌈ x ⌉⌈ x ⌉ − ⌈ x ⌉⌈ x ⌉⌈ l x − x ⌉⌈ x ⌉ + ⌈ x ⌉⌈ l x − x ⌉ , when M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , with m l + m l = m , l = 0 and l = 0 ; (8) H [ A O − ( k ) ( M )] = k h⌈ l x − x ⌉ , ⌈ x ⌉ , ⌈ x ⌉i m ⌈ l x − x ⌉ + m ⌈ x ⌉ ⌈ x ⌉⌈ l x − x ⌉ − ⌈ l x − x ⌉⌈ x ⌉⌈ x ⌉⌈ x ⌉ − ⌈ x ⌉⌈ x ⌉⌈ l x − x ⌉⌈ x ⌉ + ⌈ x ⌉⌈ l x − x ⌉ , when M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , with m l + m l = m , l = 0 and l = 0 ; (9) H [ A O − ( k ) ( M )] = k h⌈ x ⌉ , ⌈ x ⌉ , ⌈ x ⌉i m ⌈ x ⌉ + m ⌈ x ⌉ ⌈ x ⌉⌈ x ⌉ − ⌈ x ⌉⌈ x ⌉⌈ x ⌉⌈ x ⌉ − ⌈ x ⌉⌈ x ⌉⌈ x ⌉⌈ x ⌉ + ⌈ x ⌉⌈ x ⌉ when M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , with m l + m l = m , l = 0 and l = 0 . Remark 5.4.
For the case r ( M ) = 1 , we only need to consider (4-9) in Proposition5.3. Indeed, we can see the reasons by applying Theorem 3.6 and the following fact.For any ( a, b, c ) = (0 , , and l , l ∈ k , let M = a b cl a l b l cl a l b l c , C = , C ′ = . Then χ ( M, C ) = a b cl a l b l cl a l b l c = l b l a l cb a cl b l a l c and χ ( M, C ′ ) = a b cl a l b l cl a l b l c = l c l b l al c l b l ac b a . By Lemma 4.1 and Lemma 4.3, one sees that A O − ( k ) ( M ) is a Koszul Calabi-Yau DG algebra in the following cases:(1) r ( M ) = 3;(2) r ( M ) = 2, s t + s t + s t = 0, where k ( s , s , s ) T and k ( t , t , t ) T be thesolution spaces of homogeneous linear equations M X = 0 and M T X = 0,respectively;(3) M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0with l l = 0 and m l + m l = m ;(4) M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , with m l + m l = m and l l = 0;(5) M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0with m l + m l = m and l l = 0.It remains to consider the Calabi-Yau properties of A O − ( k ) ( M ) in the following4 cases:(1) Case 1: r ( M ) = 2 and s t + s t + s t = 0 where k ( s , s , s ) T and k ( t , t , t ) T be the solution spaces of homogeneous linear equations M X =0 and M T X = 0, respectively; G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 17 (2) Case 2: M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0, with m l + m l = m , l = 0 and l = 0;(3) Case 3: M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0, with m l + m l = m , l = 0 and l = 0;(4) Case 4: M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0, with m l + m l = m , l = 0 and l = 0.The proof of each case involves further classifications and complicated analysis.The main ideas of our proof is to construct the minimal semi-free resolution of A k in each case and compute the corresponding Ext-algebras. In the rest of this paper,we will allocate Section 6 and Section 7 to discuss the homological properties of A O − ( k ) ( M ) for Case 1 and Case 2 − case H ( A O − ( k ) ( M )) in Case 1 is not homologically smoothsince it is k [ ⌈ t x + t x + t x ⌉ , ⌈ s x + s x + s x ⌉ ] / ( ⌈ t x + t x + t x ⌉ ) . From H ( A O − ( k ) ( M )) we can’t judge the Calabi-Yau properties (resp. homologicallysmoothness) of A O − ( k ) ( M ). We turn to construct the minimal semi-free resolutionof A k . According to the constructing procedure of [MW1, Proposition 2.4], we willconstruct the resolution as follows.Let F = A and ε = ε : F = A → k . Define F as an extension of theDG A -module F by F = F ⊕ A e and ∂ F ( e ) = t x + t x + t x . Since t x + t x + t x ∈ B ( A ), there exist σ = q x + q x + q x ∈ A such that ∂ A ( σ ) = t x + t x + t x . Define F as an extension of the DG A -module F by F = F ⊕ A e and ∂ F ( e ) = ( t x + t x + t x ) e + σ. Case 1 .
1. If q t x + q t x + q t x B ( A ), then q t x + q t x + q t x = ∂ A ( b )+ s x + s x + s x , for some b ∈ A . For any cocycle element a + a e + a e ∈ Z ( F ), we have0 = ∂ F ( a + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + ∂ A ( a ) − a ( t x + t x + t x ) . This implies that ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 . (1)Since q t x + q t x + q t x B ( A ), one can easily check that (1) implies that a = 0 a = b ( t x + t x + t x ) a = b ( q x + q x + q x ) + b ( t x + t x + t x ) for some b , b ∈ k . So a + a e + a e = ∂ F ( b e + b e ) ∈ B ( F ). Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-free resolutionof A k . Since the proof is routine, straightforward and tedious, we omit the proofhere. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e } concentrated in degree 0,the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) = A f · e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a | a, b, c ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This implies thatTor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], the DGalgebra A in Case 1 . . Example 6.1.
The DG algebra A O − ( k ) ( M ) belongs to Case . , when M is oneof the following matrixes: (1) , (2) , (3) , (4) , (5) , (6) , (7) , (8) − − , (9) − − . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 19 Now, let us study the case that q t x + q t x + q t x ∈ B ( A ). We claim thatwe can divide it into the following two series: • Case 1 . . ∗ , when ( q t , q t , q t ) T = 0; • Case 1 . . ∗ , when ( q t , q t , q t ) T and ( t , t , t ) T are linearly independent.Indeed, when 0 = ( q t , q t , q t ) T and ( t , t , t ) T are linearly dependent, we mayas well let ( q t , q t , q t ) T = c ( t , t , t ) T for some c ∈ k × . Let q ′ = q − ct , q ′ = q − ct , q ′ = q − ct . Then ( q ′ t + q ′ t + q ′ t = 0 ∂ A ( q ′ x + q ′ x + q ′ x ) = t x + t x + t x . We can replace q , q , q by q ′ , q ′ , q ′ in the construction.6.1. Case . . ∗ . Since q t x + q t x + q t x = 0, we may choose τ = r x + r x + r x = 0, equivalently each r i = 0, such that ∂ A ( τ ) = 0. We label it “Case1 . .
1” when q x + q x + q x B ( A ). We extend F to a semi-free DG module F with F = F ⊕ A e and ∂ F ( e ) = ( t x + t x + t x ) e + σe . We claim H ( F ) = 0. Indeed, for any cocycle element a + a e + a e + a e ∈ Z ( F ), wehave 0 = ∂ F ( a + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ] − a [( t x + t x + t x ) e + σe ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + ∂ A ( a ) − a ( t x + t x + t x ) − a σ. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ = 0 . (2)Since q x + q x + q x B ( A ), it is easy to check that (2) implies, a = 0 a = c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c σ + c ( t x + t x + t x ) , for some c , c and c ∈ k . Then a + a e + a e + a e = ∂ A ( c e + c e + c e ). So H ( F ) = 0. Furthermore, we can show that F is the minimal semi-free resolutionof A k . Since the proof is routine, straightforward and tedious, we omit the proofhere. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixes in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) = A f · e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a | a, b, c, d ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A in Case 1 . . . . Example 6.2.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M isone of the following matrixes: (1) , (2) , (3) , (4) , (5) , (6) . If q t x + q t x + q t x = 0 and q x + q x + q x ∈ B ( A ), then thingswill be different from Case 1 . .
1. We must proceed our construction. Let λ = u x + u x + u x such that ∂ A ( λ ) = q x + q x + q x . We label it “Case 1 . . u t x + u t x + u t x B A . We extend F in Case 1 . . F with F = F ⊕A e and ∂ F ( e ) = ( t x + t x + t x ) e + σe + λ. Weclaim H ( F ) = 0. Indeed, for any cocycle element a + a e + a e + a e + a e ∈ G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 21 Z ( F ), we have0 = ∂ F ( a + a e + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe ]+ ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + λ ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a ( t x + t x + t x ) − a σ ] e + ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ = 0 . (3)Since u t x + u t x + u t x B A , (4) implies a = 0 a = c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( u x + u x + u x ) + c ( t x + t x + t x ) , for some c , c , c , c ∈ k . Then a + a e + a e + a e + a e = ∂ A ( c e + c e + c e + c e ) . Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-freeresolution of A k . Since the proof is routine, straightforward and tedious, we omitthe proof here. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a a = a = a = a , a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a e d c b a | a, b, c, d, e ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A in Case 1 . . . . Example 6.3.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M iseither one of the following matrixes: (1) , (2) , (3) . If q t x + q t x + q t x = 0 q x + q x + q x ∈ B ( A ) u t x + u t x + u t x ∈ B ( A ) , then we must continue the process after the construction of F in Case 1 . .
2. Let ω = v x + v x + v x such that ∂ A ( ω ) = u t x + u t x + u t x . We label it“Case 1 . .
3” when(4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x B A . We extend F in Case 1 . . F with F = F ⊕ A e and ∂ F ( e ) = ( t x + t x + t x ) e + σe + λe + 2 ω. G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 23 We claim H ( F ) = 0. Indeed, for any a + a e + a e + a e + a e + a e ∈ Z ( F ),we have0 = ∂ F ( a + a e + a e + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe ]+ ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + λ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + λe + 2 ω ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ ] e + ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ − a ω. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a λ − a ω. (4)Since (4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x B A , (4) implies a = 0 a = c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( u x + u x + u x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( u x + u x + u x ) + 2 c ( v x + v x + v x )+ c ( t x + t x + t x )for some c , c , c , c , c ∈ k . Then a + a e + a e + a e + a e + a e = ∂ A ( c e + c e + c e + c e + c e ) . Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-freeresolution of A k . Since the proof is routine, straightforward and tedious, we omitthe proof here. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e , e } concentrated indegree 0, the elements in Hom A ( F , F ) is one to one correspondence with thematrixes in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i ω λ τ σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i ω λ τ σ P i =1 t i x i A f which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a = a a = a = a = a = a ,a = a = a = a ,a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a e d c b a f e d c b a | a, b, c, d, e, f ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A in Case 1 . . . . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 25 Example 6.4.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M = . If q t x + q t x + q t x = 0 q x + q x + q x ∈ B ( A ) u t x + u t x + u t x ∈ B ( A )(4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x ∈ B A , then we must continue the process after the construction of F in Case 1 . .
3. Since( t , t , t ) = 0 and ∂ A ( q x + q x + q x ) = t x + t x + t x = 0, we have( q , q , q ) = 0. Hence q t x + q t x + q t x = 0 implies that there exist oneor two nonzero elements in { t , t , t } . By symmetry, we only need to consider thefollowing three cases:Case A: t = 0 , q = 0 t = 0 , q = 0 t = 0 , q = 0 Case B: t = 0 , q = 0 t = 0 , q = 0 t = 0 , q = 0 Case C: t = 0 , q = 0 t = 0 , q = 0 t = 0 , q = 0 . If Case A happens, then ∂ A ( t x + t x ) = 0, ∂ A ( q x ) = t x + t x and ∂ A ( u x + u x + u x ) = q x . So B ( A ) = k ( t x + t x ) ⊕ kx . Since ∂ A ( v x + v x + v x ) = u t x + u t x ∈ B ( A ) , we have u = lt , u = lt , for some l ∈ k . If u = 0, then ∂ A ( u x + u x ) = l∂ A ( t x + t x ) = 0which contradicts with ∂ A ( u x + u x ) = q x = 0. Hence u = 0. Then ∂ A ( u x + u x + u x ) = u ∂ A ( x ) + l∂ A ( t x + t x )= u q [ t x + t x ] , which contradicts with the assumption ∂ A ( u x + u x + u x ) = q x . Hence Case A is impossible to occur.If Case B happens, then we have ∂ A ( x ) = 0, ∂ A ( q x + q x ) = t x and ∂ A ( u x + u x + u x ) = ∂ A ( u x + u x ) = q x + q x . So B ( A ) = kx ⊕ k ( q x + q x ). We have ∂ A ( v x + v x + v x ) = u t x .Since ∂ A ( x ) = 0, we may choose v = 0. If u = 0, then ( v , v ) = 0 and( v , v ) = ( u t q , u t q ) since Z ( A ) = kx . Then(4 v t +2 q u ) x +(4 v t +2 q u ) x +(4 v t +2 q u ) x = 2 q u x +2 q u x ∈ B A . So u = cq , u = cq for some c ∈ k . But then we have ∂ A ( u x + u x + u x ) = ∂ A ( cq x + cq x ) + u ∂ A ( x )= ct x , which contradicts with the assumption that ∂ A ( u x + u x + u x ) = q x + q x . Hence u = 0. So u t x + u t x + u t x = 0. We may choose v = v = v = 0.Then 2 q u x + 2 q u x = (4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x ∈ B A . Since B ( A ) = kx ⊕ k ( q x + q x ), we get u = lq , u = lq for some l ∈ k . Then ∂ A ( u x + u x ) = l∂ A ( q x + q x ) = lt x , and we reach a contradiction withthe assumption that ∂ A ( u x + u x ) = ∂ A ( u x + u x + u x ) = q x + q x . Therefore, Case B is also impossible to occur.If Case C happens, then we have ∂ A ( x ) = 0, ∂ A ( q x ) = t x , and ∂ A ( u x + u x + u x ) = q x . So B ( A ) = kx ⊕ kx . Since ∂ A ( x ) = 0, we may choose u = 0 in construction.We claim that u = 0. Indeed, if u = 0, then ∂ A ( u x ) = ∂ A ( u x + u x + u x ) = q x , which contradicts with the assumption that ∂ A ( q x ) = t x . So u = 0. Inthe construction, we can choose v = v = v = 0 since u t x + u t x + u t x = 0.Then (4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x = 2 q u x . Wemay choose w = 2 u u q , w = 2 u q , w = 0and η = w x + w x + w x such that ∂ A ( η ) = 2 q u x . We extend F in Case1 . . F with F = F ⊕ A e and ∂ F ( e ) = ( t x ) e + σe + λe + η. It is straightforward for one to show that ∂ F [( t x ) e + σe + λe + ηe ] = 0 . So H ( F ) = 0. We extend F to a semi-free DG module F with F = F ⊕ A e and ∂ F ( e ) = ( t x ) e + σe + λe + ηe . We claim H ( F ) = 0. Indeed, for anycocycle element a + a e + a e + a e + a e + a e + a e + a e ∈ Z ( F ), wehave0 = ∂ F ( a + a e + a e + a e + a e + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x ) + ∂ A ( a ) e − a [( t x ) e + σ ] + ∂ A ( a ) e − a [( t x ) e + σe ] + ∂ A ( a ) e − a [( t x ) e + σe + λ ] + ∂ A ( a ) e − a [( t x ) e + σe + λe ] + ∂ A ( a ) e − a [( t x ) e + σe + λe + η ]+ ∂ A ( a ) e − a [( t x ) e + σe + λe + ηe ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x )] e + [ ∂ A ( a ) − a ( t x ) − a σ ] e + [ ∂ A ( a ) − a ( t x ) − a σ ] e + [ ∂ A ( a ) − a σ − a ( t x ) − a λ ] e + [ ∂ A ( a ) − a σ − a ( t x ) − a λ ] e + [ ∂ A ( a ) − a ( t x ) − a σ − a λ ] e + ∂ A ( a ) − a ( t x ) − a σ − a λ − a η. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x ) = 0 ∂ A ( a ) − a σ − a ( t x ) = 0 ∂ A ( a ) − a σ − a ( t x ) = 0 ∂ A ( a ) − a σ − a ( t x ) − a λ = 0 ∂ A ( a ) − a σ − a ( t x ) − a λ = 0 ∂ A ( a ) − a ( t x ) − a σ − a λ = 0 ∂ A ( a ) − a ( t x ) − a σ − a λ − a η = 0 . (5) G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 27 Since u = 0, we have u x + 3 u x B ( A ). Then (5) implies that a = 0 a = c t x a = c q x + c t x a = c q x + c t x a = c q x + c ( u x + u x ) + c t x a = c ( u x + u x ) + c q x + c t x a = c ( u x + u x ) + c q x + c η + c t x a = c ( u x + u x ) + c q x + c η + c t x , for some c , c , c , c , c , c , c ∈ k . a + a e + a e + a e + a e + a e + a e + a e = ∂ F ( c e + c e + c e + c e + c e + c e + c e ) . Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-freeresolution of A k . Since the proof is routine, straightforward and tedious, we omitthe proof here. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e , e , e , e } concentrated indegree 0, the elements in Hom A ( F , F ) is one to one correspondence with thematrixes in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by amatrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f t x σ t x σ t x λ σ t x λ σ t x η λ σ t x η λ σ t x = t x σ t x σ t x λ σ t x λ σ t x η λ σ t x η λ σ t x A f which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a = a = a = a a = a = a = a = a = a = a ,a = a = a = a = a = a ,a = a = a = a = a ,a = a = a = a , a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a e d c b a f e d c b a g f e d c b a h g f e d c b a | a, b, c, d, e, f, g, h ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A this case is a Koszul Calabi-Yau DG algebra. For the convenienceof future talk, we rename Case C to Case 1 . .
4. We can list the following twoexamples for Case 1 . . Example 6.5.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M iseither one of the following matrixes: , . Case . . ∗ . By assumption, q t x + q t x + q t x and t x + t x + t x constitute a basis of B ( A ). Let τ = r x + r x + r x ∈ A such that ∂ A ( τ ) = q t x + q t x + q t x . We label it “Case 1 . .
1” when P i =1 (4 r i t i + q i ) x i B ( A ).We extend F to a semi-free DG module F with F = F ⊕ A e and ∂ F ( e ) = ( t x + t x + t x ) e + σe + 2 τ. G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 29 We claim H ( F ) = 0. Indeed, for any a + a e + a e + a e ∈ Z ( F ), we have0 = ∂ F ( a + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ] − a [( t x + t x + t x ) e + σe + 2 τ ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ = 0 . (6)Since (4 r t + q ) x + (4 r t + q ) x + (4 r t + q ) x B ( A ), it is easy to checkthat (6) implies , a = 0 a = c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = 2 c τ + c σ + c ( t x + t x + t x ) , for some c , c and c ∈ k . Then a + a e + a e + a e = ∂ A ( c e + c e + c e ).Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-freeresolution of A k . Since the proof is routine, straightforward and tedious, we omitthe proof here. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) = A f · e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a | a, b, c, d ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A in Case 1 . . . . Example 6.6.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M isone of the following matrixes: (1) , (2) , (3) . We label it “Case 1 . .
2” when P i =1 (4 r i t i + q i ) x i ∈ B ( A ). Then things will bedifferent from Case 1 . .
1. We must proceed our process after constructing F inCase 1 . .
1. Let λ = u x + u x + u x such that ∂ A ( λ ) = (4 r t + q ) x + (4 r t + q ) x + (4 r t + q ) x . We extend F in Case 1 . . F with F = F ⊕ A e and ∂ F ( e ) = ( t x + t x + t x ) e + σe + λ. In order to get a minimal semi-freeresolution of k , we should proceed our construction by extending F . For this, weneed some analysis first. Proposition 6.7.
Let M = ( m ij ) × be a matrix which satisfies the the followingconditions: (1) r ( M ) = 2 , ∃−→ s = s s s = 0 and −→ t = t t t = 0 such that M −→ s = 0 , M T −→ t = 0 , and s t + s t + s t = 0 ; (2) ∃−→ q = q q q such that M T −→ q = t t t , which is linearly independentfrom q t q t q t ; (3) ∃−→ r = r r r such that M T −→ r = q t q t q t ; (4) ∃−→ u = u u u such that M T −→ u = r t + q r t + q r t + q . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 31 Then M belongs to one of the following types: M = a λab ec λc , M = b e a λa c λc , M = a λa c λc b e , where a, c, λ ∈ k × , e = λb and a = λc . The proof of Proposition 6.7 concerns tedious computations and complicatedmatrix analysis. To make it more readable, I would like to divide it into someparts. For simplicity, we write −→ t = t t t , −→ qt = t t t , −−−−−→ rt + q = r t + q r t + q r t + q . We show the following lemmas first.
Lemma 6.8.
Assume that M = ( m ij ) × satisfies the conditions (1) , (2) and (3) in Proposition 6.7. Then there is at least one zero in { s , s , s , t , t , t } .Proof. If the elements in { s , s , s , t , t , t } are all non-zero, then s t + s t = 0since s t + s t + s t = 0. Since M −→ s = 0 and M T −→ t = 0, M can be written by a a s a + s a − s b b s b + s b − s t a + t b − t t a + t b − t t ( s a + s a )+ t ( s b + s b ) s t By M T −→ q = −→ t , we have 2 = r ( M T ) = r ( M T , −→ t ), which implies t = − s s t − s s t . (7)Similarly, M T −→ r = −→ qt implies 2 = r ( M T ) = r ( M T , −→ rt ) and hence q t = − s s q t − s s q t ⇔ q = − s t s t q − s t s t q . (8)Since −→ qt and −→ t are linearly independent, the vectors (cid:18) q t q t (cid:19) and (cid:18) t t (cid:19) arelinearly independent. Indeed, if (cid:18) q t q t (cid:19) and (cid:18) t t (cid:19) are linearly dependent, thenthere exist λ ∈ k such that λt = q t , λt = q t , which implies q = λt , q = λt .And hence q = − s t s t q − s t s t q = − λs t s t − λs t s t = λ s t s t = λt . But then −→ qt = λ −→ t and −→ t are linearly dependent. We reach a contradiction. Sothe vectors (cid:18) q t q t (cid:19) and (cid:18) t t (cid:19) are linearly independent and hence q t = q t .On the other hand, M T −→ q = −→ t implies a q + b q + ( t a + t b )( s t q + s t q ) t s t = t (9) and a q + b q + ( t a + t b )( s t q + s t q ) t s t = t , (10)which are respectively equivalent to( a q + b q ) s t t + ( t a + t b )( s t q + s t q ) t = t s t t and ( a q + b q ) s t t + ( t a + t b )( s t q + s t q ) t = t s t t . Then ( a q t + b q t − a q t − b q t )( s t + s t )= ( s t q + s t q )( t t a + t b − t a − t t b ) ⇒ s b q t t − s b q t + s a q t − s a q t t = s b q t t − s b q t t + s a q t t − s a q t t ⇒ q t [ b s t t − b s t − a s t + a s t t ]= q t [ b s t t − a s t + a s t t − b s t ] . If b s t t − a s t + a s t t − b s t = 0, then we get q t = q t , which contradictswith q t = q t .If b s t t − a s t + a s t t − b s t = 0, then s t b − s t a = ( s t b − s t a ) t t (11)and we can show as follows that (9) and (10) are equivalent. Indeed, (10) is equiv-alent to ( a q + b q )( − s t − s t ) + ( t a + t b )( s t q + s t q ) s t = t ⇔ − a q s t − b s q t + a q s t t + b q s t t s t = t ⇔ ( q t − q t )( s t b − s t a ) s t = t ⇔ ( q t − q t )( s t b − s t a ) t t s t = t ⇔ ( q t − q t )( s t b − s t a ) s t = t . Similarly, (9) is equivalent to( a q + b q )( − s t − s t ) + ( t a + t b )( s t q + s t q ) s t = t ⇔ − a q s t − b s q t + a q s t t + b q s t t s t = t ⇔ ( q t − q t )( s t b − s t a ) s t = t . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 33 So (9) and (10) are equivalent. Then r a b t a + t b − t a b t a + t b − t ! = 1 and hence r ( M T ) = r a b t a + t b − t a b t a + t b − t s a + s a − s s b + s b − s t ( s a + s a )+ t ( s b + s b ) s t = 1 , which contradicts with r ( M ) = 2.Then we reach a conclusion that there are at least one zero in { s , s , s , t , t , t } . (cid:3) Lemma 6.9.
Assume that M = ( m ij ) × satisfies all the conditions in Proposition6.7. Then there is at least one zero in { t , t , t } . Furthermore, there are exactlyonly one zero in { t , t , t } .Proof. We will give a proof of the first part of the statement by contradiction.Assume that each t i is nonzero. Then there exists at least one zero in { s , s , s } byLemma 6 .
8. Furthermore, there is exactly one zero s i since s t + s t + s t = 0.Without the loss of generality, we assume s = 0. Then s t + s t = 0, and s , s ∈ k × . By M s = 0, we have m s + m s = 0 m s + m s = 0 m s + m s = 0 ⇔ m m m = − s s m m m . Moreover, M T t = 0 and r ( M ) = 2 imply that M can be written by M = a a s a − s b b s b − s a t + b t − t a t + b t − t s a t + s b t s t with r (cid:18) a a b b (cid:19) = 2 . By M T −→ r = −→ qt , we have r ( M T ) = r ( M T , −→ qt ), which implies − s s q t = q t orequivalently q = − s q t s t . Substitute it into M T −→ q = −→ t . We have a q − s t b q s t − a t q + b t q t = t (12)and a q − s t b q s t − a t q + b t q t = t . (13)By computations, (12) and (13) are respectively equivalent to a q s t t − b q s t − s t ( a t q + b t q ) = s t t t ⇔ a q s t t − b q s t − a q s t t − b q s t = s t t t ⇔ a q t − a q t = t t ⇔ a ( q t − q t ) = t t and a q s t t − s t b q − ( a t q + b t q ) s t = s t t ⇔ a q s t t − b q s t − a q s t t − b q s t = s t t ⇔ a q t − a q t = t t ⇔ a ( q t − q t ) = t t . Since each t i is non-zero, we have a , a and q t − q t are all non-zeros. Hence a = t t a . Then M T = a b a t + b t − t t t a b a t + b t t − t t t t a − s s b s b t s t − t t a . Since s t = − s t and s q t = − s q t , we have s q s t = s q t = s q t = s q s t , which implies s q = − s q . On the other hand, r ( M T ) = r ( M T , rt + q ) since M T −→ u = −−−−−→ rt + q . Then we have − s s (4 r t + q ) = 4 r t + q ⇔ − s r t − s q = 4 s r t + s q ⇔ − s r t = s r t ⇔ r = − s r t s t . Substitute it into M T r = qt . We get a r − b s r t s t − a t + b t t r = q t (14)and t t a r − b s r t s t − a t + b t t t t r = q t . (15)By computations, (14) and (15) are respectively equivalent to a s r t t − b s r t − ( a t + b t ) s t r = q s t t t ⇔ a s r t t − b r ( s t + s t ) − a r s t t = q s t t t ⇔ a r t − a r t = q t t ⇔ q t = a r t t − a r t t t t and a r s t t − b s r t t − ( a t + b t t ) s t t r = q s t t t ⇔ a r s t t − b r t ( s t + s t ) − a s r t t = q s t t t ⇔ a r t t − a r t t = q t t ⇔ q t = a r t t − a r t t t t . Then q t = q t , which contradicts with the assumption that −→ qt and −→ t are linearlyindependent ( One can see why q t = q t in the proof of Lemma 6.8).By the proof above, we can conclude that there is at least one zero in { t , t , t } .If there are two zeros in { t , t , t } , then −→ qt and −→ t are obviously linearly dependent,which contradicts with the condition (2) in Proposition 6.7. (cid:3) Lemma 6.10.
Assume that M = ( m ij ) × satisfies all the conditions in Proposi-tion 6.7. Then any two non-zero columns of M are linearly independent.Proof. We will give a proof by contradiction. Suppose that M admits two nonzerolinearly dependent columns. Without the loss of generality, M can be written as a ua db ub ec uc f , u ∈ k × . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 35 Since
M s = 0 and r ( M ) = 2, we have s = − us , s = 0. Then s t + s t = 0, and s , s ∈ k × since u = 0. Since M T −→ q = −→ t and −→ q = 0, we have r ( M T ) = r ( M T , −→ t ),which implies t = ut . Similarly, M T −→ r = −→ qt implies r ( M T ) = r ( M T , −→ qt ), andhence q t = uq t . We claim t = 0. Indeed, if t = 0,then −→ qt = q t uq t and −→ t = t ut are linearly dependent, which contradicts with the assumption.Since t = ut , we have t = t = 0 or t = 0 , t = 0. However, both casescontradicts with the statement of Lemma 6.9. Then we complete our proof. (cid:3) Lemma 6.11.
Assume that M = ( m ij ) × satisfies all the conditions in Propo-sition 6.7. Then there is at lest one zero in { s , s , s } . Furthermore, there areexactly two zeros in { s , s , s } .Proof. Assume that each s i is non-zero. It suffices to reach a contradiction. ByLemma 6.9, there is exactly one zero in { t , t , t } . Without the loss of generality,we let t = 0 and t , t ∈ k × . Then M can be written as M = a a s a + s a − s b b s b + s b − s b t − t b t − t t s b + t s b s t . Since M T −→ q = −→ t and M T −→ r = −→ qt , we have r ( M T , −→ t ) = r ( M T ) and r ( M T , −→ qt ) = r ( M T ) respectively. Then we get t = − s s t and q t = − s s q t . This implies that −→ qt = q t − s s q t and −→ t = t − s s t are linearly dependent. This contradictswith the condition (2) in Proposition 6.7. So there is at lest one zero in { s , s , s } .Since r ( M ) = 2 and M −→ s = 0, we can conclude that there are exactly two zeros in { s , s , s } . (cid:3) Now, let us come to the proof of Proposition 6.7.
Proof.
By Lemma 6.10 and Lemma 6.11, one sees that M admits one zero column.More precisely, s i = 0 if and only if the i -th column of M is zero. Without the lossof generality, we may let s = 0. Then we can write M = a db ec f . We have −→ s = s , −→ t = t t , with t , t ∈ k × . Since M T −→ t = 0, we have ( at + ct = 0 dt + f t = 0 , (16)which implies (cid:12)(cid:12)(cid:12)(cid:12) a cd f (cid:12)(cid:12)(cid:12)(cid:12) = 0 ⇔ af = cd. There is at least one non-zero element in { a, f, c, d } . Otherwise, r ( M ) = 1. On theother hand, t and t are both non-zeros. Hence (16) implies that ( a = 0 , c = 0 d = 0 , f = 0 or ( a = 0 , c = 0 d = 0 , f = 0 or ( a = 0 , c = 0 d = 0 , f = 0 . If ( a = 0 , c = 0 d = 0 , f = 0 , then M = db e f . We have b = 0 since r ( M ) = 2.We can take −→ t = f − d , −→ q = d − f edbf b . Then −→ qt = f d − f edb . Since −→ qt and −→ t are linearly independent, we have d b = f e . By M T −→ r = −→ qt , we cantake −→ r = e f − d befd b fd b − f edb . Then −−−−−→ rt + q = (5 f e − d b )( f e − d b ) d b f b and hence3 = r ( M T , −−−−−→ rt + q ) = r ( M T ) = 2, which contradicts with M T −→ u = −−−−−→ rt + q . Sothis case is impossible to occur.If ( a = 0 , c = 0 d = 0 , f = 0 , then M = a b ec . We have e = 0 since r ( M ) = 2.We can take −→ t = c − a , −→ q = c a − baea e . Then −→ qt = c a − abce .Since −→ qt and −→ t are linearly independent, we have a b = c e . By M T −→ r = −→ qt , wecan take −→ r = c a − bce . Then −−−−−→ rt + q = (5 c e − a b )( c e − a b ) a e a e and hence3 = r ( M T , −−−−−→ rt + q ) = r ( M T ) = 2, which contradicts with M T −→ u = −−−−−→ rt + q . Sothis case is impossible to occur.Now, lets consider the cases ( a = 0 , c = 0 d = 0 , f = 0 . Since af = cd , there exists λ ∈ k × such that d = λa, f = λc . We have M = a λab ec λc with e = λb since r ( M ) = 2. By computations, we can choose −→ t = c − a , −→ q = c e − a bea − λaba − λc e − λb . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 37 Then −→ qt = c e − a bcea − λab . Since −→ qt and −→ t are linearly independent, we have a b = c e . By M T −→ r = −→ qt , we can choose −→ r = c e − a bcea ( e − λb ) λ ( a bc − c e ) a ( e − λb ) . Then −−−−−→ rt + q = (5 c e − a b )( c e − a b ) a ( e − λb ) ( a − λc ) ( e − λb ) . Since M T −→ u = −−−−−→ rt + q , we have r ( M T , −−−−−→ rt + q ) = r ( M T ) = 2, which implies a = λc . Note that one get a b = c e when λb = e and a = λc . Therefore, M = a λab ec λc with a, c, λ ∈ k × , e = λb and a = λc .Conversely, if M = a λab ec λc with a, c, λ ∈ k × , e = λb and a = λc ,then it is straight forward to show that M satisfies the conditions (1),(2),(3),(4) inProposition 6.7 and s = 0.Similarly, M = b e a λa c λc (resp. M = a λa c λc b e ) with a, c, λ ∈ k × , e = λb and a = λc if and only if M satisfies the conditions (1),(2),(3),(4) inProposition 6.7 and s = 0 (resp. s = 0). (cid:3) Proposition 6.12.
Assume that M = ( m ij ) × satisfies all the conditions inProposition 6.7. Then there is −→ v = v v v such that M T −→ v = −−−−−→ ut + 2 rq = u t + 2 r q u t + 2 r q u t + 2 r q . Furthermore, r ( M T , −−−−−−−−−−−→ vt + 2 uq + 4 r ) = 3 = r ( M T ) = 2 , where −−−−−−−−−−−→ vt + 2 uq + 4 r = v t + 2 u q + 4 r v t + 2 u q + 4 r v t + 2 u q + 4 r . Proof.
By Proposition 6.7, M belongs to three different types. Since the proofsfor them are similar, we only need to give a detailed proof for the first case. Let M = a λab ec λc with a, c, λ ∈ k × , e = λb and a = λc . By the proof of Proposition 6.7, we can choose −→ t = c − a , −→ q = c e − a bea − λaba − λc e − λb = c a , −→ r = c e − a bcea ( e − λb ) λ ( a bc − c e ) a ( e − λb ) = ceλ ( e − λb ) caλb − e . Hence −−−−−→ rt + q = (5 c e − a b )( c e − a b ) a ( e − λb ) ( a − λc ) ( e − λb ) = (5 e − λb ) c λ ( e − λb ) . By M T −→ u = −−−−−→ rt + q , we can choose −→ u = ea (5 e − λb ) λ ( e − λb ) ( λb − e ) c ( e − λb ) . Then we have −−−−−→ ut + 2 rq = u t + 2 r q u t + 2 r q u t + 2 r q = eac (7 e − λb ) λ ( e − λb ) . Hence r ( M T , −−−−−→ ut + 2 rq ) = r ( M T ) = 2 and there exists −→ v such that M T −→ v = −−−−−→ ut + 2 rq . More precisely, we can choose −→ v = e c (7 e − λb ) λ ( e − λb ) eac (3 λb − e ) λ ( e − λb ) . Then −−−−−−−−−−−→ vt + 2 uq + 4 r = v t + 2 u q + 4 r v t + 2 u q + 4 r v t + 2 u q + 4 r == ec (37 e − eλb + λ b ) λ ( e − λb ) c a ( λb − e ) . We have r ( M T , −−−−−−−−−−−→ vt + 2 uq + 4 r ) = 3 = r ( M T ). (cid:3) By Proposition 6.12, ( u t + 2 r q ) x + ( u t + 2 r q ) x + ( u t + 2 r q ) x ∈ B ( A ). There exists ω = v x + v x + v x such that ∂ A ( ω ) = P i =1 ( u i t i + 2 r i q i ) x i .It is straightforward for one to see that( t x + t x + t x ) e + σe + 2 τ e + λe + 2 ω ∈ Z ( F ) . We extent F to a semi-free DG module F with F = F ⊕ A e and ∂ F ( e ) =( t x + t x + t x ) e + σe + 2 τ e + λe + 2 ω. One sees that X i =1 (4 v i t i + 2 u i q i + 4 r i ) x i B ( A ) . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 39 We claim H ( F ) = 0. Indeed, for any a + a e + a e + a e + a e + a e ∈ Z ( F ),we have 0 = ∂ F ( a + a e + a e + a e + a e + a e )= ∂ A ( a ) + ∂ A ( a ) e − a ( t x + t x + t x ) + ∂ A ( a ) e − a [( t x + t x + t x ) e + σ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + 2 τ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + 2 τ e + λ ] + ∂ A ( a ) e − a [( t x + t x + t x ) e + σe + 2 τ e + λe + 2 ω ]= ∂ A ( a ) e + [ ∂ A ( a ) − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a σ − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a τ − a σ − a ( t x + t x + t x )] e + [ ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ − a λ ] e + ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ − a λ − a ω. Then ∂ A ( a ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a τ − a σ − a ( t x + t x + t x ) = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ − a λ = 0 ∂ A ( a ) − a ( t x + t x + t x ) − a σ − a τ − a λ − a ω. (17)Since (4 v t + 2 q u ) x + (4 v t + 2 q u ) x + (4 v t + 2 q u ) x B A , (17) implies a = 0 a = c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( u x + u x + u x ) + c ( t x + t x + t x ) a = c ( q x + q x + q x ) + c ( u x + u x + u x ) + 2 c ( v x + v x + v x )+ c ( t x + t x + t x )for some c , c , c , c , c ∈ k . Then a + a e + a e + a e + a e + a e = ∂ A ( c e + c e + c e + c e + c e ) . Hence H ( F ) = 0. Furthermore, we can show that F is the minimal semi-freeresolution of A k . Since the proof is routine, straightforward and tedious, we omitthe proof here. By the minimality of F , the Ext-algebra E = H ( R Hom A ( k, k )) = H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ . So E is concentrated in degree 0. On the other hand,Hom A ( F , F ) ∼ = { k ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ ⊕ ke ∗ } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e , e } concentrated indegree 0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixes in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by amatrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i ω λ τ σ P i =1 t i x i = P i =1 t i x i σ P i =1 t i x i τ σ P i =1 t i x i λ τ σ P i =1 t i x i ω λ τ σ P i =1 t i x i A f which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a = a a = a = a = a = a ,a = a = a = a ,a = a = a , a = a by direct computations. Hence the the Ext-algebra E ∼ = a b a c b a d c b a e d c b a f e d c b a | a, b, c, d, e, f ∈ k ∼ = k [ x ] / ( x ) . So E is a symmetric Frobenius algebra concentrated in degree 0. This impliesthat Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By [HM, Theorem 4.2], theDG algebra A in Case 1 . . . .
2. We can list the following examples for Case 1 . . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 41 Example 6.13.
The DG algebra A O − ( k ) ( M ) belongs to Case . . , when M isone of the following matrixes: (1) , (2) , (3) , (4) − −
21 0 1 . Remark 6.14.
In summary, we have the following table, which contains a completelist of the
Ext -algebra of A O − ( k ) ( M ) in all subcases of Case .Subcases Ext -algebaCase . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) Case . . ∼ = k [ x ] / ( x ) For each subcases, the
Ext -algebra is a commutative Frobenius algebra. We showthat A O − ( k ) ( M ) in Case is a Koszul Calabi-Yau DG algebra. On the otherhand, we get counter-examples for Question 0.1, since the Ext -algebras of two quasi-isomorphic connected cochain DG algebras should be isomorphic to each other. case , case and case A O − ( k ) ( M ) for case 2, case3 and case 4. By Theorem 3.6, we have the following lemmas on its isomorphismclasses. Lemma 7.1.
Let M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , m l + m l = m , l = l = 0 . Then (1) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ) if M = 0 and M = 0 ; (2) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E ) if M = 0 , M = 0 or M = 0 , M = 0 .Proof. (1) By the assumption, we have m = 0 and M = m m .Let C = q m
00 0 q m . Then C ∈ Gl ( k ), and χ ( M, C ) = √ m
00 0 √ m m m m
00 0 m = = E + E . By Theorem 3.6, A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ). (2)If m = 0 and m = 0, then M = m E . Let C ′ = q m
00 0 1 .Then χ ( M, C ′ ) = √ m
00 0 1 m
00 0 00 0 0 m
00 0 1 = = E . If m = 0 and m = 0, then M = m E . Let C ′′ = q m .Then χ ( M, C ′′ ) = √ m m m = = E . On the other hand, let Q = . Then χ ( E , Q ) = E and so A O − ( k ) ( E ) ∼ = A O − ( k ) ( E ) by Theorem 3.6. (cid:3) Lemma 7.2.
Let M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , m l + m l = m , l = 0 and l = 0 . Then (1) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E + E + E ) if m = 0 and m = 0 ; (2) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ) if m = 0 and m = 0 ; (3) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E ) if m = 0 and m = 0 .Proof. (1)By the assumption, m = m l and M = m l m m m l l m l m .Let C = m l m l
00 0 q m l m . Then χ ( M, C ) = C − M ( c ij )= m l m l
00 0 p m l m M m l m l
00 0 m l m = = E + E + E + E + E + E ) . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 43 So A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E + E + E by Theorem 3.6.(2)By the assumption, m = m l = 0 and M = m l m . Let C ′ = l l
00 0 q l m . Then χ ( M, C ′ ) = l l
00 0 p l m m l m l l
00 0 l m = = E + E . Thus A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ) by Theorem 3.6.(3)By assumptions, m = m l = 0 and m = 0 M = m l m m l l m
00 0 0 .Let C ′′ = m l m l
00 0 q m l . Then χ ( M, C ′′ )= m l m l
00 0 p m l m l m m l l m
00 0 0 ! m l m l
00 0 m l = = E + E + E + E . Therefore, A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E ) by Theorem 3.6. (cid:3) By a similar proof, we can show the following proposition.
Lemma 7.3.
Let M = m m m l m l m l m l m l m l m , ( m , m , m ) = 0 , m l + m l = m , l = 0 and l = 0 . Then (1) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E + E + E ) if m = 0 and m = 0 ; (2) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E ) if m = 0 and m = 0 ; (3) A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E ) if m = 0 and m = 0 . Proof. (1)By the assumption, m = m l and M = m l m m m l l m l m .Let C = m l q m m l
00 0 m l . Then χ ( M, C ) = C − M ( c ij )= m l p m m l
00 0 m l M m l m m l
00 0 m l = = E + E + E + E + E + E ) . So A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E + E + E + E + E ) by Theorem 3.6.(2)By the assumption, m = m l and M = m l m m l m l . Let C ′ = m l m l . Then χ ( M, C ′ ) = m l m l m l m m l m l m l l m = = E + E + E + E . Thus A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ) by Theorem 3.6.(3)By assumptions, m = m l = 0, m = 0 and M = m
00 0 00 l m .Let C ′′ = m l m . Then χ ( M, C ′′ )= m m l m
00 0 00 l m ! m m l ! = = E + E . Therefore, A O − ( k ) ( M ) ∼ = A O − ( k ) ( E + E ) by Theorem 3.6. (cid:3) Lemma 7.4.
We have A O − ( k ) ( M ) ∼ = A O − ( k ) ( N ) if M and N belong to thefollowing cases: (1) M = E + E , N = E + E ; G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 45 (2) M = E + E + E + E , N = E + E + E + E ; (3) M = E + E + E + E + E + E , N = E + E + E + E + E + E .Proof. (1)Let C = . Then χ ( E + E , C )= = = E + E ,χ ( E + E + E + E , C )= = = E + E + E + E and χ ( E + E + E + E + E + E , C )= = = E + E + E + E + E + E . By Theorem 3.6, we finish the proof. (cid:3)
Remark 7.5.
By Lemma 7.1, Lemma 7.2, Lemma 7.3 and Lemma 7.4, it remainsto study the homological properties of A O − ( k ) ( M ) when M belongs to one of thefollowing six specific matrixes: M = , M = , M = ,M = , M = , M = . Proposition 7.6.
For any i ∈ { , , , , , } , the connected cochain DG algebra A O − ( k ) ( M i ) is a Koszul Calabi-Yau DG algebra.Proof. For briefness, we denote A i = A O − ( k ) ( M i ) , i = 1 , , · · · ,
6. We will proveone by one that each A i is a Koszul Calabi-Yau DG algebra.(1)We have ∂ A ( x ) = x + x , ∂ A ( x ) = ∂ A ( x ) = 0. According to theconstructing procedure of the minimal semi-free resolution in [MW1, Proposition2.4], we get a minimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -module such that F = A ⊕ A e ⊕ A e ⊕ A e ⊕ A e ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x x x x x x x x x x x x x x x x e e e e e e e . Let D = x x x x x x x x x x x x x x x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e , e , e , e } concentratedin degree 0, the elements in Hom A ( F , F ) is one to one correspondence with thematrixes in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by amatrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e e e e . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 47 And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a = a = a = a a = a = a = a = a = a a = a = a = a = a = a a = a = a = a a = a = a = a a = a , a = a a = a = a = 0by direct computations. Hence the algebra E ∼ = a b a c a d c b a e b c a f d e b c a g e d c b a h f g e d b c a | a, b, c, d, e, f, g, h ∈ k = E . Set ξ = X i =1 E ii ,ξ = E + E + E + E + E + E ,ξ = E + E + E + E + E + E ,ξ = E + E + E + E ,ξ = E + E + E + E ,ξ = E + E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = ξ , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = ξ ,ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = 0; ξ = ξ , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = ξ ,ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0; ξ ξ = ξ ξ = ξ , ξ = 0 , ξ ξ i = ξ i ξ = 0 , ∀ i ∈ { , , } ; ξ j ξ i = ξ i ξ j = 0 , ∀ i, j ∈ { , , , } . It is easy to check that the map ε : E → Hom k ( E , k ) defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x, y ] / ( x − y , x ) of k -algebras defined by θ : ξ → ξ → ¯ xξ → ¯ yξ → ¯ x ¯ yξ → ¯ x ξ → ¯ x ¯ yξ → ¯ x ξ → ¯ x ¯ y. is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra.(2)We have ∂ A ( x ) = x , ∂ A ( x ) = ∂ A ( x ) = 0. According to the constructingprocedure of the minimal semi-free resolution in [MW1, Proposition 2.4], we get aminimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -modulesuch that F = A ⊕ A e ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x x x x x x e e e e . Let D = x x x x x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 49 So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a a = a = a a = a = a = 0 a = a by direct computations. Hence the algebra E ∼ = a b a c a d b a e d b a | a, b, c, d, e ∈ k = E . Set ξ = X i =1 E ii ,ξ = E + E + E ,ξ = E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = ξ , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ = 0 . It is easy to check that the map ε : E → Hom k ( E , k ) defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x, y ] / ( x , xy, y ) of k -algebras defined by θ : ξ → ξ → ¯ xξ → ¯ yξ → ¯ x ξ → ¯ x . is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra.(3)We have ∂ A ( x ) = P i =1 x i = ∂ A ( x ) , ∂ A ( x ) = 0. According to the con-structing procedure of the minimal semi-free resolution in [MW1, Proposition 2.4],we get a minimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -module such that F = A ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x − x x x x − x x e e e . Let D = x − x x x x − x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixes G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 51 in M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) = A f · e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = a a = 0 a = a by direct computations. Hence the algebra E ∼ = a b a c a d b c a | a, b, c, d ∈ k = E . Set ξ = X i =1 E ii ,ξ = E + E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = ξ , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = 0 ,ξ = ξ , ξ ξ = ξ ξ = 0 , ξ = 0 . It is easy to check that the map ε : E → Hom k ( E , k )defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x, y ] / ( x , xy, y , x − y ) of k -algebras definedby θ : ξ → ξ → ¯ xξ → ¯ yξ → ¯ x . is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra.(4)We have ∂ A ( x ) = x = ∂ A ( x ) , ∂ A ( x ) = 0. According to the constructingprocedure of the minimal semi-free resolution in [MW1, Proposition 2.4], we get aminimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -modulesuch that F = A ⊕ A e ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x x − x x x x x e e e e . Let D = x x − x x x x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) f ( e ) = A f · e e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a = a a = a = a a = a = a = 0 a = a
52G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 53 by direct computations. Hence the algebra E ∼ = a b a c a d b a e d b | a, b, c, d, e ∈ k = E . Set ξ = X i =1 E ii ,ξ = E + E + E ,ξ = E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = ξ , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ = 0 . It is easy to check that the map ε : E → Hom k ( E , k )defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x, y ] / ( x , xy, y ) of k -algebras defined by θ : ξ → ξ → ¯ xξ → ¯ yξ → ¯ x ξ → ¯ x . is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra.(5)We have ∂ A ( x ) = x + x = ∂ A ( x ) , ∂ A ( x ) = 0. According to theconstructing procedure of the minimal semi-free resolution in [MW1, Proposition2.4], we get a minimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -module such that F = A ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x x − x x − x x e e e . Let D = x x − x x − x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) = A f · e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = a a = 0 a = a by direct computations. Hence the algebra E ∼ = a b a c a d c b a | a, b, c, d ∈ k = E . G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 55 Set ξ = X i =1 E ii ,ξ = E + E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = 0 , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ = 0 . It is easy to check that the map ε : E → Hom k ( E , k )defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x ] / ( x ) of k -algebras defined by θ : ξ → ξ → ¯ xξ → ¯ x ξ → ¯ x . is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra.(6)We have ∂ A ( x ) = x + x = ∂ A ( x ) , ∂ A ( x ) = 0. According to theconstructing procedure of the minimal semi-free resolution in [MW1, Proposition2.4], we get a minimal semi-free resolution f : F ≃ → k , where F is a semi-free DG A -module such that F = A ⊕ A e ⊕ A e ⊕ A e , with a differential ∂ F defined by ∂ F (1) ∂ F ( e ) ∂ F ( e ) ∂ F ( e ) = x x − x x − x x e e e . Let D = x x − x x − x x . By the minimality of F , we have H (Hom A ( F , k )) = Hom A ( F , k )= k ∗ ⊕ [ M i =1 k ( e i ) ∗ ] . So the Ext-algebra E = H (Hom A ( F , F )) is concentrated in degree 0. On theother hand, Hom A ( F , F ) ∼ = { k ∗ ⊕ ⊕ [ M i =1 k ( e i ) ∗ ] } ⊗ k F is concentrated in degree ≥
0. This implies that E = Z (Hom A ( F , F )). Since F is a free graded A -module with a basis { , e , e , e } concentrated in degree0, the elements in Hom A ( F , F ) is one to one correspondence with the matrixesin M ( k ). Indeed, any f ∈ Hom A ( F , F ) is uniquely determined by a matrix A f = ( a ij ) × ∈ M ( k ) with f (1) f ( e ) f ( e ) f ( e ) = A f · e e e . And f ∈ Z [Hom A ( F , F )] if and only if ∂ F ◦ f = f ◦ ∂ F , if and only if A f D = D A f , which is also equivalent to a ij = 0 , ∀ i < ja = a = a = a a = a a = 0 a = a by direct computations. Hence the algebra E ∼ = a b a c a d c b a | a, b, c, d ∈ k = E . Set ξ = X i =1 E ii ,ξ = E + E ,ξ = E + E ,ξ = E . One sees that { ξ i | i = 1 , , · · · , } is a k -linear basis of E and ξ i ξ = ξ ξ i , ∀ i = 1 , , · · · , ξ = 0 , ξ ξ = ξ ξ = ξ , ξ ξ = ξ ξ = 0 ,ξ = 0 , ξ ξ = ξ ξ = 0 , ξ = 0 . It is easy to check that the map ε : E → Hom k ( E , k ) G ALGEBRA STRUCTURES ON THE QUANTUM AFFINE n -SPACE O − ( k n ) 57 defined by ε : ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ ξ → ξ ∗ . is an isomorphism of left E -modules. Thus E is a commutative Frobenius algebra.Actually, the morphism θ : E → k [ x ] / ( x ) of k -algebras defined by θ : ξ → ξ → ¯ xξ → ¯ x ξ → ¯ x . is an isomorphism. Hence E is a symmetric Frobenius algebra concentrated indegree 0. This implies that Tor A ( k A , A k ) ∼ = E ∗ is a symmetric coalgebra. By[HM, Theorem 4.2], A is a Koszul Calabi-Yau DG algebra. (cid:3) Now, we can reach the following conclusion: A O − ( k ) ( M ) is a Koszul Calabi-Yau DG algebra, for any M ∈ M ( k ). We finish this section with the followingconjecture. Conjecture 7.7.
For any M ∈ M n ( k ) , A O − ( k n ) ( M ) is a Koszul Calabi-Yau con-nected cochain DG algebra. Acknowledgments
X.-F. Mao was supported by the National Natural Science Foundation of China(No. 11871326). X.-T. Wang was supported by Simons Foundation Program:Mathematics and Physical Sciences-Collaboration Grants for Mathematician (AwardNo.688403).
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E-mail address : [email protected] Department of Mathematics, Howard University, Washington DC, 20059, USA
E-mail address : [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, China
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