Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras
aa r X i v : . [ m a t h . R T ] J a n DRINFELD DOUBLES VIA DERIVED HALL ALGEBRASAND BRIDGELAND HALL ALGEBRAS
FAN XU AND HAICHENG ZHANG
Abstract.
Let A be a finitary hereditary abelian category. We give a Hall algebra pre-sentation of Kashaev’s theorem on the relation between Drinfeld double and Heisenbergdouble. As applications, we obtain realizations of the Drinfeld double Hall algebra of A via its derived Hall algebra and Bridgeland Hall algebra of m -cyclic complexes. Introduction
The Hall algebra H ( A ) of a finite dimensional algebra A over a finite field was introducedby Ringel [9] in 1990. Ringel [8, 9] proved that if A is a hereditary algebra of finite type, thetwisted Hall algebra H v ( A ), called the Ringel–Hall algebra, is isomorphic to the positivepart of the corresponding quantized enveloping algebra. In 1995, Green [3] generalizedRingel’s work to any hereditary algebra A and showed that the composition subalgebraof H v ( A ) generated by simple A -modules gives a realization of the positive part of thequantized enveloping algebra associated with A . Moreover, he introduced a bialgebrastructure on H v ( A ) via a significant formula called Green’s formula. In 1997, Xiao [13]provided the antipode on H v ( A ), and proved that the extended Ringel–Hall algebra is aHopf algebra. Furthermore, he considered the Drinfeld double of the extended Ringel–Hallalgebras, and obtained a realization of the full quantized enveloping algebra.In order to give an intrinsic realization of the entire quantized enveloping algebra viaHall algebra approach, one tried to define the Hall algebra of a triangulated category (forexample, [5], [12], [14]). Kapranov [5] considered the Heisenberg double of the extendedRingel–Hall algebras, and defined an associative algebra, called the lattice algebra, for thebounded derived category of a hereditary algebra A . By using the fibre products of modelcategories, To¨en [12] defined an associative algebra, called the derived Hall algebra, for aDG-enhanced triangulated category. Later on, Xiao and Xu [14] generalized the definitionof the derived Hall algebra to any triangulated category with some homological finitenessconditions. In particular, the derived Hall algebra DH ( A ) of the bounded derived category Mathematics Subject Classification.
Key words and phrases.
Heisenberg double; Drinfeld double; derived Hall algebra; Bridgeland Hallalgebra. of a hereditary algebra A can be defined, and it is proved in [12] that there exist certainHeisenberg double structures in DH ( A ).Recently, for each hereditary algebra A , Bridgeland [1] defined an associative algebra,called the Bridgeland Hall algebra, which is the Ringel–Hall algebra of 2-cyclic complexesover projective A -modules with some localization and reduction. He proved that thequantized enveloping algebra associated to A can be embedded into its Bridgeland Hallalgebra. This provides a beautiful realization of the full quantized enveloping algebraby Hall algebras. Afterwards, Yanagida [15] (see also [16]) showed that the BridgelandHall algebra of 2-cyclic complexes of a hereditary algebra is isomorphic to the Drinfelddouble of its extended Ringel–Hall algebras. Inspired by the work of Bridgeland, Chenand Deng [2] introduced the Bridgeland Hall algebra DH m ( A ) of m -cyclic complexes of ahereditary algebra A for each nonnegative integer m = 1. If m = 0 or m >
2, the algebrastructure of DH m ( A ) has a characterization in [17], in particular, it is proved that thereexist Heisenberg double structures in DH m ( A ).Kashaev [6] established a relation between the Drinfeld double and Heisenberg doubleof a Hopf algebra. Explicitly, he showed that the Drinfeld double is representable as asubalgebra in the tensor square of the Heisenberg double.In this paper, let A be a finitary hereditary abelian category. We first give a Hallalgebra presentation of Kashaev’s Theorem on the relation between Drinfeld double andHeisenberg double. Then we apply this presentation to the Bridgeland Hall algebra andderived Hall algebra of A .Throughout the paper, all tensor products are assumed to be over the complex numberfield C . Let k be a fixed finite field with q elements and set v = √ q ∈ C . Let A bea finitary hereditary abelian k -category. We denote by Iso ( A ) and K ( A ) the set ofisoclasses of objects in A and the Grothendieck group of A , respectively. For each object M in A , the class of M in K ( A ) is denoted by ˆ M , and the automorphism group of M is denoted by Aut ( M ). For a finite set S , we denote by | S | its cardinality, and we alsowrite a M for | Aut ( M ) | . For a positive integer m , we denote the quotient ring Z /m Z by Z m = { , , . . . , m − } . By convention, Z = Z .2. Preliminaries
In this section, we recall the definitions of Ringel–Hall algebra, Heisenberg double, andDrinfeld double (cf. [10, 13, 5]).2.1.
Hall algebras.
For objects
M, N , . . . , N t ∈ A , let g MN ··· N t be the number of thefiltrations M = M ⊇ M ⊇ · · · ⊇ M t − ⊇ M t = 0 RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 3 such that M i − /M i ∼ = N i for all 1 ≤ i ≤ t . In particular, if t = 2, g MN N is the numberof subobjects X of M such that X ∼ = N and M/X ∼ = N . One defines the Hall algebra H ( A ) to be the vector space over C with basis [ M ] ∈ Iso ( A ) and with the multiplicationdefined by [ M ] ⋄ [ N ] = X [ L ] g LMN [ L ] . By definition, it is easy to see that for each 1 < i < t , g MN ··· N t = X [ X ] g MN ··· N i − X g XN i ··· N t = X [ Y ] g YN ··· N i g MY N i +1 ··· N t . For any
M, N ∈ A , define h M, N i := dim k Hom A ( M, N ) − dim k Ext A ( M, N ) . It induces a bilinear form h· , ·i : K ( A ) × K ( A ) −→ Z , known as the Euler form . We also consider the symmetric Euler form ( · , · ) : K ( A ) × K ( A ) −→ Z , defined by ( α, β ) = h α, β i + h β, α i for all α, β ∈ K ( A ).The twisted Hall algebra H v ( A ), called the Ringel–Hall algebra , is the same vectorspace as H ( A ) but with the twisted multiplication defined by[ M ][ N ] = v h M, N i · [ M ] ⋄ [ N ] . We can form the extended Ringel–Hall algebra H ev ( A ) by adjoining symbols K α for all α ∈ K ( A ) and imposing relations K α K β = K α + β , K α [ M ] = v ( α, ˆ M ) · [ M ] K α . (2.1)Green [3] introduced a (topological) bialgebra structure on H ev ( A ) by defining the co-multiplication as follows:∆([ L ] K α ) = X [ M ] , [ N ] v h M,N i a M a N a L g LMN [ M ] K ˆ N + α ⊗ [ N ] K α , for any L ∈ A , α ∈ K ( A ) . That ∆ is a homomorphism of algebras amounts to the following crucial formula.
Theorem 2.1. (Green’s formula)
Given
M, N, M ′ , N ′ ∈ A , we have the followingformula a M a N a M ′ a N ′ X [ L ] g LMN g LM ′ N ′ a L = X [ A ] , [ A ′ ] , [ B ] , [ B ′ ] | Ext A ( A, B ′ ) || Hom A ( A, B ′ ) | g MAA ′ g NBB ′ g M ′ AB g N ′ A ′ B ′ a A a A ′ a B a B ′ . (2.2) FAN XU AND HAICHENG ZHANG
Heisenberg doubles.
Let A and B be Hopf algebras, and let ϕ : A × B → C be aHopf pairing. The Heisenberg double HD ( A, B, ϕ ) is defined to be the free product A ∗ B imposed by the following relations (with a ∈ A and b ∈ B ): b ∗ a = X ϕ ( a , b ) a ∗ b , where and elsewhere we use Sweedler’s notation ∆( a ) = P a ⊗ a .There exists a so-called Green’s pairing ϕ : H ev ( A ) × H ev ( A ) → C defined by ϕ ([ M ] K α , [ N ] K β ) = δ [ M ] , [ N ] v ( α,β ) a M , which is a Hopf pairing.Now let us apply the construction of Heisenberg double to Ringel–Hall algebras. Let H + ( A ) (resp. H − ( A )) be the Ringel–Hall algebra H ev ( A ) with each [ M ] K α rewritten as µ + M K + α (resp. µ − M K − α ). Thus, considering A = H − ( A ), B = H + ( A ) and ϕ = ϕ , weobtain the Heisenberg double Hall algebra , denoted by HD ( A ). By direct calculations,we give the characterization of HD ( A ) via generators and generating relations (with α, β ∈ K ( A ) and [ M ] , [ N ] ∈ Iso ( A )) as follows (cf. [5]): µ + M µ + N = X [ L ] v h M,N i g LMN µ + L , µ − M µ − N = X [ L ] v h M,N i g LMN µ − L ; (2.3) K + α µ + M = v ( α, ˆ M ) µ + M K + α , K − α µ − M = v ( α, ˆ M ) µ − M K − α ; (2.4) K ± α K ± β = K ± α + β , K + α K − β = v ( α,β ) K − β K + α ; (2.5) K + α µ − M = µ − M K + α , K − α µ + M = v − ( α, ˆ M ) µ + M K − α ; (2.6) µ + M µ − N = X [ X ] , [ Y ] v h ˆ N − ˆ Y , ˆ X − ˆ Y i γ XYMN K − ˆ N − ˆ Y µ − Y µ + X ; (2.7)where and elsewhere γ XYMN = a X a Y a M a N P [ L ] a L g MLX g NY L .Similarly, one defines the dual Heisenberg double Hall algebra ˇ HD ( A ), which is givenby the generators and generating relations (with α, β ∈ K ( A ) and [ M ] , [ N ] ∈ Iso ( A )) asfollows: ν + M ν + N = X [ L ] v h M,N i g LMN ν + L , ν − M ν − N = X [ L ] v h M,N i g LMN ν − L ; (2.8) K + α ν + M = v ( α, ˆ M ) ν + M K + α , K − α ν − M = v ( α, ˆ M ) ν − M K − α ; (2.9) K ± α K ± β = K ± α + β , K + α K − β = v − ( α,β ) K − β K + α ; (2.10) K − α ν + M = ν + M K − α , K + α ν − M = v − ( α, ˆ M ) ν − M K + α ; (2.11) ν − N ν + M = X [ X ] , [ Y ] v h ˆ N − ˆ Y , ˆ Y − ˆ X i γ Y XNM K +ˆ N − ˆ Y ν + X ν − Y . (2.12) RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 5
Drinfeld doubles.
Let A and B be Hopf algebras, and let ϕ : A × B → C be aHopf pairing. The Drinfeld double D ( A, B, ϕ ) is defined to be the free product A ∗ B imposed by the following relations (with a ∈ A and b ∈ B ): X ϕ ( a , b ) b ∗ a = X ϕ ( a , b ) a ∗ b . (2.13)Applying the construction of Drinfeld double to the Ringel–Hall algebras H − ( A ) and H + ( A ), we obtain the Drinfeld double Hall algebra , denoted by D ( A ), which is definedby the generators and generating relations (with α, β ∈ K ( A ), [ M ] , [ N ] ∈ Iso ( A )) asfollows: ω + M ω + N = X [ L ] v h M,N i g LMN ω + L , ω − M ω − N = X [ L ] v h M,N i g LMN ω − L ; (2.14) K + α ω + M = v ( α, ˆ M ) ω + M K + α , K − α ω − M = v ( α, ˆ M ) ω − M K − α ; (2.15) K ± α K ± β = K ± α + β , K + α K − β = K − β K + α ; (2.16) K + α ω − M = v − ( α, ˆ M ) ω − M K + α , K − α ω + M = v − ( α, ˆ M ) ω + M K − α ; (2.17) X [ X ] , [ Y ] v h ˆ M − ˆ X, ˆ M − ˆ N i γ XYMN K − ˆ M − ˆ X ω − Y ω + X = X [ X ] , [ Y ] v h ˆ M − ˆ X, ˆ N − ˆ M i γ Y XNM K +ˆ M − ˆ X ω + X ω − Y . (2.18)3. Kashaev’s theorem: Hall algebra presentation
In this section, we prove Kashaev’s theorem [6, Theorem 2] in the form of Ringel–Hallalgebras. There are some similar constructions in [4], but they are not so natural.
Theorem 3.1.
There exists an embedding of algebras I : D ( A ) ֒ → HD ( A ) ⊗ ˇ HD ( A ) defined on generators by K + α K + α ⊗ K + α , ω + M X [ M ] , [ M ] v h M ,M i a M a M a M g MM M µ + M K +ˆ M ⊗ ν + M , and K − α K − α ⊗ K − α , ω − M X [ M ] , [ M ] v h M ,M i a M a M a M g MM M µ − M ⊗ ν − M K − ˆ M . Proof.
In order to prove that I is a homomorphism of algebras, it suffices to show thatthe relations from (2 .
14) to (2 .
18) are preserved under I . We only prove the relations(2 .
14) and (2 . . X [ L ] v h M,N i g LMN I ( ω + L ) = X [ L ] , [ L ] , [ L ] v h M,N i + h L ,L i a L a L a L g LMN g LL L µ + L K +ˆ L ⊗ ν + L . FAN XU AND HAICHENG ZHANG I ( ω + M ) I ( ω + N ) = X [ M ] , [ M ] , [ N ] , [ N ] v h M ,M i + h N ,N i a M a M a N a N a M a N g MM M g NN N µ + M K +ˆ M µ + N K +ˆ N ⊗ ν + M ν + N = X [ M ] , [ M ] , [ N ] , [ N ] v x a M a M a N a N a M a N g MM M g NN N µ + M µ + N K +ˆ M + ˆ N ⊗ ν + M ν + N ( x = h M , M i + h N , N i + ( M , N ))= X [ M ] , [ M ] , [ N ] , [ N ] , [ L ] , [ L ] v x a M a M a N a N a M a N g MM M g NN N g L M N g L M N µ + L K +ˆ L ⊗ ν + L ( ∗ )( x = h M , M i + h N , N i + ( M , N ) + h M , N i + h M , N i ) . For each fixed L , L , noting that in ( ∗ ) ˆ M = ˆ M + ˆ M , ˆ N = ˆ N + ˆ N , ˆ L i = ˆ M i + ˆ N i for i = 1 ,
2, we obtain that x = h M, N i + h L , L i − h M , N i . Thus, by Green’s formula,we conclude that X [ M i ] , [ N i ] ,i =1 , v x a M a M a N a N a M a N g MM M g NN N g L M N g L M N = X [ L ] v h M,N i + h L ,L i a L a L a L g LMN g LL L and thus I ( ω + M ) I ( ω + N ) = X [ L ] v h M,N i g LMN I ( ω + L ) . Similarly, we can prove that the second relation in (2.14) is also preserved under I .Now, we come to prove that the relation in (2.18) is preserved under I . First of all,substituting γ XYMN = a X a Y a M a N P [ L ] a L g MLX g NY L into (2.18), we rewrite (2.18) as follows: X [ X ] , [ Y ] , [ L ] v h ˆ L, ˆ M − ˆ N i a X a Y a L g MLX g NY L K − ˆ L ω − Y ω + X = X [ X ] , [ Y ] , [ L ] v h ˆ L, ˆ N − ˆ M i a X a Y a L g MXL g NLY K +ˆ L ω + X ω − Y . On the one hand,LHS := X [ X ] , [ Y ] , [ L ] v h ˆ L, ˆ M − ˆ N i a X a Y a L g MLX g NY L I ( K − ˆ L ) I ( ω − Y ) I ( ω + X )= X [ X ] , [ Y ] , [ L ] , [ Y ] , [ Y ] , [ X ] , [ X ] v y a X a X a Y a Y a L g MLX g XX X g YY Y g NY L K − ˆ L µ − Y µ + X K +ˆ X ⊗ K − ˆ L ν − Y K − ˆ Y ν + X ( y = h ˆ L, ˆ M − ˆ N i + h X , X i + h Y , Y i )= X [ L ] , [ X ] , [ X ] , [ Y ] , [ Y ] v y a X a X a Y a Y a L g MLX X g NY Y L K − ˆ L µ − Y µ + X K +ˆ X ⊗ K − ˆ Y +ˆ L ν − Y ν + X ( y = y − ( Y , Y ) = h ˆ L, ˆ M − ˆ N i + h X , X i − h Y , Y i ) . RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 7
By (2.12), ν − Y ν + X = X [ A ] , [ B ] v h ˆ Y − ˆ B, ˆ B − ˆ A i γ BAY X K +ˆ Y − ˆ B ν + A ν − B = X [ A ] , [ B ] , [ C ] v h ˆ C, ˆ B − ˆ A i a A a B a C a X a Y g Y CB g X AC K +ˆ C ν + A ν − B . Thus , LHS = X [ L ] , [ X ] , [ Y ] , [ A ] , [ B ] , [ C ] v y a L a X a A a C a B a Y g MLX AC g NCBY L K − ˆ L µ − Y µ + X K +ˆ A + ˆ C ⊗ K − ˆ Y +ˆ L K +ˆ C ν + A ν − B ( y = y + h ˆ C, ˆ B − ˆ A i = h ˆ L, ˆ M − ˆ N i + h ˆ X , ˆ A + ˆ C i − h ˆ Y , ˆ B + ˆ C i + h ˆ C, ˆ B − ˆ A i )= X [ L ] , [ X ] , [ Y ] , [ A ] , [ B ] , [ C ] v y a L a X a A a C a B a Y g MLX AC g NCBY L K − ˆ L µ − Y µ + X K +ˆ A + ˆ C ⊗ K +ˆ C ν + A ν − B K − ˆ Y +ˆ L ( y = h ˆ L, ˆ M − ˆ N i + h ˆ X , ˆ A + ˆ C i − h ˆ Y , ˆ B + ˆ C i + h ˆ C, ˆ B − ˆ A i + ( ˆ L + ˆ Y , ˆ B + ˆ C )) . On the other hand,RHS := X [ X ] , [ Y ] , [ L ] v h ˆ L, ˆ N − ˆ M i a X a Y a L g MXL g NLY I ( K +ˆ L ) I ( ω + X ) I ( ω − Y )= X [ X ] , [ Y ] , [ L ] , [ X ] , [ X ] , [ Y ] , [ Y ] v z a X a X a Y a Y a L g XX X g MXL g NLY g YY Y K +ˆ L µ + X K +ˆ X µ − Y ⊗ K +ˆ L ν + X ν − Y K − ˆ Y ( z = h ˆ L, ˆ N − ˆ M i + h X , X i + h Y , Y i )= X [ L ] , [ X ] , [ X ] , [ Y ] , [ Y ] v z a X a X a Y a Y a L g MX X L g NLY Y K +ˆ X +ˆ L µ + X µ − Y ⊗ K +ˆ L ν + X ν − Y K − ˆ Y ( z = z − ( X , X ) = h ˆ L, ˆ N − ˆ M i + h Y , Y i − h X , X i ) . By (2.7), µ + X µ − Y = X [ A ] , [ B ] v h ˆ Y − ˆ B, ˆ A − ˆ B i γ ABX Y K − ˆ Y − ˆ B µ − B µ + A = X [ A ] , [ B ] , [ C ] v h ˆ C, ˆ A − ˆ B i a A a B a C a X a Y g X CA g Y BC K − ˆ C µ − B µ + A . Thus , RHS = X [ L ] , [ X ] , [ Y ] , [ A ] , [ B ] , [ C ] v z a C a A a X a L a Y a B g MCAX L g NLY BC K +ˆ X +ˆ L K − ˆ C µ − B µ + A ⊗ K +ˆ L ν + X ν − Y K − ˆ B + ˆ C ( z = z + h ˆ C, ˆ A − ˆ B i = h ˆ L, ˆ N − ˆ M i + h ˆ Y , ˆ B + ˆ C i − h ˆ X , ˆ A + ˆ C i + h ˆ C, ˆ A − ˆ B i )= X [ L ] , [ X ] , [ Y ] , [ A ] , [ B ] , [ C ] v z a C a A a X a L a Y a B g MCAX L g NLY BC K − ˆ C µ − B µ + A K +ˆ X +ˆ L ⊗ K +ˆ L ν + X ν − Y K − ˆ B + ˆ C FAN XU AND HAICHENG ZHANG ( z = h ˆ L, ˆ N − ˆ M i + h ˆ Y , ˆ B + ˆ C i − h ˆ X , ˆ A + ˆ C i + h ˆ C, ˆ A − ˆ B i + ( ˆ L + ˆ X , ˆ A + ˆ C )) . Identifying
L, X , A, C, B, Y in LHS with C, A, X , L, Y , B in RHS, respectively, we ob-tain that y = h ˆ C, ˆ M − ˆ N i + h ˆ A, ˆ X + ˆ L i − h ˆ B, ˆ Y + ˆ L i + h ˆ L, ˆ Y − ˆ X i + ( ˆ B + ˆ C, ˆ Y + ˆ L ) . Noting that in RHS ˆ M − ˆ N = ˆ X − ˆ Y = ( ˆ X − ˆ Y ) + ( ˆ X − ˆ Y ) = ( ˆ A − ˆ B ) + ( ˆ X − ˆ Y ),we have that y = h ˆ C, ˆ A − ˆ B i + h ˆ C, ˆ X i − h ˆ C, ˆ Y i + h ˆ A, ˆ X i + h ˆ A, ˆ L i − h ˆ B, ˆ Y + ˆ L i + h ˆ L, ˆ Y − ˆ X i + ( ˆ C, ˆ L ) + h ˆ C, ˆ Y i + h ˆ Y , ˆ C i + h ˆ B, ˆ Y + ˆ L i + h ˆ Y , ˆ B i + h ˆ L, ˆ B i = h ˆ C, ˆ A − ˆ B i + h ˆ A + ˆ C, ˆ X i + h ˆ A, ˆ L i + h ˆ L, ˆ Y − ˆ X i + ( ˆ C, ˆ L ) + h ˆ Y , ˆ B + ˆ C i + h ˆ L, ˆ B i and z = h ˆ L, ˆ B i − h ˆ L, ˆ A i + h ˆ L, ˆ Y − ˆ X i + h ˆ Y , ˆ B + ˆ C i − h ˆ X , ˆ A + ˆ C i + h ˆ C, ˆ A − ˆ B i + ( ˆ C, ˆ L ) + h ˆ L, ˆ A i + h ˆ A, ˆ L i + h ˆ X , ˆ A + ˆ C i + h ˆ A + ˆ C, ˆ X i = h ˆ L, ˆ B i + h ˆ L, ˆ Y − ˆ X i + h ˆ Y , ˆ B + ˆ C i + h ˆ C, ˆ A − ˆ B i + ( ˆ C, ˆ L ) + h ˆ A, ˆ L i + h ˆ A + ˆ C, ˆ X i = y . Hence, LHS = RHS, and we have proved that I is a homomorphism of algebras.Since D ( A ) ∼ = H + ( A ) ⊗ H − ( A ) as a vector space, and the restriction of I to the positive(negative) part is injective, we conclude that I is injective. Therefore, we complete theproof. (cid:3) Applications
In this section, we apply Theorem 3.1 to Bridgeland Hall algebras of m -cyclic complexesand derived Hall algebras.4.1. Bridgeland Hall algebras.
Assume that A has enough projectives, the BridgelandHall algebra of 2-cyclic complexes of A was introduced in [1]. Inspired by the work ofBridgeland, for each nonnegative integer m = 1, Chen and Deng [2] introduced theBridgeland Hall algebra DH m ( A ) of m -cyclic complexes. For m = 0 or m >
2, we recallthe algebra structure of DH m ( A ) by [17] as follows: Proposition 4.1. ([17])
Let m = 0 or m > . Then DH m ( A ) is an associative and unital C -algebra generated by the elements in { e M,i | [ M ] ∈ Iso ( A ) , i ∈ Z m } and { K α,i | α ∈ RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 9 K ( A ) , i ∈ Z m } , and the following relations: K α,i K β,i = K α + β,i , K α,i K β,j = v ( α, β ) K β,j K α,i i = j + 1 ,v − ( α, β ) K β,j K α,i i = m − j,K β,j K α,i otherwise ; (4.1) K α,i e M,j = v ( α, ˆ M ) e M,j K α,i i = j,v − ( α, ˆ M ) e M,j K α,i i = m − j,e M,j K α,i otherwise ; (4.2) e M,i e N,i = X [ L ] v h M, N i g LMN e L,i ; (4.3) e M,i +1 e N,i = X [ X ] , [ Y ] v h ˆ M − ˆ X, ˆ X − ˆ Y i γ XYMN K ˆ M − ˆ X,i e Y,i e X,i +1 ; (4.4) e M,i e N,j = e N,j e M,i , i − j = 0 , m − . (4.5) Corollary 4.2.
Let m = 0 or m > . Then for each i ∈ Z m , (1) there exists an embedding of algebras κ i : HD ( A ) ֒ → DH m ( A ) defined on genera-tors by K + α K α,i +1 , K − α K α,i , µ + M e M,i +1 , µ − M e M,i ;(2) there exists an embedding of algebras ˇ κ i : ˇ HD ( A ) ֒ → DH m ( A ) defined on genera-tors by K + α K α,i , K − α K α,i +1 , ν + M e M,i , ν − M e M,i +1 . Proof.
By Proposition 4.1 the defining relations of HD ( A ) and ˇ HD ( A ) are preservedunder κ i and ˇ κ i , respectively, we obtain that κ i and ˇ κ i are homomorphisms of algebras.According to [17, Proposition 2.7], we conclude that they are injective. (cid:3) As a first application of Theorem 3.1, we have the following
Theorem 4.3.
Let m = 0 or m > . Then for each i ∈ Z m , there exists an embedding ofalgebras ψ i : D ( A ) ֒ → DH m ( A ) ⊗ DH m ( A ) defined on generators by K + α K α,i +1 ⊗ K α,i , ω + M X [ M ] , [ M ] v h M ,M i a M a M a M g MM M e M ,i +1 K ˆ M ,i +1 ⊗ e M ,i , and K − α K α,i ⊗ K α,i +1 , ω − M X [ M ] , [ M ] v h M ,M i a M a M a M g MM M e M ,i ⊗ e M ,i +1 K ˆ M ,i +1 . Proof.
For each i ∈ Z m , by the following commutative diagram D ( A ) (cid:31) (cid:127) I / / ψ i ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ HD ( A ) ⊗ ˇ HD ( A ) (cid:127) _ κ i ⊗ ˇ κ i (cid:15) (cid:15) DH m ( A ) ⊗ DH m ( A )we complete the proof. (cid:3) Remark 4.4.
As mentioned in Introduction, there is an isomorphism ρ : D ( A ) →DH ( A ), which is defined on generators by ω + M E M a M , ω − M F M a M , K + α K α , K − α K ∗ α , where the notations E M , F M , K α and K ∗ α are the same as those in [16]. Hence, Theorem4.3 establishes a relation between the Bridgeland Hall algebra of 2-cyclic complexes andthat of m -cyclic complexes.4.2. Derived Hall algebras.
The derived Hall algebra DH ( A ) of the bounded derivedcategory of A was introduced in [12] (see also [14]). Proposition 4.5. ([12]) DH ( A ) is an associative and unital C -algebra generated by theelements in { Z [ i ] M | [ M ] ∈ Iso ( A ) , i ∈ Z } and the following relations: Z [ i ] M Z [ i ] N = X [ L ] g LMN Z [ i ] L ; (4.6) Z [ i +1] M Z [ i ] N = X [ X ] , [ Y ] q −h Y,X i γ XYMN Z [ i ] Y Z [ i +1] X ; (4.7) Z [ i ] M Z [ j ] N = q ( − i − j h N,M i Z [ j ] N Z [ i ] M , i − j > . (4.8)According to [11], we twist the multiplication in DH ( A ) as follows: Z [ i ] M ∗ Z [ j ] N = v ( − i − j h M,N i Z [ i ] M Z [ j ] N . (4.9)The twisted derived Hall algebra DH tw ( A ) is the same vector space as DH ( A ), but withthe twisted multiplication. In order to relate the modified Ringel–Hall algebra, whichis isomorphic to the corresponding Bridgeland Hall algebra if A has enough projectives,to derived Hall algebra, Lin [7] introduced the completely extended twisted derived Hallalgebra DH cetw ( A ). Definition 4.6. ([7]) DH cetw ( A ) is the associative and unital C -algebra generated by theelements in { Z [ i ] M | [ M ] ∈ Iso ( A ) , i ∈ Z } and { K [ i ] α | α ∈ K ( A ) , i ∈ Z } , and the following RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 11 relations: K [ i ] α K [ i ] β = K [ i ] α + β , K [ i ] α Z [ i ] M = v ( α, ˆ M ) Z [ i ] M K [ i ] α i = − , ,Z [ i ] M K [ i ] α otherwise; (4.10) K [ i +1] α K [ i ] β = v ( α,β ) K [ i ] β K [ i +1] α , K [ i ] α K [ j ] β = K [ j ] β K [ i ] α , | i − j | >
1; (4.11) K [ i ] α Z [ i +1] M = v − ( α, ˆ M ) Z [ i +1] M K [ i ] α i = − , ,Z [ i +1] M K [ i ] α otherwise; (4.12) K [ i ] α Z [ i − M = v − ( α, ˆ M ) Z [ i − M K [ i ] α i = − , ,Z [ i − M K [ i ] α otherwise; (4.13)For any | i − j | > , K [ i ] α Z [ j ] M = v ( − j ( α, ˆ M ) Z [ j ] M K [ i ] α i = 0 and | j | > v ( − j +1 ( α, ˆ M ) Z [ j ] M K [ i ] α i = − | j + 1 | > Z [ j ] M K [ i ] α otherwise; (4.14) Z [ i ] M Z [ i ] N = X [ L ] v h M,N i g LMN Z [ i ] L ; (4.15) Z [ i +1] M Z [ i ] N = X [ X ] , [ Y ] v −h M,N i−h
Y,X i γ XYMN Z [ i ] Y Z [ i +1] X ; (4.16) Z [ i ] M Z [ j ] N = v ( − i − j ( M,N ) Z [ j ] N Z [ i ] M , i − j > . (4.17) Remark 4.7.
In Definition 4.6, we have employed the linear Euler form, not the mul-tiplicative Euler form used in [7]; K [ i ] α and Z [ i ] M here are equal to K [ − i ] α and Z [ − i ] M in [7],respectively.Now we reformulate [7, Theorem 5.3,Corollary 5.5] as follows: Theorem 4.8.
Assume that A has enough projectives. Then there exists an isomorphismof algebras φ : DH cetw ( A ) → DH ( A ) defined on generators (with n > ) by Z [0] M e M, , K [ n ] α K α,n ,Z [ n ] M v n h M,M i e M,n n Y i =1 K ( − i ˆ M,n − i , and Z [ − n ] M v − n h M,M i e M, − n n − Y i =0 K ( − i +1 ˆ M,i − n . Remark 4.9. (1) The inverse of φ in Theorem 4.8 is the homomorphism φ − : DH ( A ) →DH cetw ( A ) defined on generators (with n >
0) by e M, Z [0] M , K α,n K [ n ] α ,e M,n v − n h M,M i Z [ n ] M n − Y i =0 K ( − n − i − ˆ M,i , and e M, − n v n h M,M i Z [ − n ] M n Y i =1 K ( − n − i ˆ M, − i . (2) Theorem 4.8 establishes the relation between the Bridgeland Hall algebra of boundedcomplexes over projectives of A and the derived Hall algebra of the bounded derivedcategory D b ( A ). In other word, one can realize the derived Hall algebra via Bridgeland’sconstruction.As a second application of Theorem 3.1, we have the following Theorem 4.10.
For each i ∈ Z , there exists an embedding of algebras ϕ i : D ( A ) ֒ →DH cetw ( A ) ⊗ DH cetw ( A ) . Explicitly, (1) if i = − , ϕ i is defined on generators by K + α K [0] α ⊗ K [ − α , ω + M X [ M ] , [ M ] v h M,M i a M a M a M g MM M Z [0] M K [0]ˆ M ⊗ Z [ − M K [ − M , K − α K [ − α ⊗ K [0] α , ω − M X [ M ] , [ M ] v h M,M i a M a M a M g MM M Z [ − M K [ − M ⊗ Z [0] M K [0]ˆ M ;(2) if i = 0 , ϕ i is defined on generators by K + α K [1] α ⊗ K [0] α , ω + M X [ M ] , [ M ] v −h ˆ M, ˆ M i a M a M a M g MM M Z [1] M K [1]ˆ M K [0]ˆ M ⊗ Z [0] M , K − α K [0] α ⊗ K [1] α , ω − M X [ M ] , [ M ] v −h ˆ M, ˆ M i a M a M a M g MM M Z [0] M ⊗ Z [1] M K [1]ˆ M K [0]ˆ M ;(3) if i < − , ϕ i is defined on generators by K + α K [ i +1] α ⊗ K [ i ] α , K − α K [ i ] α ⊗ K [ i +1] α ,ω + M X [ M ] , [ M ] v x a M a M a M g MM M Z [ i +1] M − ( i +1) Y j =1 K [ − j ]( − i + j +1 ˆ M K [ i +1]ˆ M ⊗ Z [ i ] M − i Y j =1 K [ − j ]( − i + j ˆ M x = h ˆ M , ˆ M − ˆ M i − i ( h M , M i + h M , M i ) ,ω − M X [ M ] , [ M ] v y a M a M a M g MM M Z [ i ] M − i Y j =1 K [ − j ]( − i + j ˆ M ⊗ Z [ i +1] M − ( i +1) Y j =1 K [ − j ]( − i + j +1 ˆ M K [ i +1]ˆ M y = h ˆ M , ˆ M − ˆ M i − i ( h M , M i + h M , M i );(4) if i > , ϕ i is defined on generators by K + α K [ i +1] α ⊗ K [ i ] α , K − α K [ i ] α ⊗ K [ i +1] α ,ω + M X [ M ] , [ M ] v x a M a M a M g MM M Z [ i +1] M i Y j =0 K [ j ]( − i − j ˆ M K [ i +1]ˆ M ⊗ Z [ i ] M i − Y j =0 K [ j ]( − i − j − ˆ M x = h ˆ M , ˆ M − ˆ M i − i ( h M , M i + h M , M i ) , RINFELD DOUBLES, DERIVED HALL ALGEBRAS, BRIDGELAND HALL ALGEBRAS 13 ω − M X [ M ] , [ M ] v y a M a M a M g MM M Z [ i ] M i − Y j =0 K [ j ]( − i − j − ˆ M ⊗ Z [ i +1] M i Y j =0 K [ j ]( − i − j ˆ M K [ i +1]ˆ M y = h ˆ M , ˆ M − ˆ M i − i ( h M , M i + h M , M i ) . Proof.
By the following commutative diagram D ( A ) (cid:31) (cid:127) ψ i / / ϕ i ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ DH ( A ) ⊗ DH ( A ) ∼ = φ − ⊗ φ − (cid:15) (cid:15) DH cetw ( A ) ⊗ DH cetw ( A )we complete the proof. (cid:3) References [1] T. Bridgeland, Quantum groups via Hall algebras of complexes, Ann. Math. (2013), 1–21.[2] Q. Chen and B. Deng, Cyclic complexes, Hall polynomials and simple Lie algebras, J. Algebra (2015), 1–32.[3] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. (1995),361–377.[4] M. Kapranov, Eisenstein series and quantum affine algebras, J. Math. Sci. (5) (1997),1311–1360.[5] M. Kapranov, Heisenberg doubles and derived categories, J. Algebra (1998), 712–744.[6] R. M. Kashaev, The Heisenberg double and the pentagon relation, Algebra i Analiz (4) (1996),63–74.[7] J. Lin, Modified Ringel–Hall algebras, naive lattice algebras and lattice algebras, arXiv:1808.04037v1.[8] C. M. Ringel, Hall algebras, in: S. Balcerzyk, et al. (Eds.), Topics in Algebra, Part 1, in: BanachCenter Publ. (1990), 433–447.[9] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. (1990), 583–592.[10] O. Schiffmann, Lectures on Hall algebras, Geometric methods in representation theory II, 1–141,S´emin. Congr., 24-II, Soc. Math. France, Paris, 2012.[11] J. Sheng and F. Xu, Derived Hall algebras and lattice algebras, Algebra Colloq. (03) (2012),533–538.[12] B. To¨en, Derived Hall algebras, Duke Math. J. (2006), 587–615.[13] J. Xiao, Drinfeld double and Ringel–Green theory of Hall algebras, J. Algebra (1997), 100–144.[14] J. Xiao and F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. (2) (2008),357–373.[15] S. Yanagida, A note on Bridgeland’s Hall algebra of two-periodic complexes, Math. Z. (3) (2016),973–991.[16] H. Zhang, A note on Bridgeland Hall algebras, Comm. Algebra (6) (2018), 2551–2560.[17] H. Zhang, Bridgeland’s Hall algebras and Heisenberg doubles, J. Algebra Appl. (6) (2018). Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R.China.
E-mail address : [email protected] Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal Univer-sity, Nanjing 210023, P. R. China.
E-mail address ::