BiHom-algebra was introduced by Graziani et al[1]. The concept arose from the research of algebra in group-Hom categories, and it is an important generalization of Hom-algebra[2-4]. H-pseudoalgebra was introduced by Bakalov et al[5]. It is an algebra in the pseudotensor category M*(H), and it is a generalization of the conformal algebra introduced by Kac[6]. We can also regard it as multidimensional conformal algebra. The structure of H-pseudoalgebra has some connections with mathematical physics and nonlinear equations[7-9] and it has been developed into more general forms[10-11].
The purpose of this paper is to define an algebraic system which generalizes BiHom-algebra and H-pseudoalgebra, so as to research its basic properties.
1 Preliminaries
Remark 1 Throughout this paper, we define that
1) k is a field, and the base vector spaces of all algebraic structures are over k.
2) H is a cocommutative Hopf algebra.
3) We use the sweedler notation to express the coproduct of H: Δ(h)=h1⊗h2, for any h∈H.
Definition 1[5] An associative H-pseudoalgebra is a pair (A,μ=*) which satisfies:
1) A is an H-module.
2) u∈HomH⊗H(A⊗A,(H⊗H)⊗HA) and we denote μ(a⊗b)=a*b. It is equivalent to (H-bilinearity) fa*gb=((f⊗g)⊗H1)(a*b).
3) (associative law) (a*b)*c=a*(b*c).
Remark 2 We explicitly describe the associative law of H-pseudoalgebra.
If we denote
a*
⊗gi)⊗Hei, ei*
⊗gij)⊗Heij b*
⊗li)⊗Hdi, a*
⊗lij)⊗Hdij
Then,
(a*b)*
⊗gifij2⊗gij)⊗Hei a*(b*c)
⊗hilij1⊗lilij2)⊗Hdij
Definition 2[5] A map of H-pseudoalgebras from (A,μA) to (B,μB) is defined as follows:
1) f∈HomH(A,B);
2) μB(f⊗f)=(IH⊗H⊗f)μA.
Definition 3[10] A Hom-associative H-pseudoalgebra is a triple (A,μ,α) which satisfies: 1) (A,μ) is an H-pseudoalgebra; 2) α∈HomH(A,A); 3) (a*b)*α(c)=α(a)*(b*c).
Definition 4[1] A multiplicative BiHom-associative algebra is a triple (A,α,β), which satisfies:
1) A is an algebra;
2) α,β∈Homk(A,B);
3) (a*b)*β(c)=α(a)*(b*c);
4) α(ab)=α(a)α(b), β(ab)=β(a)β(b);
5) αβ=βα.
2 BiHom-Associative H-Pseudoalgebras
Definiton 5 A BiHom-associative H-pseudoalgebra is a quadruple (A,μ=*,α,β) which satisfies:
1) A is an H-module.
2) u∈HomH⊗H(A⊗A,(H⊗H)⊗HA) and we denote u(a⊗b)=a*b;
3) α,β∈HomH(A,A);
4) αβ=βα;
5) (BiHom-associative law) α(x)*(y*z)=(x*y)*β(z).
Remark 3 A is a BiHom-associative H-pseudoalgebra.
1) If H=k, A is a BiHom-associative algebra.
2) If α=β=I, A is an associative H-pseudoalgebra.
3) If α=β, A is a Hom-associative H-pseudoalgebra.
Example 1 (A,α,β) is a finite dimensional BiHom-associative H-pseudoalgebra, H is a Hopf algebra. Then,(H⊗A,*,IH⊗α,IH⊗β)is a BiHom-associative algebra by
(f⊗a)*(g*b)=(f⊗g)⊗H(1⊗ab)
Definiton 6 A BiHom-associative H-pseudoalgebra(A,μ,α,β)is multiplicative if
(IH⊗H⊗Hα)μ=μ(α⊗α), (IH⊗H⊗Hβ)μ=μ(β⊗β)
Example 2 We take α=β=I, the BiHom-assciative H-pseudoalgebra is multiplicative.
Definition 7 Let (A,μB,αB,βB), (B,μB,αB,βB) be two (multiplicative) BiHom-associative H-pseudoalgebras, and a map of (multiplicative) BiHom-associative H-pseudoalgebras is defined as
1) f∈HomH(A,B);
2) (IH⊗H⊗Hf)μA=μB(f⊗f);
3) αBf=fαA,βBf=fβA.
Now, we introduce a method of constructing BiHom-associative H-pseudoalgebras from associative H-pseudoalgebras.
Theorem 1 1) (A,μA) is an associative H-pseudoalgebra, αA:A→A, βA:A→A are two maps of H-pseudoalgebra which satisfies
αA°βA=βA°αA
(IH⊗H⊗HαA)μA=μA(αA⊗βA)
Then, (A,μ*=(IH⊗H⊗HαA)μA,αA,βA) is a multiplicative BiHom-associative H-pseudoalgebra.
2) (B,μB) is an H-pseudoalgebra, and αB,βB:B→B are maps of H-pseudoalgebras which satisfies
αB°βB=βB°αB
(IH⊗H⊗HαB)μB=μB(αB⊗βB)
f:A→B is a map of H-pseudoalgebras which satisfies
fαA=αBf, f βA=βBf
Then, f:(A,(IH⊗H⊗HαA)μA,αA,βA)→(B,(IH⊗H⊗HαB)μB,αB,βB) is a map of multiplicative BiHom-associative H-pseudoalgebras.
Proof We denote μ*(a⊗b)=a**b. For any f,g∈H, a,b∈A,
(fa**gb)=(IH⊗H⊗HαA)μA(fa⊗gb)= (IH⊗H⊗Hα)((f⊗g)⊗H1)(a*b)= ((f⊗g)⊗H1)(IH⊗H⊗HαA)(a*b)= ((f⊗g)⊗H1)(a**b)
Then μ* is a pseudoproduct for A.
The proof of BiHom H-associative law is as follows:
a**b=αA(a)*βA(b)=(IH⊗H⊗HαA)(a*b)=
⊗gi)⊗HαA(ei)
αA(ei)**βA(c)=(IH⊗H⊗HαA)(αA(ei)*βA(c))=
(IH⊗H⊗HαA)(IH⊗H⊗HαA)(ei⊗c)=
⊗gij⊗HαA2(eij)
So,
(a**b)**βA(c)=
⊗gifij2⊗gij)⊗HαA2(eij)=
(IH⊗H⊗HαA2)((a*b)*c)=
(IH⊗H⊗HαA2)(a*(b*c))=(IH⊗H⊗HαA2)·
⊗hilji1⊗lilij2))⊗Hdij)
Similarly,
αA(a)**(b**c)=(IH⊗H⊗HαA2)·
⊗hilij1⊗lilij2)⊗Hdij)=
(a**b)**βA(c)
The proof of multiplicative for (A,μ*=(IH⊗H⊗Hα)μ,α,β) is
α(a)**α(b)=(IH⊗H⊗Hα)(a**b),β(a)**β(b)=
(IH⊗H⊗Hα)(β(a)*β(b))=
(IH⊗H⊗Hα)(IH⊗H⊗Hβ)(a*b)=
(IH⊗H⊗Hβ)(IH⊗H⊗Hα)(a*b)=
(IH⊗H⊗Hβ)(a**b)
(IH⊗H⊗Hf)(IH⊗H⊗HαA)μA(a⊗b)=
(IH⊗H⊗Hf)(αA(a)*βA(b))=
fαA(a)*fβA(b)=
αBf(a)*βBf(b) =
(IH⊗H⊗HαB)(f(a)*f(b))=
(IH⊗H⊗HαB)μB(f⊗f)(a⊗b)
Example 2 H is a cocommutative Hopf algebra and G(H) is the set of group like elements of H. H{e1, e2} is a free H-pseudoalgebra of rank 2, with pseudoproduct given by
e1*e1=a⊗He1, e2*e2=b⊗He2
e1*e2=a⊗He2, e2*e1=-a⊗He2
We define α,β:H{e1,e2}→H{e1,e2}:
α(e1)=ge1, α(e2)=ge2 β(e1)=he1, β(e2)=he2, g,h∈G(H)
satisfying
(g⊗g)a=(g⊗h)a, (g⊗g)b=(g⊗h)
Then, α,β are endomorphisms of H{e1,e2}, and(H{e1,e2},μ1=(IH⊗H⊗α)μ,α,β)is a multiplicative BiHom-associative H-pseudoalagebra with pseudoproduct μ1 given by
μ1(e1⊗e1)=(g⊗g)a⊗He1 μ1(e2⊗e2)=(g⊗g)b⊗He2 μ1(e1⊗e2)=(g⊗g)a⊗He2 μ1(e2⊗e1)=-(g⊗g)a⊗He2
Definition 8 The BiHom-associative H-pseudoalgebra (A,μ*=(IH⊗H⊗Hα)μ,α,β)is called the Yau twist of H-pseudoalgebra(A,μ) and is denoted by Aα,β.
The following is a generalization of Theorem 1.
Theorem 2 (A,μ,α～,β～) is a multiplicative BiHom-associative H-pseudoalgebra, and α,β∈HomH(A,A) satisfies
(IH⊗H⊗α)μ=μ(α⊗β),(IH⊗H⊗α)μ=μ(α⊗α) (IH⊗H⊗β)μ=μ(β⊗β)
and each of α, β, α～, β～ can commute with others.
Then, (A,μ(α⊗β), α～°α,β～°β)is a multiplicative BiHom-associative H-pseudoalgebra.
Proof We denote μ(α⊗β)(a⊗b)≡a**b.
The proof of BiHom-associative law is
(b**c)=(IH⊗H⊗Hα)(b*c)=
⊗li)⊗di)=
⊗li)⊗α(di)
α～α(a)**α(di)=αα～(a)**α(di)=
(IH⊗H⊗Hα)(αα～(a)*α(di))=
(IH⊗H⊗Hα2)(α～(a)*di)
So,
α～α(a)**(b**c)=(IH⊗H⊗H⊗Hα2)(α～(a)*(b*c))
Similarly,
(a**b)**β～β(c)=
(IH⊗H⊗H⊗Hα2)(a*b)*β～(c)
The proof of α～α(a)**α～α(b)=β～β(a)**β～β(b) is
α～α(a)**α～α(b)=
(IH⊗H⊗Hα)(α～α(a)*α～α(b))=
(IH⊗H⊗Hα)(IH⊗H⊗Hα～)(α(a)*α(b))=
(IH⊗H⊗Hαα～)(a**b)
Similarly,
β～β(a)**β～β(b)=(IH⊗H⊗Hβ～β)(a**b)
Theorem 3 (A,μA,αA,βA) is a multiplicative BiHom-associative H1-pseudoalgebra and (B,μB,αB,βB) is a multiplicative BiHom-associative H2-pseudoalgebra, and then(A⊗B,μA⊗B,αA⊗αB,βA⊗βB)is a multiplicative BiHom-associative H=H1⊗H2-pseudoalgebra by
μA⊗B((a1⊗b1)⊗(a2⊗b2))=
⊗FI⊗gi⊗GI)⊗H(ei⊗EI)
We denote
a1*
⊗gi)⊗ei,b1*
⊗lI)⊗HdI
Proof The H=H1⊗H2-module structure of A⊗B is defined as (g⊗h)(a⊗b)=ga⊗hb and the pseudoproduct of A⊗B is well defined. The H-bilinearity of μA⊗B is obvious. We define some symbols:
a2*
⊗li)⊗Hdi, αA(a1)*di=
⊗lij)⊗Hdij
a1*
⊗gi)⊗Hei, ei*βA(di)=
⊗gij)⊗Heij
b2*
⊗LI)⊗HDI, αB(b1)*Di=
⊗LIJ)⊗HDIJ
b1*
⊗GI)⊗HEI, EI*βA(DI)=
⊗GIJ)⊗HEIJ
The proof of BiHom-associative law is as follows:
(a2⊗b2)*(a3⊗b3)=
⊗HI⊗li⊗LI)⊗H(di⊗DI)·
(αA(a1)⊗αB(b2))*(di⊗DI)=
⊗HIJ⊗lij⊗LIJ)⊗H(dij⊗DIJ)
So,
(αA⊗αB)(a1⊗b1)*((a2⊗b2)*(a3⊗b3))=
⊗HIJ)⊗(hi⊗HI)(lij1⊗LIJ1)⊗
(li⊗LI)(lij2⊗LIJ2)⊗H(dij⊗DIJ)
Similarly,
((a1⊗b1)⊗(a2⊗b2))*(β(a3)⊗β(b3))=
⊗FI)(fij1⊗FIJ1)⊗(gi⊗GI)(fij2⊗FIJ2)⊗
(gij⊗GIJ)⊗H(eij⊗EIJ)
By the BiHom-associative law of (A,μA,αA,βA) and (B,μB,αB,βB),
(αA⊗αB)(a1⊗b1)*((a2⊗b2)*(a3⊗b3))=
((a1⊗b1)⊗(a2⊗b2))*(βA(a3)⊗βB(b3))
The proof of the multiplicative law is
(IH1⊗IH2⊗IH1⊗IH2⊗αA⊗αB)((a1⊗b1)*
(a2⊗b2))=(IH1⊗IH2⊗IH1⊗IH2⊗αA⊗αB)·
⊗FI⊗gi⊗GI)⊗H(ei⊗EI))=
⊗FI⊗gi⊗GI)⊗H(αA(ei)⊗αB(EI))
On the other side,
αA(a1)*αA(a2)=(IH1⊗IH1⊗αA)(a1⊗a2)=
⊗gi)⊗H1αA(ei) αB(b1)*αB(b2)=
(IH2⊗IH2⊗αB)(b1⊗b2)=
⊗GI)⊗H2αB(EI)
Therefore,
(IH1⊗IH2⊗IH1⊗IH2⊗αA⊗αB)((a1⊗b1)*
(a2⊗b2))=(αA⊗αA)(a1⊗b1)*
(αB⊗αB)(a2⊗b2)
Similarly,
(IH1⊗IH2⊗IH1⊗IH2⊗βA⊗βB)((a1⊗b1)*(a2⊗b2))= (βA⊗βB)(a1⊗b1)*(αB⊗αB)(a2⊗b2)
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