Pre-alternative algebras and pre-alternative bialgebras

TL;DR

This paper introduces pre-alternative algebras, explores their bialgebra theory, and establishes analogies with Lie bialgebras and classical Yang-Baxter equations, expanding algebraic structures.

Contribution

It defines pre-alternative algebras, develops their bialgebra theory, and introduces $PA$-equations as analogues of the classical Yang-Baxter equation.

Findings

Pre-alternative algebras decompose products into compatible parts.

Existence of a bialgebra theory for pre-alternative algebras.

Introduction of $PA$-equations as analogues of Yang-Baxter equations.

Abstract

We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform dialgebra or a pre-Lie algebra. The left and right multiplication operators of a pre-alternative algebra give a bimodule structure of the associated alternative algebra. There exists a (coboundary) bialgebra theory for pre-alternative algebras, namely, pre-alternative bialgebras, which exhibits all the familiar properties of the famous Lie bialgebra theory. In particular, a pre-alternative bialgebra is equivalent to a phase space of an alternative algebra and our study leads to what we called $PA$-equations in a pre-alternative algebra, which are analogues of the classical Yang-Baxter equation.

Tables

Table 1. Table 1. Comparison between alternative D-bialgebras and pre-alternative bialgebras

Algebras	Alternative D-bialgebras	Pre-alternative bialgebras
Matched pairs of alternative	$(A,A^{\ast },{𝔯}_{\circ }^{\ast },{𝔩}_{\circ }^{\ast },{𝔯}_{\ast }^{\ast },{𝔩}_{\ast }^{\ast })$	$(As\left(A\right),As\left(A^{\ast }\right),{𝔯}_{\prec }^{\ast },{𝔩}_{\succ }^{\ast }$,
algebras		${𝔯}_{{\prec }_{\ast }}^{\ast },{𝔩}_{{\succ }_{\ast }}^{\ast })$
Alternative algebra structures on	alternative	phase spaces
the direct sum of the alternative	analogues of Manin
algebras in the matched pairs	triples
Bilinear forms on	symmetric	skew-symmetric
the direct sum of the alternative	$⟨x+a^{\ast },y+b^{\ast }⟩$	$⟨x+a^{\ast },y+b^{\ast }⟩$
algebras in the matched pairs	$=⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩$	$=-⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩$
	invariant	symplectic forms
Double structures on	Drinfeld’s doubles	Drinfeld’s symplectic
the direct sum of the alternative		doubles
algebras in the matched pairs
Algebraic equations associated	skew-symmetric solutions	symmetric solutions
to coboundary cases	alternative YBEs in	$PA$-equations in
	alternative algebras	pre-alternative algebras
Nondegenerate solutions	symplectic forms of	2-cocycles of pre-alternative
	alternative algebras	algebras
$Al$-operators	associated to $({𝔯}_{\circ }^{\ast },{𝔩}_{\circ }^{\ast })$	associated to $({𝔯}_{\prec }^{\ast },{𝔩}_{\succ }^{\ast })$
	skew-symmetric parts	symmetric parts
Constructions from	$r=\sum _{i=1}^n\left(e_i\otimes e_i^{\ast }-e_i^{\ast }\otimes e_i\right)$	$r=\sum _{i=1}^n\left(e_i\otimes e_i^{\ast }+e_i^{\ast }\otimes e_i\right)$
pre-alternative algebras	induced bilinear forms	induced bilinear forms
	$⟨x+a^{\ast },y+b^{\ast }⟩$	$⟨x+a^{\ast },y+b^{\ast }⟩$
	$=-⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩$	$=⟨x,b^{\ast }⟩+⟨a^{\ast },y⟩$

Equations

$$(x\prec y)\prec z=x\prec (y\circ z),(x\succ y)\prec z=x\succ (y\prec z),(x\circ y)\succ z=x\succ (y\succ z),$$

$$(x,y,z{)}_r=(x\prec y)\prec z-x\prec (y\circ z);$$

$$(x,y,z{)}_m=(x\succ y)\prec z-x\succ (y\prec z);$$

$$(x,y,z{)}_l=(x\circ y)\succ z-x\succ (y\succ z),\forall x,y,z\in A,$$

$$W^{\perp }=\{x\in V|\mathfrak{B}(x,y)=0,\forall y\in W\}.$$

$$r_{12}=\sum _i{a}_i\otimes b_i\otimes 1,r_{13}=\sum _i{a}_i\otimes 1\otimes b_i,r_{23}=\sum _{i}1\otimes a_i\otimes b_i,$$

$$r_{12}\diamond r_{13}=\sum _{i,j}{a}_i\diamond a_j\otimes b_i\otimes b_j,r_{13}\diamond r_{23}=\sum _{i,j}{a}_i\otimes a_j\otimes b_i\diamond b_j,r_{23}\diamond r_{12}=\sum _{i,j}{a}_j\otimes a_i\diamond b_j\otimes b_i.$$

$$\sigma (x\otimes y)=y\otimes x,\forall x,y\in V.$$

$$⟨u^{\ast }\otimes v^{\ast },r⟩=⟨u^{\ast },T_r(v^{\ast })⟩,\forall u^{\ast },v^{\ast }\in V^{\ast },$$

$$\mathfrak{B}(u,v)=⟨T^{-1}u,v⟩,\forall u,v\in V.$$

$$⟨v_1,T^{\ast }(v_2^{\ast })⟩=⟨T(v_1),v_2^{\ast }⟩,\forall v_1\in V_1,v_2^{\ast }\in V_2^{\ast }.$$

$$⟨r_T,v_2^{\ast }\otimes v_1⟩=⟨T(v_1),v_2^{\ast }⟩,\forall v_1\in V_1,v_2^{\ast }\in V_2^{\ast }.$$

$$⟨{\rho }^{\ast }(x)v^{\ast },u⟩=⟨v^{\ast },\rho (x)u⟩,\forall x\in A,u\in V,v^{\ast }\in V^{\ast }.$$

$$(x,x,y)=(y,x,x)=0,\forall x,y,z\in A,$$

$$(x_1,x_2,y)+(x_2,x_1,y)=0,(y,x_1,x_2)+(y,x_2,x_1)=0,\forall x_1,x_2,y\in A.$$

$$L(x^2)=L(x)L(x),R(x^2)=R(x)R(x),$$

$$R(y)L(x)-L(x)R(y)=R(x\circ y)-R(y)R(x),L(y\circ x)-L(y)L(x)=L(y)R(x)-R(x)L(y).$$

$$(x_1+v_1)\ast (x_2+v_2)=x_1\circ x_2+(L(x_1)v_2+R(x_2)v_1),\forall x_1,x_2\in A,v_1,v_2\in V.$$

$$L(x\circ y)-L(x)L(y)=-L(y\circ x)+L(y)L(x)=R(x)L(y)-L(y)R(x),\forall x,y\in A.$$

$$⟨(L^{\ast }(y)R^{\ast }(x)-R^{\ast }(x)L^{\ast }(y))u^{\ast },v⟩$$

$$(x,y,z{)}_m+(y,x,z{)}_r=0,(x,y,z{)}_m+(x,z,y{)}_l=0,(y,x,x{)}_r=(x,x,y{)}_l=0,\forall x,y,z\in A,$$

$$(x,y,z{)}_m+(y,x,z{)}_r=0,(x,y,z{)}_m+(x,z,y{)}_l=0,$$

$$(x,y,z{)}_l+(y,x,z{)}_l=0,(x,y,z{)}_r+(x,z,y{)}_r=0.$$

$$x\circ y=x\succ y+x\prec y,\forall x,y\in A,$$

$$(x,x,y)$$

$$({\mathfrak{r}}_{\prec }(y){\mathfrak{l}}_{\succ }(x)-{\mathfrak{l}}_{\succ }(x){\mathfrak{r}}_{\prec }(y))z$$

$$Al(u)\circ Al(v)=Al(L(Al(u))v+R(Al(v))u),\forall u,v\in V.$$

$$u\prec v=R(Al(v))u,u\succ v=L(Al(u))v,\forall u,v\in V.$$

$$Al(u)\prec Al(v)=Al(u\prec v),Al(u)\succ Al(v)=Al(u\succ v),\forall u,v\in V.$$

$$(u\succ v)\prec w+(v\prec u)\prec w=R(Al(w))L(Al(u))v+R(Al(w))R(Al(u))v,$$

Full text

1. 1 Introduction
2. 2 Representation theory of alternative algebras and pre-alternative algebras
3. 3 Alternative D-bialgebras and an alternative analog of the classical Yang-Baxter equation
4. 4 Phase spaces of alternative algebras and matched pairs of alternative algebras
5. 5 Bimodules and matched pairs of pre-alternative algebras
6. 6 Pre-alternative bialgebras
7. 7 Coboundary pre-alternative bialgebras
8. 8 $PA$-equations and their properties
9. 9 Comparison between alternative D-bialgebras and pre-alternative bialgebras

1. Introduction

A dendriform dialgebra is a vector space $D$ together with two bilinear products denoted by $\prec,\succ:D\otimes D\rightarrow D$ such that (for any $x,y,z\in D$)

[TABLE]

where $x\circ y=x\prec y+x\succ y$. The notion of dendriform dialgebra was introduced by J.-L. Loday in 1995 as the (Koszul) dual of the associative dialgebra, which is related to periodicity phenomenons in algebraic K-theory ([L1]). It was further studied in connection with several areas in mathematics and physics, including operads ([L3]), homology ([Fra1], [Fra2]), Hopf algebras ([Cha], [Ho], [Ron]), Lie and Leibniz algebras ([Fra2]), combinatorics ([AS1], [AS2]), arithmetic ([L2]) and quantum field theory ([EG]). We recommend [L1] as a beautiful introduction and motivation of this subject.

Moreover, for any dendriform dialgebra $(D,\prec,\succ)$, the bilinear product $\circ$ defines an associative algebra. Thus, a dendriform dialgebra may be seen as an associative algebra whose multiplication can be decomposed into two coherent operations.

Furthermore, we re-examine the identity (1.1) in the following way. Let $A$ be a vector space together with two products denoted by $\prec,\succ:A\otimes A\rightarrow A$. The right associator (r-associator), middle associator (m-associator) and left associator (l-associator) are defined respectively by

[TABLE]

[TABLE]

[TABLE]

where $x\circ y=x\prec y+x\succ y$. So $(D,\prec,\succ)$ is a dendriform dialgebra if and only if all the above three “associators” are zero.

On the other hand, alternative algebras are a class of very important nonassociative algebras ([KS], [S1-2]), which are closely related to Lie algebras ([S2]), Jordan algebras ([J]) and Malcev algebras ([KS]), and so on. Motivated by the relations between associative algebras and alternative algebras (see the details in section 2), it is natural to consider certain algebra structure on an alternative algebra as an analogue of an dendriform dialgebra on an associative algebra. So we introduce a notion of pre-alternative algebra, which is one of the main objects in this paper. Just as an alternative algebra is a generalization of an associative algebra which weakens the condition of associativity, a pre-alternative algebra is a generalization of a dendriform dialgebra which weakens the conditions of $l$-associativity, $m$-associativity and $r$-associativity.

We would like to point out that there has already been a “Lie algebraic” version for relations between associative algebras and dendriform dialgebras. That is, a class of nonassociative algebras, namely pre-Lie algebras (or under other names like left-symmetric algebras, Vinberg algebras and so on, see a survey article [Bu] and the references therein) play a similar role of dendriform dialgebras. Therefore, in this sense, pre-alternative algebras are just “alternative” analogues of pre-Lie algebras or dendriform dialgebras.

Furthermore, Goncharov constructed a bialgebra theory for alternative algebras in [G], namely, alternative D-bialgebras. In this paper, we show that there exists a (coboundary) bialgebra theory for pre-alternative algebras, namely, pre-alternative bialgebras, which exhibits all the familiar properties of the famous Lie bialgebra theory of Drinfeld ([D]). As well as an alternative D-bialgebra is equivalent to an “alternative analogue” of Manin triple ([G], [CP]), a pre-alternative bialgebra is equivalent to a phase space of an alternative algebra ([K1], [Bai1]). In particular there also exists a “Drinfeld double” construction for a pre-alternative bialgebra which is previously unexpected. Moreover, There is also a clear analogue between alternative D-bialgebras and pre-alternative bialgebras. On the other hand, we would like to emphasis that the representation theory of alternative algebras and pre-alternative algebras play an essential role in establishing the bialgebra theories. We also would like to point out that both alternative D-bialgebras and pre-alternative bialgebras can be fit into the general framework of generalized bialgebras as introduced by Loday in [L4]. So it would be interesting to find the relations with Loday’s question, that is, to find good triples of operads ([L4]).

The paper is organized as follows. In section 2, we study bimodules of alternative algebras and introduce various methods to construct pre-alternative algebras. In section 3, we generalize the notion of phase space in mathematical physics ([K1]) to the realm of alternative algebras, and show that pre-alternative algebras are the natural underlying structures. In section $4$, we define and study bimodules and matched pairs of pre-alternative algebras. In section $5$, we introduce the notion of pre-alternative bialgebra which is equivalent to a phase spaces of an alternative algebra. In section $6$, we show that there is a reasonable coboundary (pre-alternative) bialgebra theory and what we study leads to what we call $PA$-equations. In section $7$, the properties of $PA$-equations are discussed. In Appendix, we prove the main results in [G] by a little different approach and we point out that there is a Drinfeld’s double construction for an alternative $D$-bialgebra, which was not given in [G].

Throughout this paper, all the algebras are finite-dimensional over a fixed base field k whose characteristic is not $2$. We give some notations as follow.

(1) Let $V$ be a vector space. Let $\mathfrak{B}:V\otimes V\rightarrow\mathbb{F}$ be a symmetric or skew-symmetric bilinear form on a vector space $V$. If $W$ is a subspace of $V$, then we define

[TABLE]

If $W\subset W^{\perp}$ ($W=W^{\perp}$ respectively), then $W$ is called isotropic (Lagrangian respectively).

(2) Let $(A,\diamond)$ be a vector space with a binary operation $\diamond:A\otimes A\rightarrow A$. Let $\mathfrak{l}_{\diamond}(x)$ and $\mathfrak{r}_{\diamond}(x)$ denote the left and right multiplication operator respectively, that is, $\mathfrak{l}_{\diamond}(x)y=\mathfrak{r}_{\diamond}(y)x=x\diamond y$ for any $x,y\in A$. We also simply denote them by $\mathfrak{l}(x)$ and $\mathfrak{r}(x)$ respectively without confusion. Moreover, let $\mathfrak{l}_{\diamond},\mathfrak{r}_{\diamond}:A\rightarrow gl(A)$ be two linear maps with $x\rightarrow\mathfrak{l}_{\diamond}(x)$ and $x\rightarrow\mathfrak{r}_{\diamond}(x)$ respectively.

(3) Let $V$ be a vector space and $r=\sum_{i}{a_{i}\otimes b_{i}}\in V\otimes V$. Set

[TABLE]

where $1$ is a scale. If in addition, there exists a binary operation $\diamond:V\otimes V\rightarrow V$ on $V$, then the operation between two $r$s is in an obvious way. For example,

[TABLE]

and so on.

(4) Let $V$ be a vector space. Let $\sigma:V\otimes V\rightarrow V\otimes V$ be the flip defined as

[TABLE]

For any $r\in V\otimes V$, $r$ is called symmetric (skew-symmetric respectively) if $r=\tau(r)$ ($r=-\tau(r)$ respectively). On the other hand, any $r\in V\otimes V$ can be identified as a linear map $T_{r}:V^{*}\rightarrow V$ in the following way:

[TABLE]

where $\langle,\rangle$ is the canonical paring between $V$ and $V^{*}$. $r\in V\otimes V$ is called nondegenerate if the above induced linear map $T_{r}$ is invertible. Moreover, any invertible linear map $T:V^{*}\rightarrow V$ can induce a nondegenerate bilinear form $\mathfrak{B}(,)$ on $V$ by

[TABLE]

Furthermore, $T$ is called symmetric (skew-symmetric respectively) if the induced bilinear form ${\mathfrak{B}}$ is symmetric (skew-symmetric respectively). Obviously, the symmetry or skew-symmetry of both $T$ and its corresponding $r\in V\otimes V$ coincides.

(5) Let $V_{1},V_{2}$ be two vector spaces and $T:V_{1}\rightarrow V_{2}$ be a linear map. Denote the dual (linear) map by $T^{*}:V_{2}^{*}\rightarrow V_{1}^{*}$ defined by

[TABLE]

On the other hand, $T$ can be identified as an element $r_{T}\in V_{2}\otimes V_{1}^{*}$ by

[TABLE]

Note that equation (1.9) is exactly the case that $V_{1}=V_{2}^{*}$. Moreover, in the above sense, any linear map $T:V_{1}\rightarrow V_{2}$ is obviously an element in $(V_{2}\oplus V_{1}^{*})\otimes(V_{2}\oplus V_{1}^{*})$.

(6) Let $A$ be an algebra and $V$ be a vector space. For any linear map $\rho:A\rightarrow gl(V)$, define a linear map $\rho^{*}:A\rightarrow gl(V^{*})$ by

[TABLE]

Note that in this case, $\rho^{*}$ is different from the one given by equation (1.11) which regards $gl(V)$ as a vector space, too.

(7) Let $V_{1}$ and $V_{2}$ be two vector spaces, we denote the elements of $V_{1}\oplus V_{2}$ by $u+v$ or $(u,v)$ for $u\in V_{1},v\in V_{1}$.

(8) Let $V$ be a vector space, we sometimes use 1 to denote the identity transformation of $V$.

2. Representation theory of alternative algebras and pre-alternative algebras

Definition 2.1**.**

An alternative algebra $(A,\circ)$ is a vector space $A$ equipped with a bilinear operation $(x,y)\rightarrow x\circ y$ satisfying

[TABLE]

where $(x,y,z)=(x\circ y)\circ z-x\circ(y\circ z)$ is the associator.**

Remark 2.2**.**

If the characteristic of the field is not 2, then an alternative algebra $(A,\circ)$ also satisfies the stronger axioms:

[TABLE]

Definition 2.3**.**

([S1]) Let $(A,\circ)$ be an alternative algebra and $V$ be a vector space. Let $L,R:A\rightarrow gl(V)$ be two linear maps. $V$ (or the pair $(L,R)$, or $(V,L,R)$) is called a representation or a bimodule of $A$ if (for any $x,y\in A$)

[TABLE]

[TABLE]

According to [S3], $(L,R)$ is a bimodule of an alternative algebra $(A,\circ)$ if and only if the direct sum $A\oplus V$ of vector spaces is turned into an alternative algebra (the semidirect sum) by defining multiplication in $A\oplus V$ by

[TABLE]

We denote it by $A\ltimes_{L,R}V$ or simply $A\ltimes V$.

Proposition 2.4**.**

If $(V,L,R)$ is a bimodule of an alternative algebra $(A,\circ)$, then $(V^{*},R^{*},L^{*})$ is a bimodule of $(A,\circ)$.

Proof.

By equations (2.3) and (2.4), we have

[TABLE]

So for any $u^{*}\in A^{*},v\in V$, we have

[TABLE]

Hence $L^{*}(y)R^{*}(x)-R^{*}(x)L^{*}(y)=L^{*}(x\circ y)-L^{*}(y)L^{*}(x)$. Similarly, $(R^{*},L^{*})$ also satisfies the other axioms defining a bimodule of $(A,\circ)$. ∎

Definition 2.5**.**

A pre-alternative algebra $(A,\prec,\succ)$ is a vector space $A$ with two bilinear products denoted by $\prec,\succ:A\otimes A\rightarrow A$ satisfying

[TABLE]

where $(x,y,z)_{r},(x,y,z)_{m},(x,y,z)_{l}$ are defined by equations (1.2)-(1.4) respectively and $x\circ y=x\succ y+x\prec y$.**

Remark 2.6**.**

(1) If the characteristic of the field is not 2, then a pre-alternative algebra $(A,\prec$, $\succ)$ satisfies the strong axioms (for any $x,y,z\in A$):

[TABLE]

[TABLE]

(2) It would be interesting to describe free pre-alternative algebras (cf. [L1]). **

Proposition 2.7**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then the product

[TABLE]

defines an alternative algebra, which is called the associated alternative algebra of $A$ and denoted by $As(A)$. $(A,\prec,\succ)$ is called a compatible pre-alternative algebra structure on the alternative algebra $As(A)$.

Proof.

In fact, for any $x,y\in A$, we have

[TABLE]

Similarly, we show that $(y,x,x)=0$. ∎

Remark 2.8**.**

Thus, a pre-alternative algebra can be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. On the other hand, it is obvious that an associative algebra is an alternative algebra and a dendriform dialgebra is a pre-alternative algebra.**

If $(A,\circ)$ is an alternative algebra, then $(A,\mathfrak{l}_{\circ},\mathfrak{r}_{\circ})$ is a bimodule of $A$. Moreover,

Proposition 2.9**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then $(A,\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$ is a bimodule of the associated alternative algebra $(As(A),\circ)$.

Proof.

In fact, for any $x,y,z\in A$, we have

[TABLE]

Similarly, $(\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$ satisfies the other axioms defining a bimodule of $(As(A),\circ)$. ∎

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then by Propositions 2.4 and 2.9, both $(A^{*},\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*})$ and $(A^{*},\mathfrak{r}_{\prec}^{*},\mathfrak{l}_{\succ}^{*})$ are bimodules of the associated alternative algebra $As(A)$.

The following terminology is motivated by the notion of $\mathcal{O}$-operator as a generalization of (the operator form of) the classical Yang-Baxter equation in [K2] (also see [Bai2]).

Definition 2.10**.**

Let $(V,L,R)$ be a bimodule of an alternative algebra $(A,\circ)$. A linear map $Al:V\rightarrow A$ is called an Al-operator associated to $(V,L,R)$ if

[TABLE]

Proposition 2.11**.**

Let $Al:V\rightarrow A$ be an $Al$-operator of an alternative algebra $(A,\circ)$ associated to a bimodule $(V,L,R)$. Then there exists a pre-alternative algebra structure on V given by

[TABLE]

Therefore $V$ is an alternative algebra as the associated alternative algebra of this pre-alternative algebra and $T$ is a homomorphism of alternative algebras. Furthermore, $Al(V)=\{Al(v)|v\in V\}\subset A$ is an alternative subalgebra of $(A,\circ)$ and there is an induced pre-alternative algebra structure on $T(V)$ given by

[TABLE]

Moreover, its associated alternative algebra structure is just the alternative subalgebra structure of $(A,\circ)$ and $Al$ is a homomorphism of pre-alternative algebras.

Proof.

We only prove one identity for $(V,\prec,\succ)$ being a pre-alternative algebra as an example, the proof of the others is similar. In fact, for any $u,v,w\in V$,

[TABLE]

[TABLE]

By equations (2.4), (2.10) and (2.11), we show that

[TABLE]

The remaining parts of the conclusion are obvious. ∎

Definition 2.12**.**

([S1]) Let $(A,\circ)$ be an alternative algebra and $(V,L,R)$ be a bimodule. A $1$-cocycle of $A$ into $V$ is a linear map $D:A\rightarrow V$ satisfying

[TABLE]

Proposition 2.13**.**

Let $(A,\circ)$ be an alternative algebra. Then the following conditions are equivalent.

(1) There exists a compatible pre-alternative algebra structure $(A,\prec,\succ)$ on $(A,\circ)$.

(2) There exists an invertible $Al$-operator.

(3) There exists a bijective $1$-cocycle.

Proof.

(3) $\Longrightarrow$ (2) If $D$ is a bijective $1$-cocycle of $(A,\circ)$ into a bimodule $(V,L,R)$, then $D^{-1}$ is an $Al$-operator associated to $(V,L,R)$.

(2)$\Longrightarrow$ (1) If $Al:V\rightarrow A$ is an invertible $Al$-operator associated to a bimodule $(V,L,R)$, then there is a compatible pre-alternative algebra structure on $A$ given by

[TABLE]

(1) $\Longrightarrow$ (3) If $(A,\prec,\succ)$ is a compatible pre-alternative algebra structure on $(A,\circ)$, then it is obvious that the identity map ${\rm id}$ is a bijective $1$-cocycle of $A$ into the bimodule $(A,\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$. ∎

Example 2.14**.**

Let $(A,\circ)$ be an alternative algebra graded by positive integers, that is, $A=\oplus_{i\in{\rm N}}A_{i}$, $A_{i}\circ A_{j}\subset A_{i+j}$. Then there is a bijective $1$-cocycle associated to the bimodule $(A,\mathfrak{l}_{\circ},\mathfrak{r}_{\circ})$ defined by $D(x_{i})=ix_{i}$ for $x_{i}\in A_{i}$. Therefore there exists a compatible pre-alternative algebra structure on $(A,\circ)$ given by

[TABLE]

Definition 2.15**.**

Let $(A,\circ)$ be an arbitrary algebra which is not necessarily associative and $\omega$ be a skew symmetric bilinear form on $A$. The bilinear form $\omega$ is called closed if $\omega$ satisfies

[TABLE]

If in addition, $\omega$ is nondegenerate, then $\omega$ is called symplectic. In particular, an alternative algebra $A$ equipped with a symplectic form is called a symplectic alternative algebra.**

Proposition 2.16**.**

Let ($A,\circ,\omega$) be an alternative algebra with a symplectic form $\omega$. Then there exists a compatible pre-alternative algebra structure “$\prec,\succ$” on $A$ given by

[TABLE]

Proof.

Define a linear map $T:A\rightarrow A^{*}$ by $\langle T(x),y\rangle=\omega(x,y),\forall x,y\in A$. Then $T$ is invertible and $T$ is a 1-cocycle of $A$ into the bimodule $(A^{*},\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*})$. So by Proposition 2.13, there is a compatible pre-alternative algebra structure “$\prec,\succ$” on $A$ given by

[TABLE]

Thus, for any $x,y\in A$,

[TABLE]

[TABLE]

Therefore the conclusion holds. ∎

3. Alternative D-bialgebras and an alternative analog of the

classical Yang-Baxter equation

Definition 3.1**.**

([G],[Z]) Let $M$ be an arbitrary variety of k-algebras and $(A,\circ)$ be an algebra in $M$ with comultiplication $\bigtriangleup$. Then $(A,\circ,\bigtriangleup)$ is called an M-bialgebra in the sense of Drinfeld if $D(A)$ belongs to $M$, where $D(A)=A\oplus A^{*}$ is equipped with the multiplication

[TABLE]

where $f\cdot a=\sum_{a}{a_{(1)}\langle f,a_{(2)}\rangle},a\cdot f=\sum_{a}{\langle f,a_{(1)}\rangle a_{(2)}},\langle f\bullet a,b\rangle=\langle f,a\circ b\rangle,\langle a\bullet f,b\rangle=\langle f,b\circ a\rangle$, $\bigtriangleup(a)=\sum_{a}{a_{(1)}\otimes a_{(2)}}$, and the multiplication $\ast$ on $A^{*}$ is induced by $\bigtriangleup$. In this case, $D(A)=A\oplus A^{*}$ is called the Drinfeld’s double for $A$. In particular, when $M$ is a variety of alternative algebras, $(A,\circ,\bigtriangleup)$ is called an alternative D-bialgebra.**

Remark 3.2**.**

According to [G], an alternative $D$-bialgebra $(A,\circ,\Delta)$ is equivalent to an “alternative analogue” of Manin triple ([CP]): there is an alternative algebra structure on the direct sum $A\oplus A^{*}$ of the underlying vector spaces of $A$ and $A^{*}$ such that both $A$ and $A^{*}$ are subalgebras and the symmetric bilinear form on $A\oplus A^{*}$ given by

[TABLE]

is invariant. Recall a bilinear form $\mathcal{B}$ on an alternative algebra $(A,\circ)$ is called invariant if

[TABLE]

On the other hand, it is easy to show that an alternative $D$-bialgebra $(A,\circ,\Delta)$ is equivalent to the fact that $(A,A^{*},\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\ast}^{*},\mathfrak{l}_{\ast}^{*})$ is a matched pair of alternative algebras in the sense of Proposition 4.7.**

Definition 3.3**.**

Let $(A,\circ)$ be an alternative algebra and $r=\sum_{i}{a_{i}\otimes b_{i}}\in A\otimes A$. Then the pair $(A,r)$ is called a coboundary alternative D-bialgebra if $(A,\circ,\bigtriangleup_{r})$, where

[TABLE]

is an alternative $D$-bialgebra.**

Theorem 3.4**.**

([G])* Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Assume that $r$ is skew-symmetric and*

[TABLE]

Then $(A,\circ,\bigtriangleup_{r})$ is an alternative $D$-bialgebra.

Definition 3.5**.**

Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Equation (3.5) is called an “alternative analog” of the classical Yang-Baxter equation in [G]. We also call it the alternative Yang-Baxter equation in $(A,\circ)$.**

Proposition 3.6**.**

Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Then $r$ is a skew-symmetric solution of the alternative Yang-Baxter equation in $(A,\circ)$ if and only if $T_{r}$ is an $Al$-operator associated to the bimodule $(A^{*},\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*})$, that is, $T_{r}$ satisfies following equation

[TABLE]

So there is a pre-alternative algebra structure on $A^{*}$ given by

[TABLE]

Moreover, its associated alternative algebra structure is exactly the alternative algebra structure on $A^{*}$ as a subalgebra of $D(A)=A\oplus A^{*}$ which is induced from the comultiplication defined by equation (3.4). We denote this alternative algebra structure on $A^{*}$ by $A^{*}(r)$.

Proof.

Let $\{e_{i},...,e_{n}\}$ be a basis of $A$ and $\{e_{i}^{*},...,e_{n}^{*}\}$ be its dual basis. Suppose that $e_{i}\circ e_{j}=\sum\limits_{k}c_{ij}^{k}e_{k}$ and $r=\sum_{i,j}{a_{ij}e_{i}\otimes e_{j}}$. Hence $a_{ij}=-a_{ji}$ and $T_{r}(e_{l}^{*})=\sum\limits_{k}a_{kl}e_{k}$. Then $r$ is a solution of the alternative Yang-Baxter equation in $(A,\circ)$ if and only if (for any $i,k,t$)

[TABLE]

The left hand side of the above equation is precisely the coefficient of $e_{i}$ in

[TABLE]

Thus the first half part of the conclusion holds. It is easy to get the other results.∎

Corollary 3.7**.**

Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Assume $r$ is skew-symmetric and there exists a nondegenerate symmetric invariant bilinear form $\mathcal{B}$ on $(A,\circ)$. Define a linear map $\varphi:A\rightarrow A^{*}$ by $\langle\varphi(x),y\rangle=\mathcal{B}(x,y)$ for any $x,y\in A$. Then $r$ is a solution of the alternative Yang-Baxter equation in $(A,\circ)$ if and only if $\tilde{T}_{r}=T_{r}\varphi:A\rightarrow A$ is an $Al$-operator associated to the bimodule $(A,\mathfrak{l}_{\circ},\mathfrak{r}_{\circ})$, that is, $\tilde{T}_{r}$ satisfies the following equation

[TABLE]

Hence there is a pre-alternative algebra structure on $A$ given by

[TABLE]

Proof.

For any $x,y\in A$, we have

[TABLE]

Hence $\varphi(\mathfrak{l}_{\circ}(x)y)=\mathfrak{r}_{\circ}^{*}(x)\varphi(y)$ for any $x,y\in A$. Similarly, $\varphi(\mathfrak{r}_{\circ}(x)y)=\mathfrak{l}_{\circ}^{*}(x)\varphi(y)$ for any $x,y\in A$. Let $a^{*}=\varphi(x),b^{*}=\varphi(y)$, then by Proposition 3.6, $r$ is a solution of the alternative Yang-Baxter equation in $(A,\circ)$ if and only if

[TABLE]

Therefore the conclusion holds.∎

Remark 3.8**.**

Equation (3.8) is exactly the Rota-Baxter relation of weight zero for an alternative algebra (cf. [Bax], [R]).**

Proposition 3.9**.**

Let $(A,\circ)$ be an alternative algebra. Let $(V,L,R)$ be a bimodule of $A$ and $(V^{*},R^{*},L^{*})$ be its dual bimodule. Let $T:V\rightarrow A$ be a linear map which can be identified as an element in $A\ltimes_{R^{*},L^{*}}V^{*}\otimes A\ltimes_{R^{*},L^{*}}V^{*}$. Then $T$ is an $Al$-operator of $A$ associated to $(V,L,R)$ if and only if $r=T-\tau T$ is a skew-symmetric solution of the alternative Yang-Baxter equation in $A\ltimes_{R^{*},L^{*}}V^{*}$.

Proof.

Let $\{e_{1},\cdots,e_{n}\}$ be a basis of $A$. Let $\{v_{1},\cdots,v_{m}\}$ be a basis of $V$ and $\{v_{1}^{*},\cdots,v_{m}^{*}\}$ be its dual basis. Set $T(v_{i})=\sum\limits_{k=1}^{n}a_{ik}e_{k},i=1,\cdots,m$. Then

[TABLE]

Therefore we have

[TABLE]

By the definition of the dual bimodule, we know

[TABLE]

Then

[TABLE]

Hence, we get

[TABLE]

So $r$ is a solution of the alternative Yang-Baxter equation in $A\ltimes_{R^{*},L^{*}}V^{*}$ if and only if $T$ is an $Al$-operator of $A$ associated to $(V,L,R)$.∎

Proposition 3.10**.**

([G])* Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Suppose that $r$ is skew-symmetric and nondegenerate. Then $r$ is a solution of the alternative Yang-Baxter equation in $A$ if and only if the bilinear form $\omega$ induced by $r$ through equation (1.10) is a symplectic form.*

Corollary 3.11**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Let $\{e_{1},\cdots,e_{n}\}$ be a basis of $A$ and $\{e_{1}^{*},\cdots,e_{n}^{*}\}$ be the dual basis. Then

[TABLE]

is a nondegenerate solution of the alternative Yang-Baxter equation in $As(A)\ltimes_{\mathfrak{r}^{*}_{\prec},\mathfrak{l}^{*}_{\succ}}A^{*}$. Moreover, the symplectic form $\omega_{p}$ induced by $r$ through equation (1.10) is given by

[TABLE]

Proof.

It follows from the fact that $T={\rm id}$ is an $Al$-operator of $As(A)$ associated to the bimodule $(A,\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$.∎

Proposition 3.12**.**

Let ($A,\circ,\omega$) be an alternative algebra with a symplectic form $\omega$. Suppose that there is a compatible pre-alternative algebra structure “$\prec,\succ$” on $A$ given by equation (2.16) and a pre-alternative algebra structure “$\prec_{*},\succ_{*}$ on $A^{*}$ given by equation (3.7), where the solution $r$ of the alternative Yang-Baxter equation in $(A,\circ)$ is induced by $\omega$ through equation (1.10). Let $a^{*}\ast b^{*}=a^{*}\prec_{*}b^{*}+a^{*}\succ_{*}b^{*}$ for any $a^{*},b^{*}\in A^{*}$. Then there is a pre-alternative algebra structure “$\prec_{0},\succ_{0}$” on $A\oplus A^{*}$ given by (for any $x,y\in A,a^{*},b^{*}\in A^{*}$)

[TABLE]

[TABLE]

Moreover, its associated alternative algebra is just the Drinfeld’s double $D(A)$ for the coboundary alternative $D$-bialgebra $(A,\circ,\Delta_{r})$

Proof.

In fact, since $r$ is invertible, it is easy to show that (for any $x,y\in A$ and $a^{*},b^{*}\in A^{*}$)

[TABLE]

So for any $z\in A,c^{*}\in A^{*}$,

[TABLE]

Similarly,

[TABLE]

where $\bullet=\prec_{0}+\succ_{0}$. Hence

[TABLE]

By a similar study, we prove that $(\prec_{0},\succ_{0})$ also satisfies equation (2.8) and the second equation in equation (2.7). ∎

Proposition 3.13**.**

Let $(A,\circ)$ be an alternative algebra.

(1) For any skew-symmetric solution $r$ of the alternative Yang-Baxter equation in $(A,\circ)$, the Drinfeld’s double $D(A)$ for the coboundary alternative $D$-bialgebra $(A,\circ,\Delta_{r})$ is isomorphic to $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$ as alternative algebras.

(2) The skew-symmetric solutions of the alternative Yang-Baxter equation in $(A,\circ)$ are in one to one correspondence with the linear maps $T_{r}:A^{*}\rightarrow A$ whose graphs

[TABLE]

are Lagrangian subalgebras of $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$ with respect to the bilinear form given by equation (3.2). Consequently every alternative subalgebra which is also a Lagrangian $graph(T_{r})$ of $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$, carries a pre-alternative algebra structure defined by (for any $a^{*},b^{*}\in A^{*}$)

[TABLE]

[TABLE]

Proof.

(1) Let $r$ be a skew-symmetric solution of the alternative Yang-Baxter equation in $(A,\circ)$. Let the product in $A^{*}(r)$ be $\ast$, then by Proposition $3.6$ we know $a^{*}\ast b^{*}=\mathfrak{r}_{\circ}^{*}(T_{r}(a^{*}))b^{*}+\mathfrak{l}_{\circ}^{*}(T_{r}(b^{*})a^{*})$, for any $a^{*},b^{*}\in A^{*}$. We claim that for any $x,y\in A,a^{*},b^{*}\in A^{*}$ we have

[TABLE]

In fact, it follows from the following computations (for any $c^{*}\in A^{*}$):

[TABLE]

where we use the fact that $\langle T_{r}(a^{*}),b^{*}\rangle=-\langle a^{*},T_{r}(b^{*})\rangle$, which follows from the fact that $r$ is skew-symmetric. Define a linear map $\lambda:(D(A)=A\oplus A^{*},\bullet)\rightarrow(A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*},\star)$ by

[TABLE]

Then we have

[TABLE]

where equations (3.6) and (3.17) are used. Furthermore, it is easy to see that $\lambda$ is bijective. Therefore $\lambda$ is an isomorphism of alternative algebras.

(2) First, we have $\lambda(A^{*}(r))=graph(T_{r})$. So $graph(T_{r})$ is a subalgebra of $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$. Since $r$ is skew-symmetric, $graph(T_{r})$ is isotropic with respect to the bilinear form defined by equation (3.2). Moreover, it has a complementary isotropic algebra $\lambda(A)=A$. So it is a Lagrangian subalgebra of $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$. Conversely, let $T:A^{*}\rightarrow A$ be a linear map whose $graph(T)$ is a Lagrangian subalgebra of $A\ltimes_{\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*}}A^{*}$. So $T$ is skew-symmetric, that is, $\langle T(a^{*}),b^{*}\rangle=-\langle T(b^{*}),a^{*}\rangle$ for any $a^{*},b^{*}\in A^{*}$. Since $graph(T)$ is a subalgebra, we have that

[TABLE]

So $T(a^{*})\circ T(b^{*})=T(\mathfrak{r}_{\circ}^{*}(T_{r}(a^{*}))b^{*}+\mathfrak{l}_{\circ}^{*}(T_{r}(b^{*}))a^{*})$. By Proposition 3.6, $T$ corresponds to certain skew-symmetric solution of the alternative Yang-Baxter equation in $(A,\circ)$. The last statement is obtained by transferring (by the isomorphism $\lambda$) the pre-alternative algebra structure of $A^{*}(r)$ to $graph(T_{r})$. ∎

4. Phase spaces of alternative algebras and matched pairs of

alternative algebras

Definition 4.1**.**

Let $(A,\circ,\omega)$ be a symplectic alternative algebra. $A$ is called an L-symplectic alternative algebra if $A$ is a direct sum of the underlying vector space of two Lagrangian subalgebras $A^{+}$ and $A^{-}$. It is denoted by $(A,\circ,A^{+},A^{-},\omega)$. Two $L$-symplectic alternative algebras $(A_{1},\circ,A_{1}^{+},A_{1}^{-},\omega_{1})$ and $(A_{2},\circ,A_{2}^{+},A_{2}^{-},\omega_{2})$ are isomorphic if there exists an isomorphism $\varphi:A_{1}\rightarrow A_{2}$ of alternative algebras such that

[TABLE]

It is easy to see that a symplectic alternative algebra $(A,\circ,\omega)$ is an $L$-symplectic alternative algebra if and only if $A$ is a direct sum of the underlying vector space of two isotropic subalgebras.

Proposition 4.2**.**

Let $(A,\circ,A^{+},A^{-},\omega)$ be an $L$-symplectic alternative algebra. Then there exists a pre-alternative algebra structure on $A$ given by equation $(2.18)$ such that $A^{+}$ and $A^{-}$ are pre-alternative subalgebras. Moreover, two $L$-symplectic alternative algebras $(A_{i},\circ,A_{i}^{+},A_{i}^{-},\omega_{i})\;(i=1,2)$ are isomorphic if and only if there exists an isomorphism of pre-alternative algebras satisfying equation $(4.1)$ which the compatible pre-alternative algebras are given by equation $(2.18)$.

Proof.

If $a,b,c\in A^{+}$, then $\omega(a\prec b,c)=\omega(a,b\circ c)=0$. Since $A^{+}$ is a Lagrangian subalgebra of $A$, we have $a\prec b\in A^{+},\forall a,b\in A^{+}$. Similar arguments apply to “$\succ$” and $A^{-}$. So the first conclusion holds. It is straightforward to get the second conclusion.∎

Definition 4.3**.**

Let $(A,\circ)$ be an alternative algebra. If there is an alternative algebra structure on the direct sum of the underlying vector space of $A$ and $A^{*}$ such that $A$ and $A^{*}$ are alternative subalgebras and the natural skew-symmetric bilinear form $\omega_{p}$ on $A\oplus A^{*}$ given by equation (3.11) is a symplectic form, then it is called a phase space of the alternative algebra $A$.

Remark 4.4**.**

The notion of phase space is borrowed from mathematical physics ([K1], [Bai1]).**

Proposition 4.5**.**

Every $L$-symplectic alternative algebra $(A,\circ,A^{+},A^{-},\omega)$ is isomorphic to a phase space of $A^{+}$.

Proof.

Since $A^{-}$ and $(A^{+})^{*}$ are identified by the symplectic form, we can transfer the alternative algebra structure on $A^{-}$ to $(A^{+})^{*}$. Hence the alternative algebra structure on $A^{+}\oplus A^{-}$ can be transferred to $A^{+}\oplus(A^{+})^{*}$. Therefore the conclusion follows. ∎

Remark 4.6**.**

By symmetry, every $L$-symplectic alternative algebra $(A,\circ,A^{+},A^{-},\omega)$ is isomorphic to a phase space of $A^{-}$.**

Proposition 4.7**.**

Let $(A,\circ)$ and $(B,\ast)$ be two alternative algebras. Suppose that there are linear maps $L_{A},R_{A}:A\rightarrow gl(B)$ and $L_{B},R_{B}:B\rightarrow gl(B)$ such that $(L_{A},R_{A})$ is a bimodule of $A$ and $(L_{B},R_{B})$ is a bimodule of $B$ and they satisfy the following conditions:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $x,y\in A,a,b\in B$, $Ass_{i}=L_{i}+R_{i},i=A,B$. Then there is an alternative algebra structure on the vector space $A\oplus B$ given by

[TABLE]

We denote this alternative algebra by $A\bowtie_{L_{A},R_{A}}^{L_{B},R_{B}}B$ or simply $A\bowtie B$. Moreover, $(A,B,L_{A},R_{A}$, $L_{B},R_{B})$ satisfying the above conditions is called a $matched$ $pair$ of alternative algebras. On the other hand, every alternative algebra which is a direct sum of the underlying vector spaces of two subalgebras can be obtained from the above way.

Proof.

Straightforward.∎

Proposition 4.8**.**

Let $(A,\prec_{1},\succ_{1})$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Suppose that there exists a pre-alternative algebra structure “$\prec_{2},\succ_{2}$” on its dual space $A^{*}$ and $(As(A^{*}),\ast)$ is the associated alternative algebra. Then there exists an $L$-symplectic alternative algebra structure on the vector space $A\oplus A^{*}$ such that $(As(A),\circ)$ and $(As(A^{*}),\ast)$ are Lagrangian subalgebras associated to the symplectic form $(3.11)$ if and only if $(As(A),As(A^{*}),\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*},\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*})$ is a matched pair of alternative algebras. Furthermore, every $L$-symplectic alternative algebra can be obtained from the above way.

Proof.

If $(As(A),As(A^{*}),\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*},\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*})$ is a matched pair of alternative algebras, then it is straightforward to show that the bilinear form (3.11) is a symplectic form of the alternative algebra $As(A)\bowtie_{\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*}}^{\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*}}As(A^{*})$. Conversely, set

[TABLE]

where $\star$ is the alternative algebra structure of $As(A)\bowtie_{\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*}}^{\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*}}As(A^{*})$. Then $(A,A^{*},L_{\circ},R_{\circ}$, $L_{\ast},R_{\ast})$ is a matched pair of alternative algebras. Note that

[TABLE]

where $x,y\in A,a^{*},b^{*}\in A^{*}$. Hence, $R_{\circ}=\mathfrak{l}_{\succ_{1}}^{*},L_{\ast}=\mathfrak{r}_{\prec_{2}}^{*}$. Similarly, $L_{\circ}=\mathfrak{r}_{\prec_{1}}^{*},R_{\ast}=\mathfrak{l}_{\succ_{2}}^{*}$. ∎

5. Bimodules and matched pairs of pre-alternative algebras

Definition 5.1**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $V$ be a vector space. Let $L_{\prec},R_{\prec},L_{\succ},R_{\succ}:A\rightarrow gl(V)$ be four linear maps. $V$ (or $(L_{\prec},R_{\prec},L_{\succ},R_{\succ})$, or $(V,L_{\prec},R_{\prec},L_{\succ}$, $R_{\succ})$) is called a *representation * or a bimodule of $A$ if (for any $x,y\in A$)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $x\circ y=x\prec y+x\succ y,L_{\circ}=L_{\succ}+L_{\prec},R_{\circ}=R_{\succ}+R_{\prec}$.**

According to [S3], $(V,L_{\prec},R_{\prec},L_{\succ},R_{\succ})$ is a bimodule of a pre-alternative algebra $(A,\prec,\succ)$ if and only if the direct sum $A\oplus V$ of vector spaces is turn into a pre-alternative algebra (the semidirect sum) by defining multiplications in $A\oplus V$ by (for any $x,y\in A,a,b\in V$)

[TABLE]

We denote it by $A\ltimes_{L_{\prec},R_{\prec},L_{\succ},R_{\succ}}V$ or simply $A\ltimes V$.

Proposition 5.2**.**

Let $(V,L_{\prec},R_{\prec},L_{\succ},R_{\succ})$ be a bimodule of a pre-alternative algebra $(A,\prec,\succ)$. Let $(As(A),\circ)$ be the associated alternative algebra.

(1)* Both $(V,L_{\succ},R_{\prec})$ and $(V,L_{\circ}=L_{\prec}+L_{\succ},R_{\circ}=R_{\prec}+R_{\succ})$ are bimodules of $(As(A),\circ)$.*

(2)* For any bimodule $(V,L,R)$ of $(As(A),\circ)$, $(V,0,R,L,0)$ is a bimodule of $(A,\prec,\succ)$.*

(3)* $(V^{*},-R_{\succ}^{*},L_{\circ}^{*},R_{\circ}^{*},-L_{\prec}^{*})$ is a bimodule of $(A,\prec,\succ)$.*

Proof.

We only give the proof of (3) as an example. The others are more or less straightforward. In fact, since $(V,L_{\circ},R_{\circ})$ is a bimodule of $As(A)$, $R_{\circ}(x^{2})=R_{\circ}(x)R_{\circ}(x)$ for any $x\in A$. Hence

[TABLE]

So for any $v\in V,u^{*}\in V^{*}$,

[TABLE]

Therefore $R_{\circ}^{*}(x\circ y+y\circ x)=R_{\circ}^{*}(x)R_{\circ}^{*}(y)+R_{\circ}^{*}(y)R_{\circ}^{*}(x)$. Similarly, $(-R_{\succ}^{*},L_{\circ}^{*},R_{\circ}^{*},-L_{\prec}^{*})$ also satisfies equations (5.2)-(5.10). ∎

Example 5.3**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then $(\mathfrak{l}_{\prec},\mathfrak{r}_{\prec},\mathfrak{l}_{\succ},\mathfrak{r}_{\succ}),(0,\mathfrak{r}_{\prec},\mathfrak{l}_{\succ},0)$, $(0,\mathfrak{r}_{\circ},\mathfrak{l}_{\circ},0),(0,\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},0),(0,\mathfrak{l}_{\succ}^{*},\mathfrak{r}_{\prec}^{*},0)$ and $(-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*})$ are bimodules of $(A,\prec,\succ)$.**

Definition 5.4**.**

Let $(A,\prec_{A},\succ_{A})$ and $(B,\prec_{B},\succ_{B})$ be two pre-alternative algebras. Suppose that there are linear maps $L_{\prec_{A}},R_{\prec_{A}},L_{\succ_{A}},R_{\succ_{A}}:A\rightarrow gl(B)$ and $L_{\prec_{B}},R_{\prec_{B}},L_{\succ_{B}},R_{\succ_{B}}:B\rightarrow gl(A)$ such that the following products on the vector space $A\oplus B$ (for any $x,y\in A,a,b\in B$)

[TABLE]

[TABLE]

define a pre-alternative algebra structure. Then $(A,B,L_{\prec_{A}},R_{\prec_{A}},L_{\succ_{A}},R_{\succ_{A}},L_{\prec_{B}},R_{\prec_{B}}$, $L_{\succ_{B}}$, $R_{\succ_{B}})$ is called a matched pair of pre-alternative algebras, and we denote this pre-alternative algebra by $A\bowtie_{L_{\prec_{A}},R_{\prec_{A}},L_{\succ_{A}},R_{\succ_{A}}}^{L_{\prec_{B}},R_{\prec_{B}},L_{\succ_{B}},R_{\succ_{B}}}B$ or simply $A\bowtie B$.**

Remark 5.5**.**

It is not hard to write down explicitly a necessary and sufficient condition satisfied by the above linear maps in a matched pair of pre-alternative like Proposition 4.7, which involves 20 equations. Since we will not use such a conclusion directly in this paper, they are omitted here. Note that $(B,L_{\prec_{A}},R_{\prec_{A}},L_{\succ_{A}},R_{\succ_{A}})$ and $(A,L_{\prec_{B}},R_{\prec_{B}},L_{\succ_{B}},R_{\succ_{B}})$ must be bimodules of $A$ and $B$, respectively.

Corollary 5.6**.**

Let $(A,B,L_{\prec_{A}},R_{\prec_{A}},L_{\succ_{A}},R_{\succ_{A}},L_{\prec_{B}},R_{\prec_{B}},L_{\succ_{B}},R_{\succ_{B}})$ be a matched pair of pre-alternative algebras. Then $(As(A),As(B),L_{\prec_{A}}+L_{\succ_{A}},R_{\prec_{A}}+R_{\succ_{A}},L_{\prec_{B}}+L_{\succ_{B}},R_{\prec_{B}}+R_{\succ_{B}})$ is a matched pair of alternative algebras.

Proof.

It follows from the relation between the pre-alternative algebra and the associated alternative algebra. ∎

Proposition 5.7**.**

Let $(A,\prec_{1},\succ_{1})$ be a pre-alternative algebra and $(As(A),\circ_{1})$ be the associated alternative algebra. Suppose there is a pre-alternative algebra structure $``\prec_{2},\succ_{2}$” on its dual space $A^{*}$ and $(As(A^{*}),\circ_{2})$ is the associated alternative algebra. Then $(As(A),As(A^{*}),\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*}$, $\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*})$ is a matched pair of alternative algebras if and only if $(A,A^{*},-\mathfrak{r}_{\succ_{1}}^{*},\mathfrak{l}_{\circ_{1}}^{*},\mathfrak{r}_{\circ_{1}}^{*}$, $-\mathfrak{l}_{\prec_{1}}^{*}$, $-\mathfrak{r}_{\succ_{2}}^{*},\mathfrak{l}_{\circ_{2}}^{*},\mathfrak{r}_{\circ_{2}}^{*},-\mathfrak{l}_{\prec_{2}}^{*})$ is a matched pair of pre-alternative algebras.

Proof.

By Corollary 5.6, we only need to prove the“only if” part of the conclusion. If $(As(A)$, $As(A^{*}),\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*},\mathfrak{r}_{\prec_{2}}^{*}$, $\mathfrak{l}_{\succ_{2}}^{*})$ is a matched pair of alternative algebras, then by Proposition 4.8, $As(A)\bowtie_{\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*}}^{\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*}}As(B)$ is an $L$-symplectic alternative algebra with the symplectic form given by equation (3.11). Hence there is a compatible pre-alternative algebra structure on $As(A)\bowtie_{\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*}}^{\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*}}As(B)$ given by equation (2.18). Then for any $x,y\in A,a^{*},b^{*}\in A^{*}$ we have

[TABLE]

[TABLE]

So $a^{*}\prec x=-\mathfrak{r}_{\succ_{2}}^{*}(a^{*})x+\mathfrak{l}_{\circ_{1}}(x)a^{*}$. Similarly,

[TABLE]

Therefore $(A,A^{*},-\mathfrak{r}_{\succ_{1}}^{*},\mathfrak{l}_{\circ_{1}}^{*},\mathfrak{r}_{\circ_{1}}^{*},-\mathfrak{l}_{\prec_{1}}^{*},-\mathfrak{r}_{\succ_{2}}^{*},\mathfrak{l}_{\circ_{2}}^{*},\mathfrak{r}_{\circ_{2}}^{*},-\mathfrak{l}_{\prec_{2}}^{*})$ is a matched pair of pre-alternative algebras.∎

6. Pre-alternative bialgebras

Theorem 6.1**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative algebra $(A,\prec,\succ)$ equipped with two comultiplications $\alpha,\beta:A\rightarrow A\otimes A$ and $(As(A),\circ)$ be the associated alternative algebra. Suppose that $\alpha^{*},\beta^{*}:A^{*}\otimes A^{*}\rightarrow A^{*}$ induce a pre-alternative algebra structure “$\prec_{*},\succ_{*}$” on its dual space $A^{*}$. Then $(As(A),As(A^{*}),\mathfrak{r}_{\prec}^{*},\mathfrak{l}_{\succ}^{*},\mathfrak{r}_{\prec_{*}}^{*},\mathfrak{l}_{\succ_{*}}^{*})$ is a matched pair of alternative algebras if and only if $\alpha,\beta$ satisfy the following eight equations (for any $x,y\in A$):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

By Proposition 4.7, we just need to prove equations (6.1)-(6.8) are equivalent to equations (4.2)-(4.9), where $(A,B,L_{A},R_{A},L_{B},R_{B})$ is replaced by $(As(A),As(A^{*}),\mathfrak{r}_{\prec}^{*},\mathfrak{l}_{\succ}^{*},\mathfrak{r}_{\prec_{*}}^{*},\mathfrak{l}_{\succ_{*}}^{*})$. As an example, we give an explicit proof of the equivalence between equations (4.2) and (6.3). The proof of the others is similar. In fact, in this case, equation (4.2) becomes

[TABLE]

where $x,y\in A,a^{*}\in A^{*}$. Let both the left-hand side and the right-hand side of the above equation act on $b^{*}\in A^{*}$, we have

[TABLE]

So equation (4.2) holds if and only if equation (6.3) holds. ∎

Definition 6.2**.**

${\rm(1)}$ Let $(A,\alpha,\beta)$ be a vector space with two comultiplications $\alpha,\beta:A\rightarrow A\otimes A$. If $(A,\alpha^{*},\beta^{*})$ becomes a pre-alternative algebra, then we call the triple $(A,\alpha,\beta)$ a pre-alternative coalgebra.

${\rm(2)}$ If $(A,\prec,\succ,\alpha,\beta)$ is a pre-alternative algebra $(A,\prec,\succ)$ with two comultiplications $\alpha,\beta:A\rightarrow A\otimes A$ such that $(A,\alpha,\beta)$ is a pre-alternative coalgebra and $\alpha,\beta$ satisfy equations (6.1)-(6.8), then $(A,\prec,\succ,\alpha,\beta)$ is called a pre-alternative bialgebra.**

Combining Propositions 4.8, 5.7 and Theorem 6.1, we have the following conclusion:

Corollary 6.3**.**

Let $(A,\prec_{1},\succ_{1})$ be a pre-alternative algebra and $(As(A),\circ_{1})$ be the associated alternative algebra. Let $\alpha,\beta:A\rightarrow A\otimes A$ be two linear maps such that $\alpha^{*},\beta^{*}:A^{*}\otimes A^{*}\subset(A\otimes A)^{*}\rightarrow A^{*}$ induce a pre-alternative algebra structure on $A^{*}$ denoted by “$\prec_{2},\succ_{2}$”, that is, $(A,\alpha,\beta)$ is a pre-alternative coalgebra. Let $(As(A^{*}),\circ_{2})$ be the associated alternative algebra of $(A^{*},\prec_{2},\succ_{2})$. Then the following conditions are equivalent:

${\rm(1)}$* $(As(A)\bowtie As(A^{*}),As(A),As(A^{*}),\omega_{p})$ is an $L$-symplectic alternative algebra (or a phase space of $As(A)$), where $\omega_{p}$ is given by equation $(3.11)$;*

${\rm(2)}$* $(As(A),As(A^{*}),\mathfrak{r}_{\prec_{1}}^{*},\mathfrak{l}_{\succ_{1}}^{*},\mathfrak{r}_{\prec_{2}}^{*},\mathfrak{l}_{\succ_{2}}^{*})$ is a matched pair of alternative algebras;*

${\rm(3)}$* $(A,A^{*},-\mathfrak{r}_{\succ_{1}}^{*},\mathfrak{l}_{\circ_{1}}^{*},\mathfrak{r}_{\circ_{1}}^{*},-\mathfrak{l}_{\prec_{1}}^{*},-\mathfrak{r}_{\succ_{2}}^{*},\mathfrak{l}_{\circ_{2}}^{*},\mathfrak{r}_{\circ_{2}}^{*},-\mathfrak{l}_{\prec_{2}}^{*})$ is a matched pair of pre-alternative algebras;*

${\rm(4)}$* $(A,\prec_{1},\succ_{1},\alpha,\beta)$ is a pre-alternative bialgebra.*

Definition 6.4**.**

Let $(A,\prec_{A},\succ_{A},\alpha_{A},\beta_{A})$ and $(B,\prec_{B},\succ_{B},\alpha_{B},\beta_{B})$ be two pre-alternative bialgebras. A homomorphism of pre-alternative bialgebras $\varphi:A\rightarrow B$ is a homomorphism of pre-alternative algebras such that

[TABLE]

Proposition 6.5**.**

Two $L$-symplectic (hence phase spaces of) alternative algebras are isomorphic if and only if their corresponding pre-alternative bialgebras are isomorphic.

Proof.

Let $(As(A)\bowtie As(A^{*}),As(A),As(A^{*}),\omega_{p})$ and $(As(B)\bowtie As(B^{*}),As(B),As(B^{*}),\omega_{p})$ be two $L$-symplectic alternative algebras and $\varphi:As(A)\bowtie As(A^{*})\rightarrow As(B)\bowtie As(B^{*})$ is the isomorphism. Then $\varphi|_{A}:A\rightarrow B$ and $\varphi|_{A^{*}}:A^{*}\rightarrow B^{*}$ are isomorphisms of pre-alternative algebras due to Proposition 4.2. Moreover, $\varphi|_{A^{*}}={(\varphi|_{A})^{*}}^{-1}$ since

[TABLE]

So $(\varphi|_{A})^{*}:B^{*}\rightarrow A^{*}$ is a homomorphism of pre-alternative algebras and then $(A,\prec_{A},\succ_{A},\alpha_{A},\beta_{A})$ and $(B,\prec_{B},\succ_{B},\alpha_{B},\beta_{B})$ are isomorphic as pre-alternative bialgebras. Conversely, let $(A,\prec_{A},\succ_{A},\alpha_{A},\beta_{A})$ and $(B,\prec_{B},\succ_{B},\alpha_{B},\beta_{B})$ be two isomorphic pre-alternative bialgebras, and $\varphi^{\prime}:A\rightarrow B$ be the isomorphism. Let $\varphi:A\oplus A^{*}\rightarrow B\oplus B^{*}$ be a linear map defined by

[TABLE]

Then it is easy to show that $\varphi$ is an isomorphism of the two $L$-symplectic alternative algebras $(As(A)\bowtie As(A^{*}),As(A),As(A^{*}),\omega_{p})$ and $(As(B)\bowtie As(B^{*}),As(B),As(B^{*}),\omega_{p})$.∎

Example 6.6**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative bialgebra. Then its dual $(A,\prec_{*},\succ_{*}$, $\gamma,\delta)$ is also a pre-alternative bialgebra, where the pre-alternative algebra structure $\prec,\succ$ on $A$ is defined by the linear maps $\gamma^{*},\delta^{*}:A\otimes A\rightarrow A$ and $\alpha^{*},\beta^{*}:A^{*}\otimes A^{*}\rightarrow A^{*}$ induce a pre-alternative algebra structure on $A^{*}$ denoted by $``\prec_{*},\succ_{*}"$. **

Example 6.7**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then $(A,\prec,\succ,\ 0,0)$ is a pre-alternative bialgebra, and the corresponding pre-alternative algebra structure on $A\oplus A^{*}$ is the semi-direct sum $A\ltimes_{-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*}}A^{*}$. Moreover, its corresponding associated alternative algebra is the semi-direct sum $As(A)\ltimes_{\mathfrak{r}_{\prec}^{*},\mathfrak{l}_{\succ}^{*}}A^{*}$ with the symplectic form $\omega_{p}$ given by equation (3.11).

7. Coboundary pre-alternative bialgebras

Definition 7.1**.**

A pre-alternative bialgebra $(A,\prec,\succ,\alpha,\beta)$ is called coboundary if the linear maps $\alpha,\beta:A\rightarrow A\otimes A$ are given by

[TABLE]

[TABLE]

where $x\circ y=x\prec y+x\succ y$, $x,y\in A$ and $r_{\prec},r_{\succ}\in A\otimes A$.**

Remark 7.2**.**

The expression of equations (7.1)-(7.2) and equations (6.1)-(6.2) looks like certain kind of “$1$-coboundary” and “1-cocycle”.

Theorem 7.3**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra with two linear maps $\alpha,\beta:A\rightarrow A\otimes A$ defined by equations (7.1) and (7.2) respectively. If $r_{\prec}=r_{\succ}=r\in A\otimes A$ and $r$ is symmetric, then $\alpha,\beta$ satisfy equations (6.1)-(6.8).

Proof.

It is obvious that $\alpha,\beta$ automatically satisfy equations (6.1) and (6.2). For equations (6.3)-(6.8), as an example, we give an explicit proof of the fact that $\alpha,\beta$ satisfy equation (6.5) since the proof of the other cases are similar. Assume $r=\sum_{i}{u_{i}\otimes v_{i}}\in A\otimes A$. After rearranging the terms suitably, we have that (note that we use the fact that $r$ is symmetric)

[TABLE]

[TABLE]

The sum of the first seven terms is zero since it equals to

[TABLE]

The sum of the 8th-13th terms is zero since it equals to

[TABLE]

The sum of the 14th-16th terms, the sum of 17th-19th terms and the sum of the last three terms are all zero obviously. ∎

Lemma 7.4**.**

Let $A$ be a vector space and $\alpha,\beta:A\rightarrow A\otimes A$ be two linear maps. Then $(A,\alpha,\beta)$ is a pre-alternative coalgebra if and only if the linear maps $S^{i}_{\alpha,\beta}:A\rightarrow A\otimes A\otimes A$ $(i\in\{1,2,3,4\})$ given by the following equations are all zero: (for any $x\in A$)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

It follows immediately from the definition of pre-alternative algebra (cf. Remark 2.6). ∎

Definition 7.5**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Let $r\in A\otimes A$. The following equations are called $PA_{j}^{i}$-equations $(i=1,2,j=1,2,3)$ respectively:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We set $PA_{j}=PA_{j}^{1}+PA_{j}^{2}$, where $j=1,2,3$. All $PA_{j}^{i}$-equations $(i=1,2,j=1,2,3)$ are called $PA$-equations.**

Proposition 7.6**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Let $r\in A\otimes A$ be symmetric. Let $\alpha,\beta:A\rightarrow A\otimes A$ be two linear maps given by equations (7.1) and (7.2) respectively, where $r_{\prec}=r_{\succ}=r$. Then $(A,\alpha,\beta)$ becomes a pre-alternative coalgebra if and only if the following equations holds: (for any $x\in A$)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

We give an explicit proof of the fact that ${\rm equation\;(7.13)}\Leftrightarrow S_{\alpha,\beta}^{1}=0$ as an example. By a similar study, we can show that

[TABLE]

In fact, set $r=\sum\limits_{i}u_{i}\otimes v_{i}$. Substituting

[TABLE]

into equation (7.3) and after rearranging the terms suitably, we divide $S_{\alpha,\beta}^{1}$ into three parts:

[TABLE]

where

[TABLE]

Since $r$ is symmetric and by Remark 2.6, we have

[TABLE]

Therefore the conclusion holds. ∎

Theorem 7.7**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $r\in A\otimes A$ be symmetric. Let $\alpha,\beta:A\rightarrow A\otimes A$ be two linear maps given by equations (7.1) and (7.2) respectively, where $r_{\prec}=r_{\succ}=r$. Then $(A,\prec,\succ,\alpha,\beta)$ is a pre-alternative bialgebra if and only if equations (7.13)-(7.16) are satisfied.

Proof.

It follows from Theorem 7.3 and Proposition 7.6.∎

Next we give a Drinfeld’s “double” construction ([CP]) for a pre-alternative bialgebra.

Theorem 7.8**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative bialgebra. Then there is a canonical pre-alternative bialgebra structure on $A\oplus A^{*}$ such that both the inclusions $i_{1}:A\rightarrow A\oplus A^{*}$ and $i_{2}:A^{*}\rightarrow A\oplus A^{*}$ into the two summands are homomorphisms of pre-alternative bialgebras, where the pre-alternative bialgebra structure on $A^{*}$ is given in Example 6.6.

Proof.

Denote the pre-alternative algebra structure on $A^{*}$ induced by $\alpha^{*}$ and $\beta^{*}$ by $\prec_{*}$ and $\succ_{*}$ respectively, and its associated alternative algebra structure by $\ast$. Let $r\in A\otimes A^{*}\subset(A\oplus A^{*})\otimes(A\oplus A^{*})$ correspond to the identity map ${\rm id}:A\rightarrow A$. Then the pre-alternative algebra structure $\prec_{\bullet},\succ_{\bullet}$ on $A\oplus A^{*}$ is given by $\mathcal{PAD}(A)=A\bowtie_{-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*}}^{-\mathfrak{r}_{\succ_{*}}^{*},\mathfrak{l}_{\ast}^{*},\mathfrak{r}_{\ast}^{*},-\mathfrak{l}_{\prec_{*}}^{*}}A^{*}$, that is

[TABLE]

We shall denote its associated alternative algebra structure by $\bullet$. Let $\{e_{i},...,e_{n}\}$ be a basis of $A$ and $\{e_{1}^{*},...e_{n}^{*}\}$ be its dual basis. Then $r=\sum\limits_{i}e_{i}\otimes e_{i}^{*}$. Next we prove that

[TABLE]

induces a (coboundary) pre-alternative bialgebra structure on $A\oplus A^{*}$. Since $r$ is not symmetric we cannot apply Theorem 7.7 directly. Instead, We shall prove that $\alpha_{\mathcal{PAD}}$ and $\beta_{\mathcal{PAD}}$ satisfy equations (6.1)-(6.8) and the conditions of Lemma 7.4. For the former, we give an explicit proof of the fact that $\alpha_{\mathcal{PAD}}$ and $\beta_{\mathcal{PAD}}$ satisfy equation (6.3) as an example (the proof of the others is similar). In fact, we only need to prove

[TABLE]

We can prove the above equation in the following cases: (I) $x,y\in A$; (II) $x,y\in A^{*}$; (III) $x\in A,y\in A^{*}$ and (IV) $x\in A^{*},y\in A$. As an example, we give an explicit proof of the first case (the proof of the others is similar). Let $x=e_{i},y=e_{j}$, then the above equation becomes

[TABLE]

The coefficient of $e_{m}\otimes e_{n}$ on the left hand side of the above equation $(*)$ is

[TABLE]

The coefficient of $e_{m}\otimes e_{n}$ on the right hand side of the equation $(*)$ is

[TABLE]

So the coefficients of $e_{m}\otimes e_{n}$ on the both sides of the equation $(*)$ are the same. Similarly, the coefficients of $e_{m}^{*}\otimes e_{n},e_{m}\otimes e_{n}^{*}$ and $e_{m}^{*}\otimes e_{n}^{*}$ on both sides of the equation $(*)$ are the same.

On the other hand, we prove that $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{i}=0,i=1,2,3,4$. As an example we give an explicit proof of the fact that $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{1}=0$. In fact, the coefficient of $e_{m}\otimes e_{n}\otimes e_{p}$ in $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{1}(e_{k})$ is

[TABLE]

Similarly, the coefficients of $e_{m}^{*}\otimes e_{n}\otimes e_{p},e_{m}\otimes e_{n}^{*}\otimes e_{p},e_{m}^{*}\otimes e_{n}^{*}\otimes e_{p},e_{m}\otimes e_{n}\otimes e_{p}^{*},e_{m}^{*}\otimes e_{n}\otimes e_{p}^{*},e_{m}\otimes e_{n}^{*}\otimes e_{p}^{*}$ and $e_{m}^{*}\otimes e_{n}^{*}\otimes e_{p}^{*}$ in $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{1}(e_{k})$ are all zero. Therefore, $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{1}(e_{k})=0$. By a similar study, $S_{\alpha_{\mathcal{PAD}},\beta_{\mathcal{PAD}}}^{1}(e_{k}^{*})=0$. Hence $\mathcal{PAD}(A)$ is a pre-alternative bialgebra.

For $e_{i}\in A$, we have

[TABLE]

Similarly we have $\beta_{\mathcal{PAD}}(e_{i})=\beta(e_{i})$, so the inclusion $i_{1}:A\rightarrow A\oplus A^{*}$ is a homomorphism of pre-alternative bialgebras. Similarly, the inclusion $i_{2}:A^{*}\rightarrow A\oplus A^{*}$ is also a homomorphism of pre-alternative bialgebras.∎

Definition 7.9**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative bialgebra. With the pre-alternative bialgebra structure given in Theorem 7.8, $A\oplus A^{*}$ is called a Drinfeld’s symplectic double of $A$. We denoted it by $\mathcal{PAD}(A)$.**

Proposition 7.10**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative bialgebra with $\alpha,\beta$ defined by equations (7.1) and (7.2), where $r_{\prec}=r_{\succ}=r\in A\otimes A$ and $r$ is a solution of $PA$-equations. Then $T_{r}$ is a homomorphism of pre-alternative bialgebras from the pre-alternative bialgebra given in Example 6.6 to $(A,\prec,\succ,\alpha,\beta)$.

Proof.

Note that $(1\otimes\alpha)r=r_{12}\prec r_{13}$ and $(\alpha\otimes 1)r=r_{13}\prec r_{23}$. Denote the pre-alternative algebra structure on $A^{*}$ induced by $\alpha^{*}$ and $\beta^{*}$ by $\prec_{*}$ and $\succ_{*}$ respectively and the pre-alternative algebra structure $\prec,\succ$ on $A$ is defined by the linear maps $\gamma^{*},\delta^{*}:A\otimes A\rightarrow A$ respectively. Then

[TABLE]

[TABLE]

where $a^{*},b^{*}\in A^{*}$. Similarly we have $T_{r}(a^{*}\succ_{*}b^{*})=T_{r}(a^{*})\succ T_{r}(b^{*})$ and $(T_{r}\otimes T_{r})\delta(a^{*})=\beta(T_{r}(a^{*}))$. So the conclusion holds.∎

8. $PA$-equations and their properties

The simplest way to satisfy the conditions of Theorem 7.7 is given as follows.

Proposition 8.1**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $r\in A\otimes A$ be symmetric. Let $\alpha,\beta:A\rightarrow A\otimes A$ be two linear maps defined by equations (7.1) and (7.2). Then $(A,\prec,\succ,\alpha,\beta)$ is a pre-alternative bialgebra if $r$ satisfies $PA$-equations.

Proposition 8.2**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Let $r\in A\otimes A$ be a symmetric solution of $PA$-equations in $A$. Then the pre-alternative algebra structure “$\prec_{\bullet},\succ_{\bullet}$” on the Drinfeld’s symplectic double $\mathcal{PAD}(A)$ is given as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $x\in A,a^{*},b^{*}\in A^{*}$, the pre-alternative algebra structure on $A^{*}$ is denoted by “$\prec_{*},\succ_{*}$” and the associated alternative algebra structure on $As(A^{*})$ is denoted by “$\ast$”.

Proof.

Let $\{e_{1},...,e_{n}\}$ be a basis of $A$ and $\{e_{1}^{*},...,e_{n}^{*}\}$ be its dual basis. Suppose that

[TABLE]

Then $T_{r}(e_{i}^{*})=\sum\limits_{k}a_{ki}e_{k}$. Thus (for any $k,l$)

[TABLE]

So equation (8.1) holds. Similarly equation (8.2) holds. Therefore

[TABLE]

Therefore equation (8.4) follows from the fact that $e_{m}\succ_{\bullet}e_{k}^{*}=\mathfrak{r}_{\circ}^{*}(e_{m})e_{k}^{*}-\mathfrak{l}_{\prec_{*}}^{*}(e_{k}^{*})e_{m}$. Similarly, we can get the other equations. ∎

Proposition 8.3**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Let $r\in A\otimes A$ be symmetric. Then $r$ is a solution of one of $PA_{j}^{i}$-equations ($i=1,2,j=1,2,3$) if and only if $T_{r}$ satisfies

[TABLE]

that is, $T_{r}$ is an $Al$-operator of $As(A)$ associated to the bimodule $(A^{*},\mathfrak{r}_{\prec}^{*},\mathfrak{l}_{\succ}^{*})$. So in this case $PA_{j}^{i}$-equations ($i=1,2,j=1,2,3$) are all equivalent. Moreover, if $r$ is a solution of one of $PA_{j}^{i}$-equations ($i=1,2,j=1,2,3$), then there is a pre-alternative algebra structure on $A^{*}$ given by

[TABLE]

The associated alternative algebra structure $As(A^{*})$ is the same as the one given by equations (8.1) and (8.2) which is induced by $r$ in the sense of coboundary pre-alternative bialgebras.

Proof.

It follows from a similar proof as of Proposition 3.6. ∎

Definition 8.4**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. A bilinear form $\mathcal{B}:A\otimes A\rightarrow\textbf{k}$ is called a 2-cocycle of $A$ if

[TABLE]

Proposition 8.5**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. Let $\mathcal{B}$ be a 2-cocycle of $(A,\prec,\succ)$. Then the bilinear form $\omega$ defined by

[TABLE]

is a closed form on $As(A)$.

Proof.

Straightforward.∎

Proposition 8.6**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $r\in A\otimes A$. Suppose that $r$ is symmetric and nondegenerate. Then $r$ is a solution of one of $PA_{j}^{i}$-equations ($i=1,2,j=1,2,3$) in $(A,\prec,\succ)$ if and only if the (nondegenerate) bilinear form $\mathcal{B}$ induced by $r$ through equation (1.10) is a 2-cocycle of $(A,\prec,\succ)$.

Proof.

Let $r=\sum_{i}a_{i}\otimes b_{i}$. Since $r$ is symmetric, we have $\sum_{i}a_{i}\otimes b_{i}=\sum_{i}b_{i}\otimes a_{i}$. Therefore $T_{r}(v^{*})=\sum_{i}\langle v^{*},a_{i}\rangle b_{i}=\sum_{i}\langle v^{*},b_{i}\rangle a_{i}$ for any $v^{*}\in A^{*}$. Since $r$ is nondegenerate, for any $x,y,z\in A$, there exist $u^{*},v^{*},w^{*}\in A^{*}$ such that $x=T_{r}(u^{*}),y=T_{r}(v^{*}),z=T_{r}(w^{*})$. Therefore

[TABLE]

Hence $\mathcal{B}$ is a 2-cocycle of $(A,\prec,\succ)$ if and only if equation (7.8) holds. By Proposition 8.3, the conclusion follows. ∎

Corollary 8.7**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $r\in A\otimes A$. Assume $r$ is symmetric and there exists a nondegenerate symmetric bilinear form $h(x,y)$ on $A$ which is associative in the sense that

[TABLE]

Define a linear map $\varphi:A\rightarrow A^{*}$ by $\langle\varphi(x),y\rangle=h(x,y)$. Then $\tilde{T}_{r}=T_{r}\varphi:A\rightarrow A$ is an $Al$-operator associated to the bimodule $(A,\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$ if and only if $r$ is a symmetric solution of $PA$-equations. In this case, $\tilde{T}_{r}$ satisfies the following equation:

[TABLE]

So we can define a pre-alternative algebra structure on $A$ by

[TABLE]

Proof.

It follows from a similar proof as of Corollary 3.7.∎

Remark 8.8**.**

In fact, a symmetric bilinear form on a pre-alternative algebra $(A,\prec,\succ)$ satisfying equation (8.11) is a 2-cocycle of $(A,\prec,\succ)$.**

By a similar proof as of Proposition 3.9, we have the following conclusion:

Proposition 8.9**.**

Let $(A,\circ)$ be an alternative algebra. Let $(V,L,R)$ be a bimodule of $A$ and $(V^{*},R^{*},L^{*})$ be its dual bimodule. Suppose that $Al:V\rightarrow A$ is an $Al$-operator associated to $(V,L,R)$. Then $r=Al+\tau Al$ is a symmetric solution of $PA$-equations in $Al(V)\ltimes_{0,L^{*},R^{*},0}V^{*}$, where $Al(V)\subset A$ is a pre-alternative algebra given by equation (2.12) and $(V^{*},0,L^{*},R^{*},0)$ is a bimodule of $Al(V)$.

Corollary 8.10**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra. Then

[TABLE]

is a symmetric solution of $PA$-equations in $A\ltimes_{0,\mathfrak{l}_{\succ}^{*},\mathfrak{r}_{\prec}^{*},0}A^{*}$, where $\{e_{1},...,e_{n}\}$ is a basis of $A$ and $\{e_{1}^{*},...,e_{n}^{*}\}$ is its dual basis. Moreover $r$ is nondegenerate and the induced $2$-cocycle $\mathcal{B}$ of $A\ltimes_{0,\mathfrak{l}_{\succ}^{*},\mathfrak{r}_{\prec}^{*},0}A^{*}$ is given by equation (3.2).

Proof.

It follows by letting $V=A,(L,R)=(\mathfrak{l}_{\succ},\mathfrak{r}_{\prec})$ and $Al={\rm id}$ in Proposition 8.9.∎

Corollary 8.11**.**

Let $(A,\prec,\succ)$ be a pre-alternative algebra and $(As(A),\circ)$ be the associated alternative algebra. If $r$ is a nondegenerate symmetric solution of $PA$-equations in $A$, then there is a new compatible pre-alternative algebra structure on $As(A)$ given by

[TABLE]

which is just the pre-alternative algebra structure given by

[TABLE]

where $\mathcal{B}$ is the nondegenerate symmetric 2-cocycle of $A$ induced by $r$ through equation (1.10).

Proposition 8.12**.**

Let $(A,\prec,\succ,\alpha,\beta)$ be a pre-alternative bialgebra arising from a symmetric solution $r$ of $PA$-equations and $A^{*},-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*},-\mathfrak{r}_{\succ_{*}}^{*},\mathfrak{l}_{\ast}^{*},\mathfrak{r}_{\ast}^{*},-\mathfrak{l}_{\prec_{*}}^{*})$ be the corresponding matched pair of pre-alternative algebras.

${\rm(1)}$* $A\bowtie_{-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*}}^{-\mathfrak{r}_{\succ_{*}}^{*},\mathfrak{l}_{\ast}^{*},\mathfrak{r}_{\ast}^{*},-\mathfrak{l}_{\prec_{*}}^{*}}A^{*}$ is isomorphic to $A\ltimes_{-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*}}A^{*}$ as pre-alternative algebras.*

${\rm(2)}$* The symmetric solutions of $PA$-equations are in one-to-one correspondence with linear maps $T_{r}:A^{*}\rightarrow A$ whose graphs are Lagrangian pre-alternative subalgebras (with respect to the bilinear form $(3.11)$) of $A\ltimes_{-\mathfrak{r}_{\succ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\circ}^{*},-\mathfrak{l}_{\prec}^{*}}A^{*}$.*

Proof.

It follows from a similar proof as of Proposition 3.13. ∎

9. Comparison between alternative D-bialgebras and pre-alternative bialgebras

The results in the previous sections allow us to compare the alternative D-bialgebras (see Appendix A) and pre-alternative bialgebras in terms of the following properties: matched pairs of alternative algebras, alternative algebra structures on the direct sum of the alternative algebras in the matched pairs, bilinear forms on the direct sum of the alternative algebras in the matched pairs, double structures on the direct sum of the alternative algebras in the matched pairs, algebraic equations associated to coboundary cases, nondegenerate solutions, $Al$-operators of alternative algebras and constructions from pre-alternative algebras. We list them in Table 1.

From Table 1 we observe that there is a clear analogy between the alternative D-bialgebras and pre-alternative bialgebras. Moreover, due to the correspondences between certain symmetries and skew-symmetries appearing in the table, we regard it as a kind of duality.

Appendix: Another approach to alternative D-bialgebras

In this section we prove the main results of [G] by using a slightly different method (in fact, we will prove some results which are a little stronger than [G]). Moreover, there will be some more results like the “Drinfeld’s double” theorem for an alternative bialgebra (Theorem A11) and a homomorphism property of an alternative bialgebra (Theorem A12), which were not presented in [G]. We omit the proofs since they are quite similar to the study of pre-alternative bialgebras.

Theorem A1  Let $(A,\circ)$ be an alternative algebra and $(A^{*},\star)$ be an alternative algebra induced by a linear map $\Delta:A\rightarrow A\otimes A$. Then $(A,\circ,\Delta)$ is an alternative $D$-bialgebras if and only if $(A,A^{*},\mathfrak{r}_{\circ}^{*},\mathfrak{l}_{\circ}^{*},\mathfrak{r}_{\star}^{*},\mathfrak{l}_{\star}^{*})$ is a matched pair of alternative algebras.

Theorem A2 Let $(A,\circ,\Delta)$ be an alternative algebra equipped with a linear map $\Delta:A\rightarrow A\otimes A$ such that $\Delta^{*}:A^{*}\otimes A^{*}\rightarrow A^{*}$ induces an alternative algebra structure on $A^{*}$. Then $(A,A^{*},\mathfrak{r}_{\circ},\mathfrak{l}_{\circ},\mathfrak{r}_{\star},\mathfrak{l}_{\star})$ is a matched pair of alternative algebras if and only if the following equations hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $x,y\in A$, the multiplication $\star$ is induced by $\Delta$ and $\mathfrak{Ass}_{\circ}=\mathfrak{l}_{\circ}+\mathfrak{r}_{\circ}$.

Remark A3 Note that equations (A.1) and (A.4) have already appeared in [G] (Lemma 2).

Definition A4 An alternative bialgebra $(A,\circ,\Delta)$ is called coboundary if there exists an $r\in A\otimes A$ such that $\Delta$ is given by

[TABLE]

Remark A5 Note that the above definition is as the same as Definition 3.3.

Lemma A6 Let $A$ be a vector space. Then a linear map $\Delta:A\rightarrow A\otimes A$ induces an alternative algebra structure on $A^{*}$ if and only if the following two equations hold: (for any $x,y\in A$)

[TABLE]

[TABLE]

Definition A7 Let $(A,\circ)$ be an alternative algebra. The following equations are called $A_{i}$-equations ($i=1,2$) in $A$ respectively:

[TABLE]

[TABLE]

Note that $A_{1}$ given by equation (A.9) is exactly $C_{A}(r)$ given by equation (3.5).

Proposition A8 *Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. If $r$ is skew-symmetric, then $A_{1}$ and $A_{2}$ are the same. *

Proposition A9 Let $(A,\circ)$ be an alternative algebra. Let $r\in A\otimes A$ be skew-symmetric. Define a linear map $\Delta:A\rightarrow A\otimes A$ by equation (A.5). Then $\Delta$ induces an alternative algebra structure on $A^{*}$ if and only if the following equations hold:

[TABLE]

[TABLE]

for any $x\in A$.

Theorem A10 ([G], Theorem 2) Let $(A,\circ)$ be an alternative algebra and $r\in A\otimes A$. Let $\Delta:A\rightarrow A$ be a linear map defined by equation (A.5). If $r$ is a skew-symmetric solution of alternative Yang-Baxter equation in $A$, then $(A,\circ,\Delta)$ is an alternative bialgebra.

Theorem A11 Let $(A,\circ,\Delta_{A})$ be an alternative bialgebra. Then there is a canonical alternative bialgebra structure on $A\oplus A^{*}$ such that the inclusion $i_{1}:A\rightarrow A\oplus A^{*}$ is a homomorphism of alternative bialgebras, which the alternative bialgebra structure on $A$ is given by $(A,\circ,-\Delta_{A})$ and the inclusion $i_{2}:A^{*}\rightarrow A\oplus A^{*}$ is a homomorphism of alternative bialgebras, which the alternative bialgebra structure on $A^{*}$ is given by $(A^{*},\ast,\Delta_{B})$, where “$\ast$” is induced by $\Delta_{A}$ and the alternative algebra structure “$\circ$” on $A$ is induced by $\Delta_{B}:A^{*}\rightarrow A^{*}\otimes A^{*}$.

Theorem A12 Let $(A,\circ,\Delta)$ be an alternative bialgebra arising from a solution $r$ of alternative Yang-Baxter equation in $A$. Then $T_{r}$ is a homomorphism of alternative bialgebras from $(A^{*},\ast,\delta)$ to $(A,\circ,-\Delta)$, where “$\ast$” is induced by $\Delta$ and the alternative algebra structure “$\circ$” on $A$ is induced by $\delta:A^{*}\rightarrow A^{*}\otimes A^{*}$.

Acknowledgements

The authors thank Professor J.-L. Loday for useful suggestion. The first author also thanks Professor V.N. Zhelyabin for kindly sending his paper to him. This work was supported in part by the National Natural Science Foundation of China (10621101), NKBRPC (2006CB805905) and SRFDP (200800550015).