Medical, Pharma, Engineering, Science, Technology and Business

College of Mathematics and Information Science, Henan Normal University,
Xinxiang 453007, Henan Province, China, **E-mail:** [email protected].edu.cn

**Received** June 26, 2009 **Revised** November 02, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.

As an infinitesimal or semiclassical structures underlying the theory of quantum groups, the notion of Lie bialgebras was introduced by Drinfeld in his remarkable report [3], where he also introduced the double Lie bialgebra as an important construction. Some years later, the theory of matched pairs of Lie algebras was introduced by Majid in [4]. Its bicrossed product (or double cross sum) is more general than Drinfeld's classical double because and need not have the same dimension and the actions need not be strictly coadjoint ones.

Since then it was found that many other structures in Hopf algebras can be constructed in the infinitesimal setting, see [5] and the references cited therein. Also in [6], Majid introduced the concept of braided Lie bialgebras and proved the bosonisation theorem (see Theorem 4.4) associating braided Lie bialgebras to ordinary Lie bialgebras. Examples of braided Lie bialgebras were also given there. On the other hand, there is a close relation between extension theory and cross product Lie bialgebras, see Masuoka [7].

Natural and interesting questions are the following: Are there any more general results on this to uniform all the existing results together? What conditions will be needed? We give answers to these questions in the present paper. Majid's and Masuoka's results will be generalized as corollaries of our main results.

This paper is organized as follows. In Section 2, we recall some definitions and fix some notations. In Section 3, we review the notion of matched pairs of Lie coalgebras as dual version of Majid's matched pairs of Lie algebras. In Section 4, we construct double cross biproduct Lie bialgebras through two braided Lie bialgebras. In Section 5, we construct bicycle bicrossproduct Lie bialgebras, which is a generalization of Masuoka's cross product Lie bialgebras. The main results are stated in Theorems 4.6 and 5.7.

Throughout this paper, all vector spaces will be over a fixed field of character zero. The identity map of a vector space V is denoted by id

**Definition 2.1** (see [8]). A *Lie coalgebra* is a vector space L equipped with a linear map called a cobracket, satisfying the co-anticommutativity and the co-Jacobi
identity:

where are twistings defined by

We would like to use the sigma notation for all to denote the cobracket, then the above conditions can be written as

In fact, (CL2) is equivalent to one of the following two equations by (CL1):

**Definition 2.2.** A Lie bialgebra H is a vector space equipped simultaneously with a Lie
algebra structure (H; [ ; ]) and a Lie coalgebra (H; *δ*) structure such that the following compatibility
condition is satisfied:

and we denote it by (H; [ ; ]; *δ*).

Note that (LB) is equal to

We can also write them as ad-actions on tensors by

Let *A;H* be both Lie algebras and Lie coalgebras; for denote maps

by

We now fix some notions.

For a Lie algebra H and a linear map such that

then (A,α) is called a left H-module. For a Lie coalgebra H and a linear map such that

then is called a left H-comodule. If H and A are Lie algebras, A is a left H-module
and then is called a left *H*-module Lie algebra. If *H* is a Lie coalgebra and *A* is a Lie algebra, *A* is a left *H*-comodule and then *A* is called a left *H*-comodule Lie
algebra. Right Lie (co)module and Lie (co)module Lie (co)algebra can be defined similarly,
see also [5].

The general theory of matched pairs of Lie algebras was introduced by Majid in his Ph.D. thesis and published in [4]. He summarized his theory in his book [5]. In this section, we review the notion of matched pairs of Lie coalgebras, which is the dual version of Majid's matched pairs of Lie algebras. We will use it to construct double matched pairs in the next section.

**Definition 3.1** (see [5, Definition 8.3.1]). Assume that A and H are Lie algebras. If (A,α)
is a left H-module, is a right A-module, and the following (BB1) and (BB2) hold,
then (or (A;H)) is called a *matched pair of Lie algebras*:

**Lemma 3.2** (see [4]). *Let (A;H) be a matched pair of Lie algebras, then we get a new Lie
algebra on the vector space* *with bracket given by*

*We denoted it by *

**Proposition 3.3** (see [5, Proposition 8.3.4]). *Assume that A and H are Lie bialgebras,
(A;H) is a matched pair of Lie algebras, A is a left H-module Lie coalgebra, and H is a
right A-module Lie coalgebra. If*

*i.e.,*

*then* *becomes a Lie bialgebra.*

**Example 3.4.** A skew-pairing between Lie bialgebras is a linear map such
that (see [5])

For such a skew-pair, we can define a right A-module of H as since the right-hand sides of

and

are equal because H is a Lie coalgebra. Similarly, there is a left H-module of A by It is easy to see that they form a matched pair of Lie algebras and the condition in Proposition 3.3 holds. In this case becomes a Lie bialgebra which we denote by This is a generalization of Drinfeld's double Lie bialgebra when

**Definition 3.5.** Two Lie coalgebras (A;H) form a matched pair of Lie coalgebras if is a left *H*-comodule and is a right A-comodule, obeying the conditions

In sigma notation, the above conditions are

**Lemma 3.6.** *Let (A;H) be a matched pair of Lie coalgebras, we define* *as the
vector space* *with Lie cobracket * *that is*,

*this makes* *into a Lie coalgebra.*

**Proof.** It is easy to show that *δ _{D}* satisfies the coanticommutitive condition. We now intend
to show By definition of

It follows from the braided Jacobi identity of A that (1) + (10) + (19) = 0. Since A is a left
H-comodule, By the
condition (BB3) of matched pair of braided Lie coalgebras, Therefore, (CL2) holds
on A. Similarly, (CL2) holds on *H*. Hence (CL2) holds on the direct sum space .

**Proposition 3.7.** *Let (A;H) be a matched pair of Lie coalgebras with both of them Lie
bialgebras, and A is left H-comodule Lie algebra, H is right A-comodule Lie algebra, such
that*

i.e.,

*then the direct sum Lie algebra structure makes into a Lie bialgebra. We call it the
double cross coproduct Lie bialgebra*.

**Proof.** The Lie coalgebra structure is as in Lemma 3.6. The Lie algebra structure is as in . We only check the condition (LB). There are four cases: and We only check the first two cases. For

Now because A is a Lie bialgebra, we get Since A is a left H-comodule Lie algebra, Hence the compatibility condition (LB) is valid on A. For

Now because H is a Lie bialgebra, we get Since *H* is a right
A-comodule Lie algebra, Hence the compatibility
condition (LB) is valid on *H*.

**Example 3.8.** A skew-copairing between Lie bialgebras is a linear map which is denoted by

such that

For such a skew-copair, we can define a right A-comodule of H by a
left *H*-module of A by It is easy to see that they form a matched pair
of Lie coalgebras and the condition in Proposition 3.7 holds. In this case becomes a
Lie bialgebra which we denote by This is a generalization of Drinfeld's codouble
Lie bialgebra when

The concept of Yetter-Drinfeld modules over Lie bialgebras was introduced by Majid in [6] (where he call it Lie crossed modules), which he used to construct biproduct Lie bialgebras. This is a Lie version of biproduct Hopf algebras introduced by Radford in [9].

**Definition 4.1.** Let H be simultaneously a Lie algebra and a Lie coalgebra. If V is a left-*H* module and left *H*-comodule, satisfying

then *V* is called a left Yetter-Drinfeld module over *H*.

We denote the category of Yetter-Drinfeld modules over *H* by . It can be shown that forms a monoidal category if H is Lie bialgebra [6].

**Definition 4.2.** Let A be simultaneously a Lie algebra and Lie coalgebra. If V is a right
A-module and right A-comodule, satisfying

then *V* is called a right Yetter-Drinfeld module over A.

We denote the category of Yetter-Drinfeld modules over A by .

The next condition was also introduced by Majid in [6], which is a modification of the condition (LB).

**Definition 4.3.** If A is a Lie algebra and a Lie coalgebra and H is a right Yetter-Drinfeld
module over A, we call H a *braided Lie bialgebra* in , if the following condition is satisfied:

(LBS) for H;

where

**Theorem 4.4** (see [6, Theorem 3.7]). *If A is a Lie bialgebra and H is a braided Lie bialgebra
in* , *then the biproduct** form an ordinary Lie bialgebra.*

In the following, we construct the double cross biproduct of braided Lie bialgebras. Firstly, we give conditions for A to be a braided Lie bialgebra in .

(LBS) for *A*;

where

**Definition 4.5.** Let *A, H* be both Lie algebras and Lie coalgebras, obeying the following
conditions:

then we call (*A, H*) a *double matched pair*.

Note that (BB5) and (BB8) have appeared in [5, Proposition 8.3.5] when Majid constructs Now we give the main result of this section.

**Theorem 4.6.** *Let (A;H) be matched pair of Lie algebras and Lie coalgebras and let (A;H)
be double matched pair, A is a braided Lie bialgebra in is a braided Lie bialgebra in , define the double cross biproduct of A and H, denoted by as
Lie algebra, as Lie coalgebra, then becomes a Lie bialgebra.*

**Proof.** First we check the axiom (LB) on For by (BB5), (BB6),
(YDB) we get the third equality below:

where

thus the fourth equality.

Next we investigate the case of (LB) on

where by (LBS) we get by (BB7) we get (11) - (12). In the next equality,

We remark at this moment how Theorem 4.6 generalized Majid's Theorem 4.4. On the
one hand, we get an ordinary Lie bialgebra through two braided Lie bialgebras A and H as
in Theorem 4.6, here A need not be a Lie bialgebra as in Theorem 4.4. On the other hand,
when the maps *β* and are zero maps, Theorem 4.6 reduces to Theorem 4.4.

This section is absolutely new, we construct bi-cycle bicrossproduct Lie bialgebras, which is a generalization of double cross biproduct. Let A and H be both simultaneously Lie algebrasand Lie coalgebras, A is a left H-module and a left H-comodule, H is a right A-module and a right A-comodule, denote maps

by

we always omit the sum notation.

Let *H* be a Lie algebra and for a left H-module *A*, an antisymmetric bilinear map σ : is a cocycle on *H* if and only if

Let A be a Lie algebra and for a right A-module H, an antisymmetric bilinear map is a cocycle on A if and only if

Let *H* be a Lie coalgebra and for a left *H*-comodule A, an antisymmetric bilinear map is a cycle on A if and only if

Let *A* be a Lie coalgebra and for a left *A*-comodule *H*, an antisymmetric bilinear map is a cycle on *H* if and only if

In the following definitions, we introduced the concept of cocycle Lie algebras and cycle Lie coalgebras, which are in fact not really ordinary Lie algebras and Lie coalgebras, but weaker structures than them.

Definition 5.1. (i) Let be an anticommutativity map on a vector space H satisfying (CC1)(we call it a cocycle over H), equipped with a anticommutativity map [ ; ] : satisfying the following cocycle Jacobi identity:

then we call *H* a left cocycle Lie algebra.

(ii) Let θ be an anticommutativity map on a vector space A satisfying (CC2)(we call it a cocycle over A), equipped with a anticommutativity map [ ; ] : satisfying the following cocycle Jacobi identity:

then we call A a right cocycle Lie algebra.

(iii) Let P be an anticocommutativity map on a vector space H satisfying (CC3) (we call it a cycle over H), equipped with a anticommutativity map satisfying the following cycle co-Jacobi identity:

then we call H a left cycle Lie coalgebra.

(iv) Let Q be an anticocommutativity map on a vector space A satisfying (CC4) (we call it a cocycle over A), equipped with a anticommutativity map satisfying the following cycle co-Jacobi identity:

then we call A a right cycle Lie coalgebra.Although the structure of cocycle Lie algebras and cycle Lie coalgebras may be interesting, we do not intend to devote on it. What we need is only the conditions from (CC5) to (CC8) when both sides of them become zero for H and A are ordinary Lie algebras and Lie coalgebras.

**Definition 5.2.** A cocycle cross product system is a pair of Lie algebras A and H, where A
is a left H-module, H is a right A-module, is a cocycle onis a cocycle on A and the following conditions are satisfied:

**Lemma 5.3.** *Let (A;H) be a cocycle cross product system of Lie algebras, then* *becomes a Lie algebra with brackets given by*

**Proof. **We should see

In fact,

By (TM2) we get the result. The other cases can be easily checked too.

**Definition 5.4.** A cycle cross coproduct system is a pair of Lie coalgebras A and H, where
A is left H-comodule, H is right A-comodule, is a cycle on is a cocycle on is a cycle over H and the following conditions are satisfied:

**Lemma 5.5.** *Let (A;H) be a cycle cross coproduct system of Lie algebras. Define*

*as the vector space with the Lie cobracket*

that is,

this make into a Lie coalgebra.

Proof. We only check the braided co-Jacobi identity on A. By definition of δ_{D} we have

It follows from the condition (CC8) on *A* that by (CC3) we haveby (TM3) we get

by (TBB3) we have

Therefore, (CL2) holds on *A*. Similarly, (CL2) holds on *H*.

The following conditions are needed by the next theorem. Note that (TBB5), . . . , (TBB8) are extended from (BB5), . . . , (BB8); (TLB3) and (TLB4) are extended from (LBS); (TYD) from (YDB). Here (TLB1) and (TLB2) are new ones.

**Definition 5.6.** (i) A *left bi-cycle braided Lie bialgebras* H is simultaneously a left cocycle
Lie algebra and a left cycle Lie coalgebra satisfying the above condition (TLB4).

(ii) A *right bi-cycle braided Lie bialgebras* A is simultaneously a right cocycle Lie a algebra
and right cycle Lie coalgebra satisfying the above condition (TLB3).

The next theorem says that we can get an ordinary Lie bialgebra from two bi-cycle braided Lie bialgebras.

**Theorem 5.7.** *Let (A;H) be a cocycle cross product system and a cycle cross coproduct
system, then the cocycle cross product Lie algebra and cycle cross coproduct Lie coalgebra
t together to form an ordinary Lie bialgebra if the conditions (TBB5){(TBB8), (TLB1){
(TLB4), and (TYB) are satisfied. We call it the bi-cycle bicrossproduct Lie bialgebra and
denote it by* .

**Proof.** We investigate the case of (LB) on

Denote the right-hand side terms by (a), (b),...., (h):

Then by (TLB3) we get

by (TBB7) we get

by (TLB1) we get

We investigate the case of (LB) on

Denote the right-hand side terms by

Then by (TLB4) we get

by (TBB8) we get

by (TLB2) we get

We now check the axiom (LB) on we get the equality below:

Denote the right-hand side terms by

Then by (TBB5) we get

by (TBB6) we get

by (TYB) we get

When the maps are zeros in Theorem 5.7, we get Theorem 4.6. When the maps are zeros in Theorem 5.7, we get [7, Proposition 1.8]. In fact, any maps of can be zero and then we get other types of propositions. What we need to do is to let the corresponding maps in the conditions in Theorem 5.7 be zero, we left this to the reader.

After finishing the work, the author finds that the notion of "braided Lie bialgebra" has been introduced earlier by Sommerhäuser in his unpublished paper [10] to give another construction of symmetrizable Kac-Moody algebras, where he calls it "Yetter-Drinfeld Lie algebra." On the other hand, our construction is somewhat the Lie version of the construction of bialgebra in [1, 2], so this paper can have another name "Cross Product Lie Bialgebras."

The author would like to thank the referee for very helpful comments and suggestions.

- Bespalov Y, Drabant B (1999) Cross product bialgebras. I J Algebra 219: 466-505.
- Bespalov Y, Drabant B (2001) Cross product bialgebras. II J Algebra 240: 445-504.
- Drinfel'd VG (1987) Quantum groups. In \Proceedings of the International Congress of Mathematicians 1: 798-820.
- Majid S (1990) Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Paci c J Math 141: 311-332.
- Majid S (1995) Foundations of Quantum Groups. Cambridge University Press, Cambridge.
- Majid S (2000) Braided Lie bialgebras. Paci c J Math 192: 329-356.
- Masuoka A (2000) Extensions of Hopf algebras and Lie bialgebras. Trans Amer Math Soc 352: 3837-3879.
- Michaelis W (1980) Lie coalgebras. Adv Math 38: 1-54.
- Radford DE (1985) The structure of Hopf algebras with a projection. J Algebra 92: 322-347.
- Sommerhäauser Y (1996) Kac-Moody algebras. Presented at the Ring Theory Conference, Miskolc, Hungary.

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