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Journal of Generalized Lie Theory and Applications
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Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras

Tao ZHANG

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

Received June 26, 2009 Revised November 02, 2009

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Abstract

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.

Introduction

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 image as an important construction. Some years later, the theory of matched pairs of Lie algebras image was introduced by Majid in [4]. Its bicrossed product (or double cross sum) image is more general than Drinfeld's classical double image because image and image 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 image

Preliminaries

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

image

where image are twistings defined by

image

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

image

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

image

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:

image

and we denote it by (H; [ ; ]; δ).

Note that (LB) is equal to

image

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

image

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

image

by

image

We now fix some notions.

For a Lie algebra H and a linear map image such that

image

image

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

image

then image is called a left H-comodule. If H and A are Lie algebras, A is a left H-module and image thenimage 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 imageimage 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].

Matched pairs of Lie algebras and Lie coalgebras

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, image is a right A-module, and the following (BB1) and (BB2) hold, then image (or (A;H)) is called a matched pair of Lie algebras:

image

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 image with bracket given by

image

We denoted it by image

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

image

i.e.,

image

then image becomes a Lie bialgebra.

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

image

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

image

and

image

are equal because H is a Lie coalgebra. Similarly, there is a left H-module of A by imageimage 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 image becomes a Lie bialgebra which we denote byimage This is a generalization of Drinfeld's double Lie bialgebra whenimage

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

image

In sigma notation, the above conditions are

image

image

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

image

this makes image into a Lie coalgebra.

Proof. It is easy to show that δD satisfies the coanticommutitive condition. We now intend to show image By definition of δD we have

image

image

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

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

image

i.e.,

image

then the direct sum Lie algebra structure makes image 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 image. We only check the condition (LB). There are four cases: image and image We only check the first two cases. For imafe

image

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

image

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

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

image

such that

image

For such a skew-copair, we can define a right A-comodule of H by image a left H-module of A by image 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 image becomes a Lie bialgebra which we denote by image This is a generalization of Drinfeld's codouble Lie bialgebra when image

Yetter-Drinfeld modules and double cross biproduct

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

image

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

We denote the category of Yetter-Drinfeld modules over H by image. It can be shown that image 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

image

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

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

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 image, if the following condition is satisfied:

image (LBS) for H;

where

image

Theorem 4.4 (see [6, Theorem 3.7]). If A is a Lie bialgebra and H is a braided Lie bialgebra in image, then the biproductimage 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 image.

image (LBS) for A;

where

image

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

image

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 image 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 image is a braided Lie bialgebra in image, define the double cross biproduct of A and H, denoted by image as Lie algebra, image as Lie coalgebra, then image becomes a Lie bialgebra.

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

image

image

where

image

thus the fourth equality.

Next we investigate the case of (LB) on image

image

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

image

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 imageare zero maps, Theorem 4.6 reduces to Theorem 4.4.

Bicycle bicrossproduct Lie bialgebras

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

image

by

image

we always omit the sum notation.

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

image

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

image

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

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

image

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 [ ; ] : image satisfying the following cocycle Jacobi identity:

image

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 [ ; ] : image satisfying the following cocycle Jacobi identity:

image

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 image satisfying the following cycle co-Jacobi identity:

image

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 image satisfying the following cycle co-Jacobi identity:

image

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, image is a cocycle onimageis a cocycle on A and the following conditions are satisfied:

image

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

image

Proof. We should see

image

In fact,

image

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, image is a cycle on image is a cocycle onimage is a cycle over H and the following conditions are satisfied:

image

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

image

as the vector space image with the Lie cobracket

image

that is,

image

this make image into a Lie coalgebra.

Proof. We only check the braided co-Jacobi identity on A. By definition of δD we have

image

image

image

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

image

by (TBB3) we have

image

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.

image

image

image

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 image.

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

image

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

image

Then by (TLB3) we get

image

by (TBB7) we get

image

by (TLB1) we get

image

We investigate the case of (LB) on image

image

Denote the right-hand side terms by image

image

image

Then by (TLB4) we get

image

by (TBB8) we get

image

by (TLB2) we get

image

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

image

Denote the right-hand side terms by image

image

Then by (TBB5) we get

image

by (TBB6) we get

image

by (TYB) we get

image

When the maps image are zeros in Theorem 5.7, we get Theorem 4.6. When the mapsimage are zeros in Theorem 5.7, we get [7, Proposition 1.8]. In fact, any maps of imageimage 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.

Added remarks

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."

Acknowledgements

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

References

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