Download 3-Lie bialgebras (Lb,Cd) and (Lb,Ce)

Mathematica Aeterna, Vol. 6, 2016, no. 2, 159 - 163
3-Lie bialgebras (Lb, Cd) and (Lb, Ce)
BAI Ruipu
College of Mathematics and Information Science,
Hebei University, Baoding, 071002, China
email: [email protected]
LIN Lixin
College of Mathematics and Information Science,
Hebei University, Baoding, 071002, China
email: [email protected]
Abstract
In this paper, we continue to study the structure of four dimensional
3-Lie bialgebras. We discuss the existence of 3-Lie bialgebras of types
(Lb , Cd ) and (Lb , Ce ). It is proved that there do not exist 3-Lie bialgebras of types (Lb1 , Ce ), (Lb2 , Ce ) and (Lb2 , Cd ). There exists only one
class of 3-Lie bialgebras of type (Lb , Cd ), that is (Lb1 , Cd , ∆1 ) ( Theorem
3.4 ).
2010 Mathematics Subject Classification: 17B05 17D30
Keywords: 3-Lie algebra, 3-Lie coalgebra, 3-Lie bialgebra.
1
Introduction
In 1985, Filippov provided n-Lie algebras [1]. Since then, the n-Lie algebra,
especially, the 3-Lie algebra attracts more and more attention, and it is widely
used in mathematics, mathematical physics and string theory. And some n-ary
algebras such as n-Hom algebras, n-supper algebras, n-Rota-Baxter algebras,
etc. are provided and studied (see [2, 3, 4]). Authors in paper [5] introduced
3-Lie coalgebras and 3-Lie bialgebras, and constructed some 4-dimensional 3Lie bialgebras. In paper [6], 4-dimensional 3-Lie bialgebras of type (Lb , Cb)
are classified. It is proved that there exist seven classes of 4-dimensional 3-Lie
bialgebras of type (Lb , Cb ). In this paper, we discuss 3-Lie bialgebras of types
(Lb , Cd ) and (Lb , Ce ), that is, the 3-Lie algebras with one-dimensional derived
algebra and 3-Lie coalgebras with three and four dimensional derived algebra,
160
Bai Ruipu, Lixin Lin
which is denoted by (Lb , Cd ) and (Lb , Ce ), respectively. And we suppose that
3-Lie algebras and 3-Lie coalgebras are over a field F of characteristic zero,
and omit the zero multiplication of basis vectors in 3-Lie algebras and 3-Lie
coalgebras.
2
Preliminaries
A 3-Lie algebra [1] is a vector space L endowed with a linear multiplication
µ : L∧3 → L satisfying that, for all x, y, z, u, v ∈ L,
µ(u, v, µ(x, y, z)) = µ(x, y, µ(u, v, z))+µ(y, z, µ(u, v, x))+µ(z, x, µ(u, v, y)).
For defining 3-Lie coalgebras, we need to define following linear mapps
ωi : L ⊗ L ⊗ L ⊗ L ⊗ L → L ⊗ L ⊗ L ⊗ L ⊗ L, 1 ≤ i ≤ 3, by
ω1 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x3 ⊗ x4 ⊗ x1 ⊗ x2 ⊗ x5 ,
ω2 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x4 ⊗ x5 ⊗ x1 ⊗ x2 ⊗ x3 ,
ω3 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x5 ⊗ x3 ⊗ x1 ⊗ x2 ⊗ x4 .
A 3-Lie coalgebra (L, ∆) [5] is a vector space L with a linear map ∆ :
L → L ⊗ L ⊗ L satisfying
Im(∆) ⊂ L ∧ L ∧ L, and (1 − ω1 − ω2 − ω3 )(1 ⊗ 1 ⊗ ∆)∆ = 0,
where 1 : L⊗5 → L⊗5 is identity.
Let (L1 , ∆1 ) and (L2 , ∆2 ) be 3-Lie coalgebras. If there is a linear isomorphism ϕ : L1 → L2 satisfying (ϕ ⊗ ϕ ⊗ ϕ)(∆1 (e)) = ∆2 (ϕ(e)), for all e ∈ L1 ,
then (L1 , ∆1 ) is isomorphic to (L2 , ∆2 ), and ϕ is called a 3-Lie coalgebra isoP
P
morphism, where (ϕ ⊗ ϕ ⊗ ϕ) (ai ⊗ bi ⊗ ci ) = ϕ(ai ) ⊗ ϕ(bi ) ⊗ ϕ(ci ).
i
i
A 3-Lie bialgebra [5] is a triple (L, µ, ∆) such that
(1) (L, µ) is a 3-Lie algebra with the multiplication µ : L ∧ L ∧ L → L,
(2) (L, ∆) is a 3-Lie coalgebra with ∆ : L → L ∧ L ∧ L,
(3) ∆ and µ satisfy the following identities, for x, y, u, v, w ∈ L,
(3)
(3)
∆µ(x, y, z) = ad(3)
µ (x, y)∆(z) + adµ (y, z)∆(x) + adµ (z, x)∆(y),
(3)
(3)
where ad(3)
µ (x, y), adµ (z, x), adµ (y, z) : L ⊗ L ⊗ L → L ⊗ L ⊗ L are linear
(3)
maps defined by (similar for ad(3)
µ (z, x) and adµ (y, z))
ad(3)
µ (x, y)(u ⊗ v ⊗ w) = (adµ (x, y) ⊗ 1 ⊗ 1)(u ⊗ v ⊗ w)
+(1 ⊗ adµ (x, y) ⊗ 1)(u ⊗ v ⊗ w) + (1 ⊗ 1 ⊗ adµ (x, y))(u ⊗ v ⊗ w)
= µ(x, y, u) ⊗ v ⊗ w + u ⊗ µ(x, y, v) ⊗ w + u ⊗ v ⊗ µ(x, y, w).
Two 3-Lie bialgebras (L1 , µ1 , ∆1 ) and (L2 , µ2 , ∆2 ) are called equivalent if
there exists a linear isomorphism f : L1 → L2 such that
(1) f : (L1 , µ1 ) → (L2 , µ2 ) is a 3-Lie algebra isomorphism,
(2) f : (L1 , ∆1 ) → (L2 , ∆2 ) is a 3-Lie coalgebra isomorphism, that is,
∆2 (f (x)) = (f ⊗ f ⊗ f )∆1 (x) for all x ∈ L1 .
Lemma 2.1[1] Let (L, µ) be a 4-dimensional 3-Lie algebra with dim L1 6= 0, 2,
and e1 , e2 , e3 , e4 be a basis of L. Then L is isomorphic to one and only one of the
following
3-Lie bialgebras (Lb , Cd ) and (Lb , Ce )
161
Lb1 . µ(e2 , e3 , e4 ) = e1 . Lb2 . µ(e1 , e2 , e3 ) = e1 .
Ld . µd (e2 , e3 , e4 ) = e1 , µd (e1 , e3 , e4 ) = e2 , µd (e1 , e2 , e4 ) = e3 .
Le . µe (e2 , e3 , e4 ) = e1 , µe (e1 , e3 , e4 ) = e2 , µe (e1 , e2 , e4 ) = e3 , µe (e1 , e2 , e3 ) = e4 .
Lemma 2.2 [5] Let L be a vector space over F , and µ : L ⊗ L ⊗ L → L be a
3-ary linear map. Then (L, µ) is a 3-Lie algebra if and only if (L∗ , µ∗ ) is a 3-Lie
coalgebra with µ∗ : L∗ → L∗ ⊗ L∗ ⊗ L∗ , where µ∗ is the dual map of µ.
3
3-Lie bialgebras of types (Lb, Cd ) and (Lb, Ce)
First We give the classification of 3-Lie coalgebras of the types (L, Cd ) and (L, Ce ).
Lemma 3.1 Let (L, ∆) be a 4-dimensional 3-Lie coalgebra with m-dimensional
derived algebra (3 ≤ m ≤ 4), and e1 , e2 , e3 , e4 be a basis of L. Then L isomorphic
to one and only one of the following
Cd . ∆d (e1 ) = e2 ∧ e3 ∧ e4 , ∆d (e2 ) = e1 ∧ e3 ∧ e4 , ∆d (e3 ) = e1 ∧ e2 ∧ e4 ;
Ce . ∆e (e1 ) = e2 ∧ e3 ∧ e4 , ∆e (e2 ) = e1 ∧ e3 ∧ e4 , ∆e (e3 ) = e1 ∧ e2 ∧ e4 ,
∆e (e4 ) = e1 ∧ e2 ∧ e3 .
Proof The result follows from Lemma 2.1, Lemma 2.2 and a direct computation,
we omit the computation process.
For convenience, in the following, for a 3-Lie bialgebra (L, µ, ∆), if the 3-Lie
algebra (L, µ) is the case (L, µbi ) in Lemma 2.1 and the 3-Lie coalgebra (L, ∆) is
the case (L, ∆d ) and (L, ∆e ) in Lemma 3.1, then the 3-Lie bialgebra (L, µbi , ∆d ) and
(L, µbi , ∆e ) are simply denoted by (Lbi , Cd ) and (Lbi , Ce ), which are called the 3-Lie
bialgebras of type (Lb , Cd ), and (Lb , Ce ), respectively.
For a given 3-Lie algebra L, in order to find all the 3-Lie bialgebra structures on
L, we should find all the 3-Lie coalgebra structures on L which are compatible with
the 3-Lie algebra L. Although a permutation of a basis of L gives isomorphic 3-Lie
coalgebra, but it may lead to the non-equivalent 3-Lie bialgebra.
Theorem 3.2 There do not exist 3-Lie bialgebras of the types (Lb1 , Ce ) and
(Lb2 , Ce ).
Proof By Lemma 2.1 and 2.2, we need to verify that 3-Lie algebras Lb1 and Lb2
are incompatible with the following six isomorphic 3-Lie coalgebras of the type Ce :
(1).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e1 ∧e3 ∧e4 , ∆(e3 ) = e1 ∧e2 ∧e4 , ∆(e4 ) = e1 ∧e2 ∧e3 ;
(2).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e1 ∧e3 ∧e4 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e1 ∧e3 ;
(3).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e3 ∧e1 ∧e4 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e3 ∧e1 ;
(4).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e1 ∧e4 ∧e3 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e1 ∧e3 ;
(5).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e4 ∧e1 ∧e3 , ∆(e3 ) = e2 ∧e4 ∧e1 , ∆(e4 ) = e2 ∧e1 ∧e3 ;
(6).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e4 ∧e3 ∧e1 , ∆(e3 ) = e2 ∧e4 ∧e1 , ∆(e4 ) = e2 ∧e3 ∧e1 .
Here we only check the case (1) is incompatible with the 3-Lie algebra Lb1 . Since
∆µb1 (e2 , e3 , e4 ) = ∆(e1 ) = e2 ∧ e3 ∧ e4 , but
(3)
(3)
(3)
adµb1 (e2 , e3 )∆(e4 ) + adµb1 (e3 , e4 )∆(e2 ) + adµb1 (e4 , e2 )∆(e3 ) = 0,
we obtain that
(3)
(3)
(3)
∆µb1 e2 , e3 , e4 ) 6= adµb1 (e2 , e3 )∆(e4 ) + adµb1 (e3 , e4 )∆(e2 ) + adµb1 (e4 , e2 )∆(e3 ).
Therefore, the case (1) of the type Ce is incompatible with the 3-Lie algebra Lb1 .
Similar discussions for others cases, we get the result.
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Bai Ruipu, Lixin Lin
Theorem 3.3 There does not exist 3-Lie bialgebras of the type (Lb2 , Cd ).
Proof By Lemma 2.1 and 2.2, we need to verify that 3-Lie algebra Lb2 is incompatible with following twenty-four isomorphic 3-Lie coalgebras of the type Cd :
(1).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ;
(2).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ;
(3).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ;
(4).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ;
(5).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ;
(6).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ;
(7).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ;
(8).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ;
(9).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ;
(10).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ;
(11).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ;
(12).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ;
(13).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ;
(14).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ;
(15).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ;
(16).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ;
(17).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ;
(18).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ;
(19).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ;
(20).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ;
(21).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e2 ∧ e3 ∧ e1 ;
(22).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ;
(23).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e2 ∧ e3 ∧ e1 ;
(24).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 .
The discussion is completely similar to Theorem 3.2. We omit the computing process.
Theorem 3.4 The only non-equivalent 3-Lie bialgebras of the type (Lb1 , Cd ) is
(Lb1 , Cd , ∆1 ) : ∆1 (e2 ) = e1 ∧ e4 ∧ e3 , ∆1 (e3 ) = e1 ∧ e4 ∧ e2 , ∆1 (e4 ) = e1 ∧ e3 ∧ e2 .
Proof We need to discuss the compatibility of twenty-four isomorphic 3-Lie
coalgebras of the type Ccd in Theorem 3.3 with the 3-Lie algebra Lb1 .
By a direct computation, only cases (19), (20), (21), (22), (23) and (24) are
compatible with the 3-Lie algebra Lb1 , respectively. Thanks to isomorphisms of
3-Lie bialgebras:
(19) → (22) : f (e1 ) = −e1 , f (e2 ) = e3 , f (e3 ) = e2 , f (e4 ) = e4 ;
(21) → (23), (23) → (24) : f (e1 ) = e1 , f (e2 ) = e√
3 , f (e3 ) = e4 , f (e
√4 ) = e2 ;
−1e3 , f (e4 ) = −1e
(19) → (20) : f (e1 ) = e1 , f (e2 ) = −e2 , f (e3 ) = √
√ 4;
(19) → (21) : f (e1 ) = e1 , f (e2 ) = −e2 , f (e3 ) = − −1e3 , f (e4 ) = − −1e4 ;
we get that the only non-equivalent 3-Lie bialgebras of the type (Lb1 , Cd ) is (Lb1 , Cd , ∆1 ).
Acknowledgements
The first author (R.-P. Bai) was supported in part by the Natural Science Foundation (11371245) and the Natural Science Foundation of Hebei Province (A2014201006).
3-Lie bialgebras (Lb , Cd ) and (Lb , Ce )
163
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