# A Kind of Infinite-Dimensional Novikov Algebras and Its Realizations

Authors: Liangyun Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We construct a kind of infinite-dimensional Novikov algebras and give its realization by hyperbolic sine functions and hyperbolic cosine functions.


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## Body

## 1. Introduction

Novikov algebras were introduced in connection with Hamiltonian operators in the formal variational calculus and the Poisson brackets of hydrodynamic type. They were used to construct the Virasoro-type Lie algebras. So the study of Novikov algebras is interesting in both mathematics and mathematical physics.When Gel’fand and Diki [1, 2] and Gel’fand and Dorfman [3] studied the following operator:
(1)Hij=∑k‍cijkuk(1)+dijkuk(0)ddx,cijk∈ℂ,dijk=cijk+cjik,
they gave the definition of Novikov algebras. Concretely, let cijk be the structural coefficients, and let a product of L=L(e0,e1,…) be ∘ such that
(2)ei∘ej=∑‍cijkek.
For any a,b,c∈L, the product is Hamilton operator if and only if ∘ satisfies
(3)(a∘b)∘c=(a∘c)∘b,(a∘b)∘c+c∘(a∘b)=(c∘b)∘a+a∘(c∘b).Ma presented many new soliton hierarchies of commuting bi-Hamiltonian evolution equations from the so-called Novikov algebras [4–6]. In 1987, Zel’manov [7] began to study Novikov algebras and proved that the dimension of finite-dimensional simple Novikov algebras over a field of characteristic zero is one. In algebras, what are paid attention to by mathematician are classifications and structures, but so far we have not got the systematic theory for general Novikov algebras. In 1992, Osborn [8–10] had finished the classification of infinite simple Novikov algebras with nilpotent elements over a field of characteristic zero and finite simple Novikov algebras with nilpotent elements over a field of characteristic p>0. In 1995, Xu [10–13] developed his theory and got the classification of simple Novikov algebras over an algebraically closed field of characteristic zero. Bai and Meng [14–16] did a series of researches on low dimensional Novikov algebras, such as the structure and classification. We construct two kinds of Novikov algebras [17]. Recently, people obtained some properties in Novikov superalgebras [18, 19]. In this paper, we construct an infinite-dimensional Novikov algebra and give its realization by hyperbolic sine functions and hyperbolic cosine functions.Definition 1 (see [17]).
Let(𝒜,∘) be an algebra over F such that
(4)a∘(b∘c)-(a∘b)∘c=b∘(a∘c)-(b∘a)∘c,(5)(a∘b)∘c=(a∘c)∘b,∀a,b,c∈𝒜,
and then 𝒜 is called a Novikov algebra over F.Remark 2.
An algebra𝒜 is called a left symmetric algebra if it only satisfies (4). It is clear that left symmetric algebras contain Novikov algebras.Remark 3.
(1) If(𝒜,∘) is a left symmetric algebra satisfying
(6)[a,b]=a∘b-b∘a,∀a,b∈𝒜,
then (𝒜,[,]) is a Lie algebra. Usually, it is called an adjoining Lie algebra.
(2) Let(𝒜,·) be a commutative algebra, and then (𝒜,d0,∘) is a Novikov algebra if d0 is a derivation of 𝒜 with a bilinear operator ∘ such that
(7)a∘b=a·d0(b),∀a,b∈𝒜.


## 2. Main Results

Lemma 4.
Let{b0,a1,b1,a2,b2,…an,bn,…} be a basis of the linear space 𝒜 over a field F of characteristic p≠2 satisfying
(8)aman=12(bm+n-bm-n),bmbn=12(bm+n+bm-n),ambn=bnam=12(am+n+am-n),
where b-m=bm, a-m=-am. Then 𝒜 is a commutative and associative algebra.Proof.
It is clear that𝒜 is a commutative algebra over F:
(9)(ak,an,am)=ak(anam)-(akan)am=12ak(bm+n-bn-m)-12(bk+n-bk-n)am=14(ak+m+n+ak-m-n-ak+n-m-ak-n+m-am+k+n-am-k-n+am+k-n+am-k+n)=0.
Similarly, we have that (bk,bn,bm)=(ak,an,bm)=(ak,bn,am)=(bk,an,am) = (bk,bn,am)=(bk,an,bm)=(ak,bn,bm)=0. Then (a,b,c)=0, ∀a,b,c∈𝒜. The result follows.Corollary 5.
b
0 of Lemma 4 is a unity of 𝒜.Lemma 6.
Let𝒜 be a commutative and associative algebra satisfying Lemma 4. Then the following statements hold:(1)
IfD0 is a linear transformation of 𝒜 such that
(10)D0(an)=nbn,n=1,2,3,…,D0(bn)=nan,n=0,1,2,…,thenD0 is a derivation of 𝒜.(2)
IfaD0 is a linear transformation of 𝒜 such that
(11)(aD0)(b)=aD0(b),∀a,b∈𝒜,thenaD0 is a derivation of 𝒜.(3)
𝒟1={aD0∣a∈𝒜} is a subalgebra of Lie algebra Der𝒜.Proof.
(1) We have(12)D0(anam)=D0(12(bn+m-bn-m))=12((m+n)an+m-(n-m)an-m),D0(an)am+anD0(am)=nbnam+manbm=n2(an+m-an-m)+m2(an+m-am-n)=12((m+n)am+n-(n-m)an-m).
SoD0 is a derivation of 𝒜.
(2) For∀a,b,c∈𝒜, we have
(13)(aD0)(bc)=aD0(bc)=aD0(b)c+abD0(c)=(aD0)(b)c+b(aD0)(c),
so aD0 is a derivation of 𝒜.
(3) For∀a,b,c∈𝒜, we have
(14)[aD0,bD0](c)=(aD0)(bD0)(c)-(bD0)(aD0)(c)=aD0(b)D0(c)-bD0(a)D0(c)=(aD0(b)-bD0(a))D0(c).
Then [aD0,bD0]=(aD0(b)-bD0(a))D0∈𝒟1, and so (3) holds.Theorem 7.
Let𝒜 be a commutative and associative algebra satisfying Lemma 4, and let a be an element of 𝒜. If D0 satisfies Lemma 6 and ∘ satisfies
(15)b∘c=baD0(c),∀b,c∈𝒜,
then the following statements hold:(1)
(𝒜,aD0,∘) is a Novikov algebra.(2)
(𝒜,aD0,[,]) is an adjoining Lie algebra of (𝒜,aD0,∘) and [,] such that
(16)[b,c]=a(bD0(c)-cD0(b)),∀b,c∈𝒜.Proof.
(1) By Lemma6, aD0 is a derivation of the commutative algebra 𝒜. So (𝒜,aD0,∘) is a Novikov algebra by Remark 3(2).
(2)(𝒜,aD0,[,]) is an adjoining Lie algebra of (𝒜,aD0,∘) by Remark 3(1). For ∀b,c∈𝒜,∃a∈𝒜, we have
(17)[b,c]=b∘c-c∘b=baD0(c)-caD0(b)=a(bD0(c)-cD0(b))
since 𝒜 is commutative. Hence we obtain the desired result.Letb0 be a unity of 𝒜. If we set a=b0 in Theorem 7, then an∘am=anb0D0(am)=an(mbm)=(m/2)(am+n+an-m). Similarly, we obtain the following corollary.Corollary 8.
Let𝒜 be a commutative and associative algebra satisfying Lemma 4. Then the following statements hold:
(18)an∘am=m2(an+m+an-m),bn∘bm=m2(an+m+am-n),an∘bm=m2(bn+m-bn-m),bn∘am=m2(bn+m+bn-m),[an,am]=12(m-n)an+m+12(m+n)an-m,[bn,bm]=12(m-n)an+m-12(m+n)an-m,[an,bm]=12(m-n)bn+m-12(n+m)bn-m,[bn,am]=12(m-n)bn+m+12(m+n)bn-m.We have the following: letsinhx=(ex-e-x)/2, coshx=(ex+e-x)/2, and let the field F be assumed R or C. We will construct Novikov algebras over the linear space which is generated by sinhx and coshx.First, let𝒯 be a linear space generated by {sinhmx,coshnx∣m,n∈N} over F.Lemma 9.
𝒯 satisfying the above product is a commutative associative algebra.Proof.
Since the above product is commutative and associative, we only need𝒯 to be closed for the product. In fact,
(19)sinhmxsinhnx=12[cosh(m+n)x-cosh(m-n)x],coshmxcoshnx=12[cosh(m+n)x+cosh(m-n)x],sinhmxcoshnx=12[sinh(m+n)x+sinh(m-n)x].
So 𝒯 is a commutative and associative algebra.Lemma 10.
Let𝒯 be a linear space generated by {sinhmx,coshnx∣m,n∈N} over F, and then {1,sinhmx,coshnx∣m,n∈N0} is a basis of 𝒯.Proof.
For∀n∈N0, suppose that there are c0,ai,bj∈F, i,j∈N0 such that
(20)c0+a1sinhx+b1coshx+⋯+ansinhnx+bncoshnx=0.
We take derivative for (20) such that its derivative order is 2k-1  (k∈N0), and put x=0. Then we have
(21)a1+22k-1a2+⋯+n2k-1an=0.
Let k=1,2,…,n, and then we obtain the following system of n linear equations:
(22)a1+2a2+⋯+nan=0a1+23a2+⋯+n3an=0⋮a1+22n-1a2+⋯+n2n-1an=0.
If a1,…,an are seen to be unknown, then the coefficient matrix of (22) is the Vandermonde matrix whose determinant is not 0, so ai=0, i=1,…,n.
We take derivative for (20) such that its derivative order is 2k (k∈N0), and put x=0. Then we have
(23)b1+22kb2+⋯+n2kbn=0.
Let k=1,2,…,n, and then we obtain the following system of n linear equations:
(24)b1+22b2+⋯+n2bn=0b1+24b2+⋯+n4bn=0⋮b1+22nb2+⋯+n2nbn=0.
If b1,…,bn are seen to be unknown, then the coefficient matrix of (24) is the Vandermonde matrix whose determinant is not 0, so bi=0, i=1,…,n. Since, for any i∈N0, ai=0 and bi=0 satisfy (20), we have c0=0. Hence {1, sinhx, coshx,…, sinhnx, coshnx} are linearly independent for any n∈N0, and then {1,sinhnx,coshmx∣n,m∈N0} are linearly independent and so they form a basis of 𝒯 as desired.Theorem 11.
Let𝒜1, 𝒜2 be commutative and associative algebras over F. If φ: 𝒜1→𝒜2 is an isomorphism and D1∈Der𝒜1, then the following statements hold:(1)
D2:=φD1φ-1∈Der𝒜2,(2)
φ: (𝒜1,D1,∘)→(𝒜2,D2,∘) is also an isomorphism of Novikov algebras.Proof.
(1) For anya,b∈𝒜1, we have
(25)(φD1φ-1)(φ(a)φ(b))=(φD1φ-1)(φ(ab))=φD1(ab)=φ(D1(a)b+aD1(b))=φ(D1(a))φ(b)+φ(a)φ(D1(b))=(φD1φ-1)(φ(a))φ(b)+φ(a)(φD1φ-1)(φ(b)).
So (1) holds.
(2) For anya,b∈𝒜1, we have
(26)φ(a∘b)=φ(aD1(b))=φ(a)φ(D1(b))=φ(a)(φD1φ-1)(φ(b))=φ(a)D2(φ(b))=φ(a)∘φ(b).
So (2) holds.Theorem 12.
Let𝒜 be a commutative and associative algebra over F satisfying Lemma 4, let D0 be its derivation satisfying (10), and let 𝒯 be a commutative and associative algebra over F satisfying Lemmas 9 and 10. If φ:𝒜→𝒯 satisfies
(27)φ(bm)=coshmx,m=0,1,2,…,φ(an)=sinhnx,n=1,2,…,
then the following statements hold:(1)
φ is an isomorphism of commutative and associative algebras,(2)
φD0φ-1=d/dx,(3)
φ:(𝒜,aD0,∘)→(𝒯,φ(a)(d/dx),∘) is an isomorphism of Novikov algebras.Proof.
It is clear by Lemma10, (8), and (19).
(2) By Lemma6, we have
(28)φD0φ-1(sinhnx)=φD0(an)=φ(nbn)=ncoshnx=dsinhnxdx,φD0φ-1(coshnx)=φD0(bn)=φ(nan)=nsinhnx=dcoshnxdx.
So (2) holds.
(3) It is clear thatφ(aD0)φ-1=φ(a)d/dx. By (27) and (10), we have
(29)φ(aD0)φ-1(sinhnx)=φ(aD0)(an)=φ(aD0(an))=φ(anbn)=φ(a)φ(nbn)=φ(a)ncoshnx=φ(a)d(sinhnx)dx.
Similarly, we have φ(aD0)φ-1(coshnx)=φ(a)d(coshnx)/dx. So φ(aD0)φ-1=φ(a)d/dx.
By Theorems7 and 11 and Remark 3(2), we have
(30)φ(b∘c)=φ(baD0(c))=φ(b)φ(aD0(c))=φ(b)[φ(aD0)φ-1(φ(c))]=φ(b)φ(a)ddx(φ(c))=φ(b)∘φ(c),∀b,c∈𝒜.
So φ:(𝒜0,aD0,∘)→(𝒯,φ(a)(d/dx),∘) is an isomorphism of Novikov algebras.



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