Abstract and Applied Analysis

Volume 2013 (2013), Article ID 270937, 5 pages

http://dx.doi.org/10.1155/2013/270937

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

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 24 May 2013; Accepted 16 July 2013

Academic Editor: T. Raja Sekhar

Copyright © 2013 Liangyun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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

#### 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: they gave the definition of Novikov algebras. Concretely, let be the structural coefficients, and let a product of be such that For any , the product is Hamilton operator if and only if satisfies

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 . 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 be an algebra over such that
and then is called a Novikov algebra over .

*Remark 2. *An algebra 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 is a left symmetric algebra satisfying
then is a Lie algebra. Usually, it is called an adjoining Lie algebra.

(2) Let be a commutative algebra, and then is a Novikov algebra if is a derivation of with a bilinear operator such that

#### 2. Main Results

Lemma 4. *Let be a basis of the linear space over a field of characteristic satisfying
**
where , . Then is a commutative and associative algebra.*

*Proof. *It is clear that is a commutative algebra over :
Similarly, we have that = . Then , . The result follows.

Corollary 5. * of Lemma 4 is a unity of .*

Lemma 6. *Let be a commutative and associative algebra satisfying Lemma 4. Then the following statements hold:*(1)* If is a linear transformation of such that
* *then is a derivation of .*(2)* If is a linear transformation of such that
* *then is a derivation of .*(3)* is a subalgebra of Lie algebra .*

*Proof. *(1) We have

So is a derivation of .

(2) For , we have
so is a derivation of .

(3) For , we have
Then , and so (3) holds.

Theorem 7. *Let be a commutative and associative algebra satisfying Lemma 4, and let be an element of . If satisfies Lemma 6 and satisfies
**
then the following statements hold:*(1)* is a Novikov algebra.*(2)* is an adjoining Lie algebra of and such that
*

*Proof. *(1) By Lemma 6, is a derivation of the commutative algebra . So is a Novikov algebra by Remark 3(2).

(2) is an adjoining Lie algebra of by Remark 3(1). For , we have
since is commutative. Hence we obtain the desired result.

Let be a unity of . If we set in Theorem 7, then . Similarly, we obtain the following corollary.

Corollary 8. *Let be a commutative and associative algebra satisfying Lemma 4. Then the following statements hold:
*

We have the following: let , , and let the field be assumed or . We will construct Novikov algebras over the linear space which is generated by and .

First, let be a linear space generated by over .

Lemma 9. * satisfying the above product is a commutative associative algebra.*

*Proof. *Since the above product is commutative and associative, we only need to be closed for the product. In fact,
So is a commutative and associative algebra.

Lemma 10. *Let be a linear space generated by over , and then is a basis of .*

*Proof. *For , suppose that there are , such that
We take derivative for (20) such that its derivative order is , and put . Then we have
Let , and then we obtain the following system of linear equations:
If are seen to be unknown, then the coefficient matrix of (22) is the Vandermonde matrix whose determinant is not , so , .

We take derivative for (20) such that its derivative order is , and put . Then we have
Let , and then we obtain the following system of linear equations:
If are seen to be unknown, then the coefficient matrix of (24) is the Vandermonde matrix whose determinant is not , so , . Since, for any , and satisfy (20), we have . Hence , , ,, , are linearly independent for any , and then are linearly independent and so they form a basis of as desired.

Theorem 11. *Let , be commutative and associative algebras over . If : is an isomorphism and , then the following statements hold:*(1)*,*(2)*: is also an isomorphism of Novikov algebras.*

*Proof. *(1) For any , we have
So (1) holds.

(2) For any , we have
So (2) holds.

Theorem 12. *Let be a commutative and associative algebra over satisfying Lemma 4, let be its derivation satisfying (10), and let be a commutative and associative algebra over satisfying Lemmas 9 and 10. If satisfies
**
then the following statements hold:*(1)* is an isomorphism of commutative and associative algebras,*(2)*,*(3)* is an isomorphism of Novikov algebras.*

*Proof. *It is clear by Lemma 10, (8), and (19).

(2) By Lemma 6, we have
So (2) holds.

(3) It is clear that . By (27) and (10), we have
Similarly, we have . So .

By Theorems 7 and 11 and Remark 3(2), we have
So is an isomorphism of Novikov algebras.

#### Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions on this paper. This paper supported by NNSF of China (no. 11171055), NSF of Jilin province (No. 201115006), Scientific Research Foundation for Returned Scholars Ministry of Education of China, and the Fundamental Research Funds for the Central Universities.

#### References

- I. M. Gel'fand and L. A. Diki, “Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations,”
*Functional Analysis and Its Applications*, vol. 30, pp. 77–113, 1975. View at Google Scholar - I. M. Gel'fand and L. A. Diki, “A Lie algebra structure in the formal calculus of variations,”
*Functional Analysis and Its Applications*, vol. 10, pp. 16–22, 1976. View at Google Scholar - I. M. Gel'fand and I. Y. Dorfman, “Hamiltonian operators and algebraic structures associated with them,”
*Functional Analysis and Its Applications*, vol. 13, pp. 248–262, 1979. View at Google Scholar - W. X. Ma, “Some Hamiltonian operators in infinite-dimensional Hamiltonian systems,”
*Acta Mathematicae Applicatae Sinica*, vol. 13, no. 4, pp. 484–496, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. X. Ma, “Complexiton solutions to the Korteweg-de Vries equation,”
*Physics Letters A*, vol. 301, no. 1-2, pp. 35–44, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Z. Tu and W. X. Ma, “An algebraic approach for extending Hamiltonian operators,”
*Journal of Partial Differential Equations*, vol. 5, no. 1, pp. 43–56, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. I. Zel'manov, “A class of local translation-invariant Lie algebras,”
*Soviet Mathematics—Doklady*, vol. 35, no. 1, pp. 216–218, 1987. View at Google Scholar - J. M. Osborn, “Simple Novikov algebras with an idempotent,”
*Communications in Algebra*, vol. 20, no. 9, pp. 2729–2753, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Osborn, “Infinite-dimensional Novikov algebras of characteristic $0$,”
*Journal of Algebra*, vol. 167, no. 1, pp. 146–167, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xu, “Hamiltonian operators and associative algebras with a derivation,”
*Letters in Mathematical Physics*, vol. 33, no. 1, pp. 1–6, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xu, “On simple Novikov algebras and their irreducible modules,”
*Journal of Algebra*, vol. 185, no. 3, pp. 905–934, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xu, “Novikov-Poisson algebras,”
*Journal of Algebra*, vol. 190, no. 2, pp. 253–279, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xu, “Variational calculus of supervariables and related algebraic structures,”
*Journal of Algebra*, vol. 223, no. 2, pp. 396–437, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Bai and D. Meng, “The realization of non-transitive Novikov algebras,”
*Journal of Physics A*, vol. 34, no. 33, pp. 6435–6442, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Bai and D. Meng, “A Lie algebraic approach to Novikov algebras,”
*Journal of Geometry and Physics*, vol. 45, no. 1-2, pp. 218–230, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Bai and D. Meng, “On the Novikov algebra structures adapted to the automorphism structure of a Lie group,”
*Journal of Geometry and Physics*, vol. 45, no. 1-2, pp. 105–115, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Chen, Y. Niu, and D. Meng, “Two kinds of Novikov algebras and their realizations,”
*Journal of Pure and Applied Algebra*, vol. 212, no. 4, pp. 902–909, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Kang and Z. Chen, “Novikov superalgebras in low dimensions,”
*Journal of Nonlinear Mathematical Physics*, vol. 16, no. 3, pp. 251–257, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Zhu and Z. Chen, “Novikov superalgebras with ${A}_{0}={A}_{1}{A}_{1}$,”
*Czechoslovak Mathematical Journal*, vol. 60(135), no. 4, pp. 903–907, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet