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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 270937, 5 pages
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.
We construct a kind of infinite-dimensional Novikov algebras and give its realization by hyperbolic sine functions and hyperbolic cosine functions.
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  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  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 . 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 ). 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.
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.
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