LPNG代数的泛中心扩张
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摘要
交换结合代数很常见的一类代数，李代数是现代数学前'『什领域中具有重要地位的学科之一，它包含了很多漂亮的结果Novikov代数在物理和数学上都有应用，它在解决Yang-Baxter方程方面具有重要作用LPNG代数是在交换结合代数，李代数和Tqov'ikov代数的基础上定义的一种具有三种代数结构的代数，满足4个相容性条件，并且某两个代数间能分别构成Tqov'ikov-POlSSOn代数和Gel’fand-Dorfman代数本文第二节给出了LPNG代数的定义以及其子代数，理想等基本定义，在第三节给出了LPNG代数的具体例子，第四节研究了LPNG代数的中心扩张和泛中心扩张的基本性质，得到了LPNG代数A存在泛覆盖充分必要条件为A是完全的最后一节研究了LPNG代数导子和自同构提升的条件
Commutative associative algebras are a class of common algebras,Lie algebras play an important rolein the forward fields of modern mathematics , which include many nice results.Novikov algebras are appliedin physics and mathematics,it also play an important role in solving Yang-Baxter equations. LPNG algebrais defined on the basis of commutative associative algebra,Lie algebra and Novikov algebra, it has threealgebraic structure and satisfies four compatibility conditions, certain two algebraic between it can sepa-rately formed Novikov-Poisson algebras and Gel’fand-Dorfman algebras. In the second section we givethe definition ,subalgebra , ideal and other basic definitions of LPNG algebra.In the third section we givethe examples of LPNG algebra.In the forth section we study the central extensions and universal centralextensions of LPNG algebra, we obtain the universal covering of LPNG algebra A is exist if and only if A isperfect. In the last section we acquire the conditions of lifting automorphisms and derivations.
Commutative associative algebras are a class of common algebras,Lie algebras play an important rolein the forward fields of modern mathematics , which include many nice results.Novikov algebras are appliedin physics and mathematics,it also play an important role in solving Yang-Baxter equations. LPNG algebrais defined on the basis of commutative associative algebra,Lie algebra and Novikov algebra, it has threealgebraic structure and satisfies four compatibility conditions, certain two algebraic between it can sepa-rately formed Novikov-Poisson algebras and Gel’fand-Dorfman algebras. In the second section we givethe definition ,subalgebra , ideal and other basic definitions of LPNG algebra.In the third section we givethe examples of LPNG algebra.In the forth section we study the central extensions and universal centralextensions of LPNG algebra, we obtain the universal covering of LPNG algebra A is exist if and only if A isperfect. In the last section we acquire the conditions of lifting automorphisms and derivations.
引文
[1] Balinskii A A , Novikov S P. Poisson brackets of hydrodynamic type ,Frobenius algebras and Lie algebras[J].Soviet Math Dokl,1985 , 32(1):228-231.
[2] Marshall Osborn J. Novikov algebras[J]. Nova J Algebra Geom,1992, 1:1-14.
[3] Gel’fand I M,Dikii L A. Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebrasof the Kortweg-de Vries equations[J]. Russian Math Surveys,1975, 30(5): 77-113 .
[4] Gel’fand I M,Dikii L A. A Lie algebra structure in a formal variational calculation[J].Funkts AnalPrilozh,1976,10(1): 18-25.
[5] Zel’manov E I. On a class of local translation invarlant Lie algebras[J]. Soviet Math Dokl, 1987, 35(1): 216-218.
[6] Marshall Osborn J. Simple Novikov algebras with an idempotent[J].Commun Algebra, 1992, 20(9): 2729-2753.
[7] Xu Xiaoping. On simple Novikov algebras and their irreducible modules[J]. J Algebra ,1996, 185:905-934.
[8] Xu Xiaoping. Novikov-Poisson algebras[J].J Algebra, 1997, 190:253-279.
[9] Gel’fand I M, Dikii L A. Hamiltonian operators and algebraic structures related to them[J]. Funct Anal Appl1979, 13(4): 248-262 .
[10] Xu Xiaoping.Quadratic conformal superalgebras[J]. J Algebra, 2000, 231(1):1-38.
[11] Casas J M.Noncommutative Leibniz Poisson algebras[J]. Communications in Algebra, 2006, 34(7):2507-2530.
[12] Su Yucai, Xu Xiaoping.Central simple Poisson algebras[J]. Sci China Ser A, 2004, 47(2):245-263.
[13]靳全勤，佟清Toroidal李代数上的Poisson代数结构[J]数学年刊，2007，28A(1)：57—70
[14] Bai R P,Zhang Y Y Meng D J.The central extension of n-Lie algebra[J]. Chinese Ann Math AerA, 2006,27(4):491-502.
[15] Zhang Qingcheng,Zhang Yongzheng.Derivations and extensions of Lie color algebras[J]. Acta MathematicaScientia, 2008, 28B(4):933-948.
[16] Du? A,Derivations,invariant forms and central extensions of orthosymplectic Lie superalgebras[D]. DissertationDeparment of Mathematics and Statistics, 2002.
[17] Iohara K, Koga Y.Central extensions of Lie superalgebras[J]. Comment Math Helv, 2001,76(1):110-154.
[18] Casas J M.Homology with trivial coe?cients and universal central extensions of algebras with bracket[J]. CommAlgebra, 2007,35(8):2431-2449.
[19] Neher E.An introduction to universal central extensions of Lie superalgebras[J]. Groups Rings and Hopf Alge-bras, 2003,555:141-166.
[2] Marshall Osborn J. Novikov algebras[J]. Nova J Algebra Geom,1992, 1:1-14.
[3] Gel’fand I M,Dikii L A. Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebrasof the Kortweg-de Vries equations[J]. Russian Math Surveys,1975, 30(5): 77-113 .
[4] Gel’fand I M,Dikii L A. A Lie algebra structure in a formal variational calculation[J].Funkts AnalPrilozh,1976,10(1): 18-25.
[5] Zel’manov E I. On a class of local translation invarlant Lie algebras[J]. Soviet Math Dokl, 1987, 35(1): 216-218.
[6] Marshall Osborn J. Simple Novikov algebras with an idempotent[J].Commun Algebra, 1992, 20(9): 2729-2753.
[7] Xu Xiaoping. On simple Novikov algebras and their irreducible modules[J]. J Algebra ,1996, 185:905-934.
[8] Xu Xiaoping. Novikov-Poisson algebras[J].J Algebra, 1997, 190:253-279.
[9] Gel’fand I M, Dikii L A. Hamiltonian operators and algebraic structures related to them[J]. Funct Anal Appl1979, 13(4): 248-262 .
[10] Xu Xiaoping.Quadratic conformal superalgebras[J]. J Algebra, 2000, 231(1):1-38.
[11] Casas J M.Noncommutative Leibniz Poisson algebras[J]. Communications in Algebra, 2006, 34(7):2507-2530.
[12] Su Yucai, Xu Xiaoping.Central simple Poisson algebras[J]. Sci China Ser A, 2004, 47(2):245-263.
[13]靳全勤，佟清Toroidal李代数上的Poisson代数结构[J]数学年刊，2007，28A(1)：57—70
[14] Bai R P,Zhang Y Y Meng D J.The central extension of n-Lie algebra[J]. Chinese Ann Math AerA, 2006,27(4):491-502.
[15] Zhang Qingcheng,Zhang Yongzheng.Derivations and extensions of Lie color algebras[J]. Acta MathematicaScientia, 2008, 28B(4):933-948.
[16] Du? A,Derivations,invariant forms and central extensions of orthosymplectic Lie superalgebras[D]. DissertationDeparment of Mathematics and Statistics, 2002.
[17] Iohara K, Koga Y.Central extensions of Lie superalgebras[J]. Comment Math Helv, 2001,76(1):110-154.
[18] Casas J M.Homology with trivial coe?cients and universal central extensions of algebras with bracket[J]. CommAlgebra, 2007,35(8):2431-2449.
[19] Neher E.An introduction to universal central extensions of Lie superalgebras[J]. Groups Rings and Hopf Alge-bras, 2003,555:141-166.