Some Discussions On The Structures Of Four-dimensional F-manifold Algebras

An algebra with a Lie algebra structure and a commutative associative algebra structure that are connected by the Hertling-Manin connection is referred to as an-manifold algebra.The classification of an-manifold algebra has become a significant research content in algebra as an important subject of study.The classification of low-dimensional（1≤9）≤3)-manifold algebra has been studied in the complex number field,however the classification of four-dimensional-manifold algebra has yielded no relevant results.On the basis of the classification of the four-dimensional commutative associative algebras and properties of Lie algebras,the structure of the four-dimensional-manifold algebra over the complex number field is investigated.Several structures of a Poisson algebra can be described by the structure of an-manifold algebra,because a Poisson algebra is a particular case of an-manifold algebra.First,we demonstrate the equivalence of the-manifold algebra(2（?）₀+A_2,1,·,[,])using the characteristic matrix of the commutative associative algebra in combination with significant mathematical techniques like the iterative method.Furthermore,we obtain the relations satisfied by the structure constants of the Lie brackets by applying the Jacobi identity and properties of Lie algebras.The dimension of derivative algebras of Lie algebras in the-manifold algebra(2（?）₀+A_2,1,·,[,])is then discussed by using fundamental transformations.Second,according to discussions of the-manifold algebra together with the definition of a Poisson algebra,it is not difficult to determine the dimension of derivative algebras of Lie algebras in the Poisson algebra.Because the proof is similar,we finally only list the equivalent conditions and related results of the other-manifold algebras,including Poisson algebras.Moreover,for each type of-manifold algebras,we investigate whether there exists an-manifold algebra which is not a Poisson algebra.It should be emphasized that this work only studies a portion of the-manifold algebra’s structure and does not obtain a full understanding of the clas-sification.