Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let R be a finite Lie conformal algebra.We investigate the conformal deriva-tion algebra CDer(R),conformal triple derivation algebra CTDer(R)and generalized con-formal triple derivation algebra GCTDer(R),focusing mainly on the connections among these derivation algebras.We also give a complete classification of(generalized)con-formal triple derivation algebras on all finite simple Lie conformal algebras.In partic-ular,CTDer(R)=CDer(R),where R is a finite simple Lie conformal algebra.But for GCDer(R),we obtain a conclusion that is closely related to CDer(R).Finally,we introduce the definition of a triple homomorphism of Lie conformal algebras.Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

Conformal biderivations of loop W(a,b)Lie conformal algebra

We study conformal biderivations of a Lie conformal algebra.First,we give the definition of a conformal biderivation.Next,we determine the conformal biderivations of loop W(a,b)Lie conformal algebra,loop Virasoro Lie conformal algebra,and Virasoro Lie conformal algebra.Especially,all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.

W（a, b） Lie Conformal Algebra and Its Conformal Module of Rank One

For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W（a, b） conformal algebra for some a, b and its conformal module of rank one.

The Lie Conformal Algebra of a Block Type Lie Algebra

Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.

Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.

Leibniz Conformal Algebras of Rank Two

We classify all Leibniz conformal algebras of rank two.

Cohomology of Lie Conformal Algebra Vir× Cur g

In this article,we compute cohomology groups of the semisimple Lie conformal algebra S=Vir × Cur g with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra g.

Algebraic Solution for the Forward Displacement Analysis of the General 6-6 Stewart Mechanism

The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.

Extending Structures for Gel'fand-Dorfman Bialgebras

Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.

Cohomology of Associative H-Pseudoalgebras

We define the cohomology of associative H-pseudoalgebras,and we show that it describes module extensions,abelian pseudoalgebra extensions,and pseudoalgebra first-order deformations.The same results for the special case of associative conformal algebras are also described in details.

A 3D GIS spatial data model based on conformal geometric algebra

We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.

Complex brackets and balanced complex 1st-order difference polynomials in 4-dimensional Minkowski space

This paper investigates complex brackets and balanced complex 1st-order difference (BCD) polynomials. Then we propose an algorithm of O(n log n) complexity to check the equality of brackets. It substitutes exponential algorithms before. Also, BCD polynomials have some usages in geometric calculation.