DERIVATIONS AND EXTENSIONS OF LIE COLOR ALGEBRA

In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.

COMPARISON OF HOMOLOGIES AND AUTOMATIC EXTENSIONS OF INVARIANT DISTRIBUTIONS

Let G be a reductive Nash group,acting on a Nash manifold X.Let Z be a G-stable closed Nash submanifold of X and denote by U the complement of Z in X.Letχbe a character of G and denote by g the complexified Lie algebra of G.We give a sufficient condition for the natural linear map H_(k)(g,S(U)×χ)→H_k(g,S(X)×χ)between the Lie algebra homologies of Schwartz functions to be an isomorphism.For k=0,by considering the dual,we obtain the automatic extensions of g-invariant(twisted by-χ)Schwartz distributions.

An Introduction to the Theory of Field Extensions

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

The Almost Split Sequences for Trivial Extensions of Hereditary Algebras

Let A be a basic hereditary artin algebra and R = A Q be the trivial extension of A by its minimal injective cogenerator Q. We construct some right （left） almost split morphisms and irreducible morphisms in modR through the corresponding morphisms in modA. Furthermore, we can determine its almost split sequences in modR.

H-separable Hopf Galois Extensions and Azumaya Algebra

Let H be a finite dimensional semisimple Hopf algebra over a field and A an H-module algebra. In this paper, we characterize any H-separable Galois extension of an Azumaya algebra. Assuming that A/AH is an H-separable extension, we prove that A/AH is H-Galois and AH is Azumaya if and only if A#H is an Azumaya Z-algebra, where Z is the center of A#H(not necessarily C(A)H).

ALGEBRAIC EXTENSION OF *-A OPERATOR

In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊-A operator, then Weyl＇s theorem holds for f（T）, where f is an analytic functions on some neighborhood of σ（T） and not constant on each of the components of its domain.

CONSTRUCTING POINTED WEAK HOPF ALGEBRAS BY ORE-EXTENSION

The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra from a Clifford monoid S =[Y; Gα. φα,β]by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H（S, n, c, x, a, b） through factoring At by a semilattice graded weak Hopf ideal.

Quasidiagonal Extension of AT-algebras

Let A and B be C^＊-algebras. An extension of B by A is a short exact sequence O→A→E→B→O. （＊） Suppose that A is an AT-algebra with real rank zero and B is any AT-algebra. We prove that E is an AT-algebra if and only if the extension （＊） is quasidiagonal.

On non-abelian extensions of 3-Leibniz algebras

In this paper,we study non-abelian extensions of 3-Leibniz algebras through Maurer-Cartan elements.We construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions in 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements in this differential graded Lie algebra.And also the Leibniz algebra structure on the space of fundamental elements of 3-Leibniz algebras is analyzed.It is proved that the non-abelian extension of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras.

Gorenstein dimensions for weak Hopf-Galois extensions

The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.

具有高阶导子Lie-Yamaguti代数的上同调

研究具有高阶导子的Lie-Yamaguti代数,称之为LieYHDer对。首先给出LieYHDer对的上同调,然后研究了LieYHDer对的中心扩张,根据上同调考虑LieYHDer的形变。

轨形上哈密顿向量场的中心扩张

研究轨形上的辛向量场和哈密顿向量场,得到轨形上哈密顿向量场李代数的中心扩张,并计算该中心扩张的2-cocycle.

Differential Homological Algebra and General Relativity

In 1916, F.S. Macaulay developed specific localization techniques for dealing with “unmixed polynomial ideals” in commutative algebra, transforming them into what he called “inverse systems” of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theory of such systems, using methods of homological algebra that were giving rise to “differential homological algebra”, replacing unmixed polynomial ideals by “pure differential modules”. The use of “differential extension modules” and “differential double duality” is essential for such a purpose. In particular, 0-pure differential modules are torsion-free and admit an “absolute parametrization” by means of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure differential modules, introducing a “relative parametrization” where the potentials should satisfy compatible “differential constraints”. We recently noticed that General Relativity is just a way to parametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator in order to obtain a “minimum parametrization” by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. In order to make this difficult paper rather self-contained, these unusual purely mathematical results are illustrated by many explicit examples, two of them dealing with contact transformations, and even strengthening the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory. They also bring additional doubts on the origin and existence of gravitational waves.

MORITA CONTEXT OF WEAK HOPF ALGEBRAS

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.

Revisiting the Computation of Cohomology Classes of the Witt Algebra Using Conformal Field Theory and Aspects of Conformal Algebra

In this article, we revisit some aspects of the computation of the cohomology class of H2 (Witt, C)?using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate’s exploration of conformal field theory and to gain insight on the subject from a mathematical perspective.

On Two Classes of Extended 3-Lie Algebras

In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras H, M, the (μ, ρ, β)-extension of H through M is given, and the necessary and sufficient conditions for the (μ, ρ, β)-extension algebra of H through M being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.

Crossed products for Hopf group-algebras

First,the group crossed product over the Hopf group-algebras is defined,and the necessary and sufficient conditions for the group crossed product to be a group algebra are given.The cleft extension theory of the Hopf group algebra is introduced,and it is proved that the crossed product of the Hopf group algebra is equivalent to the cleft extension.The necessary and sufficient conditions for the crossed product equivalence of two Hopf groups are then given.Finally,combined with the equivalence theory of the Hopf group crossed product and cleft extension,the group crossed product constructed by the general 2-cocycle as algebra is determined to be isomorphic to the group crossed product of the 2-cocycle with a convolutional invertible map of the 2-cocycle.The unit property of a general 2-cocycle is equivalent to the convolutional invertible map of the 2-cocycle,and the combination condition of the weak action is equivalent to the convolutional invertible map of the 2-cocycle and the combination condition of the weak action.Similarly,crossed product algebra constructed by the general 2-cocycle is isomorphic to the Hopfπ-crossed product algebra constructed by the 2-cocycle with a convolutional invertible map.

Algebraic Points of Any Degree l with (l ≥ 9) over Q on the Affine Equation Curve C3 (11): y11 = x3(x-1)3

In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C3 (11): y11 = x3 (x-1)3. This result is a special case of quotients of Fermat curves Cr,s (p) : yp = xr(x-1)s, 1 ≤ r, s, r + s ≤ p-1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C1(11)(K) of degree at most 2 on Q.

（Co）Homology and Universal Central Extension of Hom-Leibniz Algebras

Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory （such as （co）homology groups, universal central extensions） of Hom-Leibniz algebras based on some works of Loday and Pirashvili.

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A × DA in terms of the combinatorics of the quiver and relations is determined.