Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A(m) be the m-replicated algebra of A.We prove that the representation dimension of A(m) is at most 3,and that the dominant dimension of A(m) is at least m.

THE BSE PROPERTY FOR SOME VECTOR-VALUED BANACH FUNCTION ALGEBRAS

In this paper,X is a locally compact Hausdorff space and A is a Banach algebra.First,we study some basic features of C0(X,A)related to BSE concept,which are gotten from A.In particular,we prove that if C0(X,A)has the BSE property then A has so.We also establish the converse of this result,whenever X is discrete and A has the BSE-norm property.Furthermore,we prove the same result for the BSE property of type I.Finally,we prove that C0(X,A)has the BSE-norm property if and only if A has so.

Discussion on the Homology Theory of Lie Algebras

Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).

AGGREGATE SPECIAL FUNCTIONS TO APPROXIMATE PERMUTING TRI-HOMOMORPHISMS AND PERMUTING TRI-DERIVATIONS ASSOCIATED WITH A TRI-ADDITIVEψ-FUNCTIONAL INEQUALITY IN BANACH ALGEBRAS

In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.

Implicative and Fuzzy Implicative Ideals of Lattice Implication Algebras

In this paper, we defined the concept of implicative and fuzzy implicative ideals of lattice implication algebras, and discussed the properties of them. And then, we pointed out the relations between implicative ideal and LI _ideal, implicative iedal and implicative filter, implicative ideal and fuzzy implicative ideal, fuzzy implicative ideal and fuzzy implicative filter, and fuzzy implicative ideal and fuzzy LI _ideal.

Crossed modules of Lie color algebras

The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.

Enveloping algebras of generalized H-Hom-Lie algebras

Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.

π-quasitriangular group-cograded multiplier Hopf algebras

Let G be a group and （A, B） be a pair of multiplier Hopf algebras, where B is regular G-cograded. Let π be a crossing action of G on B, D^π=A^cop∝B=＋p∈GDπ^p with Dπ^p=A^cop∝Bp, is the Drinfeld double of the pair （A, B）, and then the deformation D^π becomes a multiplier Hopf algebra. B×A can be considered as a subalgebra of M（D^π×D^π）, the image of element b×a in B×A is （1∝b）×（a∝1） in M（D^π×D^π）. Let W =∑αWα∈ M（B×A） be a π-canonical multiplier for the pair （A, B） with Wα∈M（Bα×A） for all α∈G. The image of W in M（D^π×D^π）is a π-quasitriangular structure over D^π.

ON THE PRODUCT OF RIESZ ALGEBRAS AND REPRESENTATION

Let { E i∶i∈I } be a family of Archimedean Riesz algebras.The product of Riesz algebras is denoted by Π i∈I E i .The main result in this paper is the following conclusion:there exists a completely regular Hausdorff space X such that Π i∈I E i is Riesz algebra isomorphic to C(X) if and only if for every i∈I there exists a completely regular Hausdorff space X i such that E i is Riesz algebra isomorphic to C(X i) .

Morita context of weak Hopf group coalgebras

The concept of weak Hopf group coalgebras is a natural generalization of the notions of both weak Hopf algebras（quantum groupoids） and Hopf group coalgebras.Let π be a group.The Morita context is considered in the sense of weak Hopf π-coalgebras.Let H be a finite type weak Hopf π-coalgebra,and A a weak right π-H-comodule algebra.It is constructed that a Morita context connects A#H＊ which is a weak smash product and the ring of coinvariants AcoH.This result is the generalization of that of Wang＇s in the paper ＂Morita contexts,π-Galois extensions for Hopf π-coalgebras＂ in 2006.Furthermore,the result is important for constructing weak π-Galois extensions.

On the Adjoint Semigroup of Implicative BCK-algebras

We prove that the adjoint semigroup of an implicative BCK algebra is an upper semilattice, and the adjoint semigroup of an implicative BCK algebra with condition(s) is a generalized Boolean algebra. Moreover we prove the adjoint semigroup of a bounded implicative BCK algebra is a Boolean algebra.

Invariant Determinants in Banach Algebras

The invariant determinants ha Banach algebras are discussed and a necessary and sufficient condition for those integral traced unital Barach algebra(A,τ) admitting a G-invari- ant determinant is obtained,where G is a group of trace preserving automorphisms of A.

Determinant Theory in Banach Algebras

A determinant theory is developed for Banach algebras and a characterization of those traced unital Banach algebras admitting a determinant is given.

The Structures of the AR Sequences of Tilted Algebras

In this paper,We give the forms of AR sequences of a tilted algebra with terms all belonging to x(A_T),or all belonging to y(A_T).The sink maps of a tilted algebra which end at the indecompos able projective modules and the source maps of starting at the indecomposable injective modules are also obtained.These results together with the connecting sequecnes given in [3] determine the AR quiver of the tilted algcbra,morever,this can be done directly from the AR quiver of the correspond ing hereditary algebra.

DERIVATION ALGEBRAS OF THE MODULAR LIE SUPERALGEBRAS W AND S OF CARTAN-TYPE

Let F be a field and char F = p > 3. In this paper the derivation algebras of Lie superalgebras W and S of Cartan-type over F are determined by the calculating method.

EQUIVALENCE OF x-HOPF ALGEBRAS

Some basic properties of the antipode of an N0I-graded x-Hopf algebra are studied. Also, several equivalent conditions of an N0I-graded x-bialgebra (x-Hopf algebra) are given.

DE MORGAN ALGEBRAS WITH DOUBLE DEMI-PSEUDOCOMPLEMENTATION

The variety ddpM of de Morgan algebras with double demi-pseudocomplementation consists of those algebras （L; ∧ , ∨ , , , ＋ , 0, 1） of type （2, 2, 1, 1, 1, 0, 0） where （L; ∧ , ∨ , , 0, 1） is a de Morgan algebra, （L; ∧ , ∨ , , ＋ , 0, 1） is a double demi-p-lattice and the operations x → x , x → x and x → x ＋ are linked by the identities x = x , x ＋ = x ＋ and x ＋ = x ＋ . In this paper, we characterize congruences on a ddpM-algebra, and give a description of the subdirectly irreducible algebras.

TWISTED PRODUCTS AND SMASH PRODUCTS OVER WEAK HOPF ALGEBRAS

This paper gives a sufficient and necessary condition for twisted products to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar's major result and I.Boca's result.

Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.

MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.