Mal’tsevh 0-algebras

Let Φ be an associative commutative ring with unity, 1/6 ∈ Φ, write A for a Mal’tsev algebra over Φ, suppose that on A, the function h(y, z, t, x, x)=2[{yz, t, x}x+{yx, z, x}t], where {x, y, z}=(xy)z−(xz)y+2x(yz), is defined, and assume that H(A) is a fully invariant ideal of A generated by the function h. The algebra A satisfying an identity h(y, z, x, x, x)=0 [h(y, z, t, x, x)=0] is called a Mal’tsev h₀-algebra (h-algebra). We prove that in any Mal’tsev h₀-algebra, the inclusion H(A)·A₂⊆Ann A holds withAnnA the annihilator of A. This means that any semiprime h₀-algebra A is an h-algebra. Every prime h₀-algebra A is a central simple algebra over the quotient field Λ of the center of its algebra of right multiplications, R(A), and is either a 7-dimensional non-Lie algebra or a 3-dimensional Lie algebra over Λ. Supported by RFFR grant No. 94-01-00381-a. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 214–227, March–April, 1996.     

Every Akivis Algebra is Linear

An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

方程xy+yz+zx=n的正整数解

本文用 Ｓｉｅｇｅｌ－Ｔａｔｕｚａｗａ定理证明了：当ｎ＞１．２×１０~１１时,至多有两个正 整数ｎ。使方程ｘｕ＋ｙｚ＋ｚｘ＝ｎ无适合（ｘ,ｙ,ｚ）＝１且０＜ｘ＜ｙ＜ｚ的解（ｘ,ｙ,ｚ）, 并给出类数为２的二次域与多项式表素数的一个结果．     

Trivial nuclear ideals of alternative algebras

Let A be a free alternative Φ-algebra, where Φ is an associative commutative ring with 1, containing 1/6, and g(y, z, t, v, x, x)=2[J_({[y, z], t, x}_, x, v)+J_({[y,x], z, x}_, t, v)], where [x, y]=xy−yx, J_(x, y, z)=[[x, y], z]+[[z, x], y]+[[y, z], x], {x, y, z}_=J_(x, y, z)+3[x, [y, z]]. We construct trivial nuclear ideals of A, that is, nonzero ideals with zero multiplication, lying in the associative center of A. In particular, it is shown that if G and B are fully invariant ideals of A on k≥7 free generators, generated by a function g and by double commutators, respectively, then GB+BG is a nuclear ideal of A. This implies that an unmized alternative algebra satisfies GB=BG=0. If an unmixed algebra is finitely generated, then G=0. In addition, we prove that if R is an unmixed solvable alternative algebra then (R^N)²=0 for some N. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 97–115, January–February, 1997.     

关于方程xy+yz+zx=n的正整数解

本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１，２，４，６，１０，１８，２２，３０，４２，５８，７０，７８，１０２，１３０，１９０，２１０，３３０，４６２时，方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ，ｙ，ｚ）．     

Finiteness conditions in Krull subrings of a ring of polynomials

LetR be a Krull subring of a ring of polynomialsk[x ₁, …, x_n] over a fieldk. We prove that ifR is generated by monomials overk thenr is affine. We also construct an example of a non-affine Krull ringR, such thatk[x, xy]⊂R⊂k[x, y], and a non-Noetherian Krull ringS, such thatk[x, xy, z]⊂S⊂k[x, y, z].     

Kernel of the second order Cauchy difference on groups

Let (G, ·) be a group, (H, +) be an abelian group, and ${f:G\rightarrow H}$ . The second order Cauchy difference of f is $$C^{(2)}f(x,y,z)=f(xyz)-f(xy)-f(yz)-f(xz)+f(x)+f(y)+f(z).$$ The functional equation $$C^{(2)}f(x,y,z)=0$$ is studied. We present its general solution on free groups. Solutions on other selected groups are also given.     

A fully invariant ideal of alternative algebras

Let ϕ be an associative commutative ring with 1, containing 1/6, and A be an alternative ϕ-algebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements of the form h(y, z, t, x, x)=[{[y, z], t, x}-, x]+[{[y, x], z, x}-, t], where [x, y]=xy−yx, {x, y, z}-=[[x, y], z]−[[x, z], y]+2[x,[y, z]]. Here we consider an ideal Q=H∩D and prove that Q⁴=0 in the algebra A. If A is unmixed, then HD=0, DH=0, and Q²=0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q=0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x)=0. We also show that a fully invariant ideal H₀ of the unmixed algebra A, generated by all elements of the form h(x, z, t, x, x), lies in its associative center and H₀∩D=0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x)=0. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 323–340, May–June, 1997.     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Free zeropotent left distributive systems

We prove a result conjectured by J.Je[zcirc]ek in [9],namely that a zeropotent left self-distributive system need not be 3-trivial, i.e.,the identity x(yz)=0 does not follow from the identities x(yz)=(xy)(xz),xx=0 and x0=0x=0.The argument is a general scheme possibly working for various identities in the context of self-distributivity.     

Solutions of the three-dimensional sine-Gordon equation

We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009.     

Partial Difference Equations Analogous to the Cauchy-Riemann Equations and Related Functional Equations On Rings and Fields

Let (S, #, *) be an algebraic structure where # and * are binary operations with identities on the set S. Let (G, +) be an abelian group. We consider the functional equation (i) $$f(x * t, y)+ g(x, y\ \sharp\ t) = h(x, y)\ {\rm for\ all}\ x, y, t \in S,$$ where ?,g,h :S × S → G. As an application of (i) we solve $$f(x + t, y)- f(x, y) = -b(f(x, y+t)- f(x,y))\ {\rm for\ all}\ x, y, t \in S,$$ where ? :S × S → K (a field), and b ∈ K is a constant and b ≠ 0, ±1. If b = i, the pure imaginary unit, S = R and K = C, then the above equation may be considered as a discrete analogue of the Cauchy-Riemann equations. When (R, +, ?) is a commutative ring with 1, the functional equation (ii) $$\phi(y+xt)-\phi(xy+xt)=\phi(y+x)-\phi(xy+x)$$ for all x,y,t ∈ R, where ? : R → G, is basic to the general solutions of (i). We solve (ii) on certain rings and fields.     

Weak Tensor Category and Related Generalized Hopf Algebras

There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, a） and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32（2）： 397-441 （2004）. Our aim in this paper is to generalize tensor category L = （L, ×, I, a, l, r） by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized （bialgebras） Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.     

Graded automorphism group of TKK Lie algebra over semilattice

Every extended affine Lie algebra of type A ₁ and nullity ν with extended affine root system R(A ₁, S), where S is a semilattice in ℝ ^ν , can be constructed from a TKK Lie algebra T (J (S)) which is obtained from the Jordan algebra J (S) by the so-called Tits-Kantor-Koecher construction. In this article we consider the ℤ ⁿ -graded automorphism group of the TKK Lie algebra T (J (S)), where S is the “smallest” semilattice in Euclidean space ℝ ⁿ .     

On groups of automorphisms of a class of surfaces

In this note we describe the group of automorphisms of a commutative algebra with three generatorsx, y andz satisfying a relationxy= P(z), whereP(z) is a polynomial. The author was visiting Bar-Ilan and Tel-Aviv Universities while working on this project. He is deeply thankful for hospitality rendered.     

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

A Characterization of Lie Algebras of Skew-Symmetric Elements

A characterization of Lie algebras of skew-symmetric elements of associative algebras with involution is obtained. It is proved that a Lie algebra L is isomorphic to a Lie algebra of skew-symmetric elements of an associative algebra with involution if and only if L admits an additional (Jordan) trilinear operation {x,y,z} that satisfies the identities $$\{x,y,z\}=\{z,y,x\},$$ $$[[x,y],z]=\{x,y,z\}-\{y,x,z\},$$ $$[\{x,y,z\},t]=\{[x,t],y,z\}+\{x,[y,t],z\}+\{x,y,[z,t]\},$$ $$\{\{x,y,z\},t,v\}=\{\{x,t,v\},y,z\}-\{x,\{y,v,t\},z\}+\{x,y,\{z,t,v\}\},$$ where [x,y] stands for the multiplication in L.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.     

一类特殊的Koszul Calabi-Yau DG代数

假设一个连通上链DG代数A的基分次代数A~#或者同调分次代数H(A)是由一次元素x,y生成的代数kx,y/(xy+yx).本文证明A是Koszul Calabi-Yau DG代数.