论文标题
免费替代代数中心的新元素
New elements in the center of free alternative algebra
论文作者
论文摘要
在自由替代代数中发现了一系列新的中央元素。更确切地说，让$ alt [x] $和$ smalc [x] \ subset alt [x] $成为免费的替代代数，而免费的特殊Malcev代数在一组自由发电机$ x $上的特征性0领域，然后让$ f（x，x，y，y，x_1，x_1，x_1，x_1，\ ldots，x_n）\ in multial in multial smalc [x]代数。然后，元素$ u_n = u_n（x，x_1，\ ldots，x_n）= f（x^2，x，x，x_1，\ ldots，x_n）-f（x，x，x^2，x_1，\ ldots，x_n）$位于algebra $ alt [x] $的中心。 Elements $ u_n（x，x_1，\ ldots，x_n）$被唯一定义为给定$ n $的标量，并且它们在变量上是偏斜的$ x_1，\ ldots，x_n $。此外，$ u_n = 0 $ for $ n = 4m+2，\，4m+3 $。和$ u_n \ neq 0 $ for $ n = 4m，4m+1 $。由元素$ u_ {4m}，\，u_ {4m+1} $生成的理想位于代数$ alt [x] $的关联中心，并具有微不足道的乘法。
A new series of central elements is found in the free alternative algebra. More exactly, let $Alt[X]$ and $SMalc[X]\subset Alt[X]$ be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators $X$, and let $f(x,y,x_1,\ldots,x_n)\in SMalc[X]$ be a multilinear element which is trivial in the free associative algebra. Then the element $u_n=u_n(x,x_1,\ldots,x_n)=f(x^2,x,x_1,\ldots,x_n)-f(x,x^2,x_1,\ldots,x_n)$ lies in the center of the algebra $Alt[X]$. The elements $u_n(x,x_1,\ldots,x_n)$ are uniquely defined up to a scalar for a given $n$, and they are skew-symmetric on the variables $x_1,\ldots,x_n$. Moreover, $u_n=0$ for $n=4m+2,\,4m+3$. and $u_n\neq 0$ for $n=4m,4m+1$. The ideals generated by the elements $u_{4m},\,u_{4m+1}$ lie in the associative center of the algebra $Alt[X]$ and have trivial multiplication.
