AI 中文总结
本文为自由结合代数中的Jordan元素提出一个多项式判据，通过构造$U_n$和$q_n$，将元素属于Jordan子代数等价于单一方程$a\\,q_n(U_n)=0$，并给出显式投影及低度维数。
AI 中文摘要
设$A=\Phi\langle X\rangle$为特征零域上的自由结合代数，$J$为由$X$和$1$生成的$A^{(+)}$的Jordan子代数。对每个$n\geq1$，我们构造一个元素$U_n\in\mathbb Q[\mathfrak S_n]$，其在$A_n$上的像恰好为$J_n$。若在多重线性分量$V_n$上有\\[ \det(tI-U_n|_{V_n})=t^{e_n}q_n(t),\qquad q_n(0)\ne0,\\] 则\\[ a\in J_n\quad\Longleftrightarrow\quad a\\,q_n(U_n)=0.\\] 因此该判据给出了识别Jordan元素的有限算法：在每一度构造$U_n$和$q_n$并检验单一方程$a\\,q_n(U_n)=0$。此外，$I-q_n(U_n)/q_n(0)$是$A_n$到$J_n$的投影。我们在度至多四时显式给出该投影，记录至度八的多重线性维数，并在每个固定多重齐次分量上表述相应的判据。
英文摘要
Let $A=Φ\langle X\rangle$ be a free associative algebra over a field of characteristic zero, and let $J$ be the Jordan subalgebra of $A^{(+)}$ generated by $X$ and $1$. For every $n\geq1$ we construct an element $U_n\in\mathbb Q[\mathfrak S_n]$ whose image on $A_n$ is exactly $J_n$. If \[ \det(tI-U_n|_{V_n})=t^{e_n}q_n(t),\qquad q_n(0)\ne0, \] on the multilinear component $V_n$, then \[ a\in J_n\quad\Longleftrightarrow\quad a\,q_n(U_n)=0. \] Thus the criterion gives a finite algorithm for recognizing Jordan elements: in each degree one constructs $U_n$ and $q_n$ and tests the single equation $a\,q_n(U_n)=0$. Moreover, $I-q_n(U_n)/q_n(0)$ is a projection of $A_n$ onto $J_n$. We give the projection explicitly in degrees at most four, record the multilinear dimensions through degree eight, and formulate the analogous criterion on each fixed multihomogeneous component.
