自由积的迹根与余中心
Trace radicals and cocenters of free products
AI总结：
研究含幺结合代数\(A\)和\(B\)自由积\(A*B\)的迹根，通过结合余中心分解与有限维表示构造，证明\(A*B\)的迹根是\(A\)和\(B\)迹根的直和，得出\(A*B\)是迹RFD的充要条件是\(A\)和\(B\)都是迹RFD。
AI中文摘要：
我们称一个含幺结合代数\(A\)为迹剩余有限维的，如果它的元素可由有限维表示分离，且其余中心\(A/[A,A]\)可由相应的迹泛函分离。对于RFD代数\(A\)和\(B\)，我们证明了\(A*B\)的迹根（所有有限维迹泛函的公共核）是\(A\)和\(B\)的迹根的直和，这意味着\(A*B\)是迹RFD当且仅当\(A\)和\(B\)都是迹RFD。证明结合了明确的余中心分解与有限维表示的构造，其迹能检测长度大于一的非零字类。
英文摘要：
We call a unital associative algebra $A$ trace residually finite-dimensional if its elements are separated by finite-dimensional representations and its cocenter $A/[A,A]$ is separated by the corresponding trace functionals. For RFD algebras $A$ and $B$, we prove that the trace radical of $A*B$, the common kernel of all finite-dimensional trace functionals, is the direct sum of the trace radicals of $A$ and $B$, which implies that $A*B$ is trace RFD if and only if both $A$ and $B$ are trace RFD. The proof combines an explicit cocenter decomposition with a construction of finite-dimensional representations whose traces detect nonzero classes of words of length greater than one.
