Grassmann代数的极限T-子空间与n元中心多项式
Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra
摘要：
设F<X>是集合X = {x_1，x_2，.}上的域F上的自由酉结合代数. F的向量子空间V<X>称为T-子空间（或T-空间），如果V在F的所有自同态下闭<X>。F中的T-子空间V<X>是极限的，如果每个较大的T-子空间W \gneqq V都是可生成的（作为T-子空间），但V本身不是。最近小布兰登，Koshlukov，Krasilnikov和Silva证明了在特征p>2的无限域F上，无限维Grassmann代数G的中心多项式的T-子空间C（G）是极限T-子空间。证明了F中的这个极限T-子空间<X>是唯一的，即F中除C（G）外不存在极限T-子空间<X>.在本文中，我们证明了情况并非如此。在特征p>2的无限域F上的代数F中构造了无穷多个极限T-子空间R_k（k \ge 1）<X>.对于每个k \ge 1，极限T-子空间R_k由Grassmann代数G的2k元中心多项式产生.
英文摘要：
Let F<X> be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F<X> is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F<X>. A T-subspace V in F<X> is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brandão Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F<X> is unique, that is, there are no limit T-subspaces in F<X> other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F<X> over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G.
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