由传递矩阵定义的一类新的超矩阵代数
A new class of supermatrix algebras defined by transitive matrices
摘要：
给出了矩阵传递性的一个自然定义。利用基环R的自同态d和中心Z（R）上的可迁n × n矩阵，构造了由n × n超矩阵构成的全n × n矩阵代数M_{n}（R）的子代数M_{n}（R，d，T）.如果d的n次幂是恒等式，并且T满足一些额外的条件，那么我们证明了R嵌入到M_{n}（R，d，T）中.另一个结果是M_{n}（R，d，T）相对于取（前）伴随是封闭的。如果R是李幂零的，A在M_{n}（R，d，T）中，那么使用预伴随和相应的行列式和特征多项式，得到A在固定环Fix（d）中具有右系数的Cayley-Hamilton恒等式.单位的本原n次根的存在和d的n次幂是单位元的条件保证了Fix（d）上李幂零R的右可积性。本文给出了Grassmann代数上的新的超矩阵代数。
英文摘要：
We give a natural definition for the transitivity of a matrix. Using an endomorphism d of a base ring R and a transitive nxn matrix over the center Z(R), we construct the subalgebra M_{n}(R,d,T) of the full nxn matrix algebra M_{n}(R) consisting of the so called nxn supermatrices. If the n-th power of d is the identity and T satisfies some extra conditions, then we exhibit an embedding of R into M_{n}(R,d,T). An other result is that M_{n}(R,d,T) is closed with respect to taking the (pre)adjoint. If R is Lie nilpotent and A is in M_{n}(R,d,T), then the use of the preadjoint and the corresponding determinants and characteristic polynomials yields a Cayley-Hamilton identity for A with right coefficients in the fixed ring Fix(d). The presence of a primitive n-th root of unity and the condition that the n-th power of d is the identity guarantee the right integrality of a Lie nilpotent R over Fix(d). We present essentially new supermatrix algebras over the Grassmann algebra.
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