Grassman semialgebras and the Cayley-Hamilton theorem

Matrix representations of finitely generated Grassmann algebras and some consequences

We prove that the m-generated Grassmann algebra can be embedded into a 2^{m-1}x2^{m-1} matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for nxn matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.

Polynomial identities for matrices over the Grassmann algebra

We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ Márki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper.

关于n×n矩阵一般凯莱-哈密尔顿恒等式的左系数和右系数

On the left and right coefficients of the general Cayley-Hamilton identities for an nxn matrix

本文研究了n×n矩阵在任意单位环上的凯莱-哈密尔顿恒等式，证明了矩阵系数C(i)和D(i)的奇分量关系及奇分量的表达式。

Cayley algebras give rise to $q$-Fano planes over certain infinite fields and $q$-covering designs over others

Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every two-dimensional subspace of $V$ is contained in exactly one element $B$ of $\mathfrak{B}$. The question of existence of $q$-Fano planes over any field has been open since the 1970s and has attracted considerable attention in the special case that $F$ is finite. Here we show the existence of $2$-$(7, 3, 1)_F$-subspace designs over certain infinite fields $F$, including …

Cayley-Dickson Algebras and Finite Geometry

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence…