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Axes In Non-Associative Algebras, Louis Rowen, Yoav Segev
Axes In Non-Associative Algebras, Louis Rowen, Yoav Segev
Turkish Journal of Mathematics
Fusion rules are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes of respective types $(λ,δ)$ and $(λ',δ')$ must be at …
On Elements Whose Moore-Penrose Inverse Is Idempotent In A ${\Ast}$-Ring, Haiyang Zhu, Jianlong Chen, Yukun Zhou
On Elements Whose Moore-Penrose Inverse Is Idempotent In A ${\Ast}$-Ring, Haiyang Zhu, Jianlong Chen, Yukun Zhou
Turkish Journal of Mathematics
In this paper, we investigate the elements whose Moore-Penrose inverse is idempotent in a ${\ast}$-ring. Let $R$ be a ${\ast}$-ring and $a\in R^\dagger$. Firstly, we give a concise characterization for the idempotency of $a^\dagger$ as follows: $a\in R^\dagger$ and $a^\dagger$ is idempotent if and only if $a\in R^{\#}$ and $a^2=aa^*a$, which connects Moore-Penrose invertibility and group invertibility. Secondly, we generalize the results of Baksalary and Trenkler from complex matrices to ${\ast}$-rings. More equivalent conditions which ensure the idempotency of $a^\dagger$ are given. Particularly, we provide the characterizations for both $a$ and $a^\dagger$ being idempotent. Finally, the equivalent conditions under which …