Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops

Jimmie Lawson

Displaying similar documents to “Clifford algebras, Möbius transformations, Vahlen matrices, and $B$ -loops”

On multiplication groups of left conjugacy closed loops

Aleš Drápal (2004)

Commentationes Mathematicae Universitatis Carolinae

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A loop $Q$ is said to be left conjugacy closed (LCC) if the set ${L_{x}; x \in Q}$ is closed under conjugation. Let $Q$ be such a loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ , respectively, and let $Inn Q$ be its inner mapping group. Then there exists a homomorphism $ℒ \to Inn Q$ determined by $L_{x} \mapsto R_{x}^{- 1} L_{x}$ , and the orbits of $[ℒ, ℛ]$ coincide with the cosets of $A (Q)$ , the associator subloop of $Q$ . All LCC loops of prime order are abelian groups.

Pseudoautomorphisms of Bruck loops and their generalizations

Mark Greer, Michael Kinyon (2012)

Commentationes Mathematicae Universitatis Carolinae

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We show that in a weak commutative inverse property loop, such as a Bruck loop, if $α$ is a right [left] pseudoautomorphism with companion $c$ , then $c$ [ $c^{2}$ ] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism,...

A class of Bol loops with a subgroup of index two

Petr Vojtěchovský (2004)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group and $C_{2}$ the cyclic group of order $2$ . Consider the $8$ multiplicative operations $(x, y) \mapsto {(x^{i} y^{j})}^{k}$ , where $i$ , $j$ , $k \in {- 1, 1}$ . Define a new multiplication on $G \times C_{2}$ by assigning one of the above $8$ multiplications to each quarter $(G \times {i}) \times (G \times {j})$ , for $i, j \in C_{2}$ . We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M (G, 2)$ .

Displaying similar documents to “Clifford algebras, Möbius transformations, Vahlen matrices, and B -loops”

On multiplication groups of left conjugacy closed loops

Pseudoautomorphisms of Bruck loops and their generalizations

A class of Bol loops with a subgroup of index two

Displaying similar documents to “Clifford algebras, Möbius transformations, Vahlen matrices, and $B$ -loops”