PlanetPhysics/R Algebroid

R-algebroid

If $𝖦$ is a [[../QuantumOperatorAlgebra5/|groupoid]] (for example, considered as a [[../Cod/|category]] with all [[../TrivialGroupoid/|morphisms]] invertible) then we can construct an $R$-algebroid , $R𝖦$ as follows. The [[../TrivialGroupoid/|object]] set of $R𝖦$ is the same as that of $𝖦$ and $R𝖦(b,c)$ is the free $R$-module on the set $𝖦(b,c)$, with [[../Cod/|composition]] given by the usual bilinear rule, extending the composition of $𝖦$.

Alternatively, one can define $\overline{R}𝖦(b,c)$ to be the set of [[../Bijective/|functions]] Failed to parse (unknown function "\lra"): {\displaystyle \mathsf{G}(b,c)\lra R} with finite support, and then we define the \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

$(f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}.$

• As it is very well known, only the second construction is natural for the [[../CoIntersections/|topological]] case, when one needs to replace 'function' by 'continuous function with compact support' (or \emph{locally compact support} for the [[../HotFusion/|QFT]] extended symmetry sectors), and in this case $R\cong ℂ$~. The point made here is that to carry out the usual construction and end up with only an algebra rather than an [[../Algebroids/|algebroid]], is a procedure analogous to replacing a groupoid $𝖦$ by a [[../TrivialGroupoid/|semigroup]] $G^{\prime }=G\cup \{0\}$ in which the compositions not defined in $G$ are defined to be $0$ in $G^{\prime }$. We argue that this construction removes the main advantage of groupoids, namely the spatial component given by the set of objects.
• More generally, an [[../RCategory/|R-category]] is similarly defined as an extension to this R- algebroid [[../PreciseIdea/|concept]].