algebroid structures and extended symmetries
0.1 Algebroid Structures and Algebroid Extended Symmetries
Definition 0.1.
An algebroid structure $A$ will be specifically defined to mean either a ring, or more generally, any of the specifically defined algebras^{}, but with several objects instead of a single object, in the sense specified by Mitchell (1965). Thus, an algebroid has been defined (Mosa, 1986a; Brown and Mosa 1986b, 2008) as follows. An $R$-algebroid $A$ on a set of “objects” ${A}_{0}$ is a directed graph^{} over ${A}_{0}$ such that for each $x,y\in {A}_{0},A(x,y)$ has an $R$-module structure^{} and there is an $R$-bilinear function
$$\circ :A(x,y)\times A(y,z)\to A(x,z)$$ |
$(a,b)\mapsto a\circ b$ called “composition^{}” and satisfying the associativity condition, and the existence of identities^{}.
Definition 0.2.
A pre-algebroid has the same structure as an algebroid and the same axioms except for the fact that the existence of identities ${1}_{x}\in A(x,x)$ is not assumed. For example, if ${A}_{0}$ has exactly one object, then an $R$-algebroid $A$ over ${A}_{0}$ is just an $R$-algebra. An ideal in $A$ is then an example of a pre-algebroid.
Let $R$ be a commutative ring. An $R$-category $\mathrm{A}$ is a category equipped with an $R$-module structure on each hom set such that the composition is $R$-bilinear. More precisely, let us assume for instance that we are given a commutative ring $R$ with identity. Then a small $R$-category–or equivalently an $R$-algebroid– will be defined as a category enriched in the monoidal category of $R$-modules, with respect to the monoidal structure of tensor product^{}. This means simply that for all objects $b,c$ of $\mathcal{A}$, the set $\mathcal{A}(b,c)$ is given the structure of an $R$-module, and composition $\mathcal{A}(b,c)\times \mathcal{A}(c,d)\u27f6\mathcal{A}(b,d)$ is $R$–bilinear, or is a morphism^{} of $R$-modules $\mathcal{A}(b,c){\otimes}_{R}\mathcal{A}(c,d)\u27f6\mathcal{A}(b,d)$.
If $\U0001d5a6$ is a groupoid^{} (http://planetmath.org/Groupoids) (or, more generally, a category) then we can construct an $R$-algebroid $R\U0001d5a6$ as follows. The object set of $R\U0001d5a6$ is the same as that of $\U0001d5a6$ and $R\U0001d5a6(b,c)$ is the free $R$-module on the set $\U0001d5a6(b,c)$, with composition given by the usual bilinear rule, extending the composition of $\U0001d5a6$.
Alternatively, one can define $\overline{R}\U0001d5a6(b,c)$ to be the set of functions $\U0001d5a6(b,c)\u27f6R$ with finite support^{}, and then we define the convolution product^{} as follows:
$$(f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}.$$ | (0.1) |
As it is very well known, only the second construction is natural for the topological case, when one needs to replace ‘function’ by ‘continuous function^{} with compact support’ (http://planetmath.org/SmoothFunctionsWithCompactSupport) (or locally compact support^{} for the QFT (http://planetmath.org/QFTOrQuantumFieldTheories) extended http://planetmath.org/?op=getobj&from=books&id=153symmetry^{} sectors), and in this case $R\cong \u2102$ . The point made here is that to carry out the usual construction and end up with only an algebra rather than an algebroid, is a procedure analogous to replacing a groupoid (http://planetmath.org/Groupoids) $\U0001d5a6$ by a 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 (http://planetmath.org/Groupoids), namely the spatial component given by the set of objects.
Remarks: One can also define categories of algebroids, $R$-algebroids, double algebroids , and so on. A ‘category’ of $R$-categories is however a super-category^{} (http://planetmath.org/Supercategory) $\mathbb{S}$, or it can also be viewed as a specific example of a metacategory (http://planetmath.org/AxiomsOfMetacategoriesAndSupercategories) (or $R$-supercategory, in the more general case of multiple operations^{}–categorical ‘composition laws’– being defined within the same structure, for the same class, $C$).
References
- 1 I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review. SIGMA 5 (2009), 051, 70 pages. $arXiv:0904.3644$, $doi:10.3842/SIGMA\mathrm{.2009.051}$, http://www.emis.de/journals/SIGMA/2009/051/Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Title | algebroid structures and extended symmetries |
Canonical name | AlgebroidStructuresAndExtendedSymmetries |
Date of creation | 2013-03-22 18:13:55 |
Last modified on | 2013-03-22 18:13:55 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 50 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 81T25 |
Classification | msc 81T18 |
Classification | msc 81T13 |
Classification | msc 81T10 |
Classification | msc 81T05 |
Classification | msc 81R50 |
Classification | msc 55U35 |
Synonym | extensions of quantum operator algebras |
Related topic | HamiltonianAlgebroids |
Related topic | QFTOrQuantumFieldTheories |
Related topic | LieAlgebroids |
Related topic | RCategory |
Related topic | RAlgebroid |
Related topic | AxiomsOfMetacategoriesAndSupercategories |
Related topic | MonoidalCategory |
Related topic | Groupoids |
Related topic | ETAS |
Defines | algebroid structure |
Defines | convolution product |
Defines | pre-algebroid |
Defines | algebroid extended symmetries |
Defines | set of functions with finite support |