algebroid structures and extended symmetries
0.1 Algebroid Structures and Algebroid Extended Symmetries
Definition 0.1.
An algebroid structure 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 -algebroid on a set of “objects”
is a directed graph
over such that for each has an -module structure
and there is an -bilinear
function
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 is not assumed. For example, if has exactly one object, then an -algebroid over is just an -algebra. An ideal in is then an example of a pre-algebroid.
Let be a commutative ring. An -category is a category equipped with an -module structure on each hom set such that the composition is -bilinear. More precisely, let us assume for instance that we are given a commutative ring with identity. Then a small -category–or equivalently an -algebroid– will be defined as a category enriched in the monoidal category of -modules, with respect to the
monoidal structure of tensor product. This means simply that for all objects of , the set is given the structure of an -module, and composition is –bilinear, or is a morphism
of -modules .
If is a groupoid (http://planetmath.org/Groupoids) (or, more generally, a category)
then we can construct an -algebroid as
follows. The object set of is the same as that of
and is the free -module on the
set , with composition given by the usual
bilinear rule, extending the composition of .
Alternatively, one can define to be the
set of functions with finite support, and
then we define the convolution product
as follows:
(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 . 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) by a semigroup in which the
compositions not defined in are defined to be in . 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, -algebroids, double algebroids , and so on.
A ‘category’ of -categories is however a super-category (http://planetmath.org/Supercategory) , or it can also be viewed as a specific example of a metacategory (http://planetmath.org/AxiomsOfMetacategoriesAndSupercategories) (or
-supercategory, in the more general case of multiple operations
–categorical ‘composition laws’– being defined within the same structure, for the same class, ).
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. , , 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 |