algebroid structures and extended symmetries

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

∈

A

0

,

A

⁢

(

x

,

y

)

has an

R

-module

structure

and there is an

R

-

bilinear
function

∘

:

A

(

x

,

y

)

×

A

(

y

,

z

)

→

A

(

x

,

z

)

(

a

,

b

)

↦

a

∘

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

∈

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

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

𝒜

, the set

𝒜

⁢

(

b

,

c

)

is given the structure of an

R

-module, and composition

𝒜

⁢

(

b

,

c

)

×

𝒜

⁢

(

c

,

d

)

⟶

𝒜

⁢

(

b

,

d

)

is

R

–bilinear, or is a

morphism

of

R

-modules

𝒜

⁢

(

b

,

c

)

⊗

R

𝒜

⁢

(

c

,

d

)

⟶

𝒜

⁢

(

b

,

d

)

.

If

𝖦

is a

groupoid

(

http://planetmath.org/Groupoids

) (or, more generally, a category)
then we can

construct

an

R

-algebroid

R

⁢

𝖦

as
follows. The object set of

R

⁢

𝖦

is the same as that of

𝖦

and

R

⁢

𝖦

⁢

(

b

,

c

)

is the free

R

-module on the
set

𝖦

⁢

(

b

,

c

)

, with composition given by the usual
bilinear rule, extending the composition of

𝖦

.

Alternatively, one can define

R

¯

⁢

𝖦

⁢

(

b

,

c

)

to be the
set of functions

𝖦

⁢

(

b

,

c

)

⟶

R

with

finite support

, and
then we define the

convolution product

as follows:

(

f

*

g

)

⁢

(

z

)

=

∑

{

(

f

⁢

x

)

⁢

(

g

⁢

y

)

∣

z

=

x

∘

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=153

symmetry

sectors), and in
this case

R

≅

ℂ

. 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

)

𝖦

by a semigroup

G

′

=

G

∪

{

0

}

in which the
compositions not defined in

G

are defined to be

0

in

G

′

. 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

)

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.

a

⁢

r

⁢

X

⁢

i

⁢

v

:

0904.3644

,

d

⁢

o

⁢

i

:

10.3842

/

S

⁢

I

⁢

G

⁢

M

⁢

A

⁢

.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

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