Grassmann-Hopf algebras and coalgebras

Grassmann-Hopf algebras and coalgebras

0.1

Definitions of Grassmann-Hopf Al/gebras, Their Dual

Co-Algebras, and Grassmann–Hopf Al/gebroids

Let

V

be a (complex)

vector space

,

dim

𝒞

⁡

V

=

n

, and let

{

e

0

,

e

1

,

…

,

}

with

identity

e

0

≡

1

, be the

generators

of a Grassmann (exterior)

algebra

Λ

*

⁢

V

=

Λ

0

⁢

V

⊕

Λ

1

⁢

V

⊕

Λ

2

⁢

V

⊕

⋯

(0.1)

subject to the

relation

e

i

⁢

e

j

+

e

j

⁢

e

i

=

0

. Following Fauser
(2004) we append this algebra with a Hopf

structure

to obtain a
‘co–gebra’ based on the interchange (or

‘tangled duality’

):

(

objects/points

,

𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚𝑠

)

↦

(

𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚𝑠

,

objects/points.

)

This leads to a

tangle duality

between an

associative

(unital algebra)

𝒜

=

(

A

,

m

)

, and an associative (unital) ‘co–gebra’

𝒞

=

(

C

,

Δ

)

:

i

the binary

product

A

⊗

A

⁢

⟶

𝑚

⁢

A

, and

ii

the

coproduct

C

⁢

⟶

Δ

⁢

C

⊗

C

,
where the Sweedler notation (Sweedler, 1996), with respect to an
arbitrary basis is adopted:

Δ

⁢

(

x

)

=

∑

r

a

r

⊗

b

r

=

∑

(

x

)

x

(

1

)

⊗

x

(

2

)

=

x

(

1

)

⊗

x

(

2

)

Δ

⁢

(

x

i

)

=

∑

i

Δ

i

j

⁢

k

=

∑

(

r

)

a

(

r

)

j

⊗

b

(

r

)

k

=

x

(

1

)

⊗

x

(

2

)

Here the

Δ

i

j

⁢

k

are called ‘

section

coefficients’. We have then a

generalization

of associativity to coassociativity:

C

→

Δ

C

⊗

C

↓

Δ

↓

id

⊗

Δ

C

⊗

C

→

Δ

⊗

id

C

⊗

C

⊗

C

(0.2)

inducing a tangled duality between an associative (unital algebra

𝒜

=

(

A

,

m

)

, and an associative (unital) ‘co–gebra’

𝒞

=

(

C

,

Δ

)

. The idea is to take this structure
and combine the

Grassmann algebra

(

Λ

*

⁢

V

,

∧

)

with the
‘co-gebra’

(

Λ

*

⁢

V

,

Δ

∧

)

(the ‘tangled dual’)
along with the

Hopf algebra

compatibility rules: 1) the product
and the unit are ‘co–gebra’

morphisms

, and 2) the coproduct and
counit are algebra morphisms.

Next we consider the following ingredients:

(1)

the graded switch

τ

^

⁢

(

A

⊗

B

)

=

(

-

1

)

∂

⁡

A

⁢

∂

⁡

B

⁢

B

⊗

A

(2)

the counit

ε

(an algebra morphism) satisfying

(

ε

⊗

id

)

⁢

Δ

=

id

=

(

id

⊗

ε

)

⁢

Δ

(3)

the antipode

S

.

The

Grassmann-Hopf algebra

H

^

thus consists of–is defined by– the

septet

H

^

=

(

Λ

*

⁢

V

,

∧

,

id

,

ε

,

τ

^

,

S

)

.

Its generalization to a

Grassmann-Hopf algebroid

is
straightforward by considering a

groupoid

𝖦

, and then
defining a

H

∧

-

𝐴𝑙𝑔𝑒𝑏𝑟𝑜𝑖𝑑

as a

quadruple

(

G

⁢

H

,

Δ

,

ε

,

S

)

by modifying the Hopf
algebroid definition so that

H

^

=

(

Λ

*

⁢

V

,

∧

,

id

,

ε

,

τ

^

,

S

)

satisfies

the standard
Grassmann-Hopf algebra axioms stated above. We may also say that

(

H

⁢

G

,

Δ

,

ε

,

S

)

is a

weak C*-Grassmann-Hopf
algebroid

when

H

∧

is a unital C*-algebra (with

𝟏

).
We thus set

𝔽

=

ℂ

. Note however
that the tangled-duals of Grassman-Hopf algebroids retain both the
intuitive interactions and the dynamic

diagram

advantages of their
physical, extended

symmetry

representations

exhibited by the
Grassman-Hopf al/gebras and co-gebras over those of either weak
C*- Hopf algebroids or weak Hopf C*- algebras.

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,

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3

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.,

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17

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89

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7

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21

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13

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11

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52

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Clifford Algebras

. Konstanz,
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17

B. Fauser: Grade

Free product

Formulae from Grassmann–Hopf Gebras.
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arXiv:0709.4364v2 [quant–ph]

Title

Grassmann-Hopf algebras and coalgebras

Canonical name

GrassmannHopfAlgebrasAndCoalgebrasgebras

Date of creation

2013-03-22 18:10:55

Last modified on

2013-03-22 18:10:55

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

52

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81T18

Classification

msc 81T13

Classification

msc 55Q25

Classification

msc 81T10

Classification

msc 16W30

Classification

msc 81T05

Classification

msc 57T05

Classification

msc 15A75

Synonym

tangled-dual Grassmann-Hopf co-algebra

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WeakHopfCAlgebra

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TopicEntryOnTheAlgebraicFoundationsOfQuantumAlgebraicTopology

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AlgebraicFoundationsOfQuantumAlgebraicTopology

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QuantumAlgebraicTopologyOfCWComplexRepresentationsNewQATResult

Defines

Grassmann-Hopf algebra

Defines

dual Grassmann-Hopf co-algebra and gebra or tangled algebra

Defines

observable operator algebra

Defines

Grassman-Hopf algebroid

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