SPINORS IN BRAIDED GEOMETRY

BANACH CENTER PUBLICATIONS, VOLUME 37 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1996

SPINORS IN BRAIDED GEOMETRY

M I ´ C O DU R DE V I ´ C Instituto de Matematicas, UNAM

Area de la Investigacion Cientifica, Circuito Exterior Ciudad Universitaria, M´ exico DF, CP 04510 M´ exico

E-mail: micho@matem.unam.mx

Z B I G N I E W O Z I E W I C Z

Facultad de Estudios Superiores Cuautitlan, UNAM Apartado Postal #25 Cuautitlan Izcalli, CP 54700, M´ exico

E-mail: oziewicz@servidor.unam.mx

Abstract. Let V be a C-space, σ ∈ End(V

) be a pre-braid operator and let F ∈ lin(V

, C ). This paper offers a sufficient condition on (σ, F ) that there exists a Clifford algebra C`(V, σ, F ) as the Chevalley F -dependent deformation of an exterior algebra C`(V, σ, 0) ≡ V

(σ).

If σ 6= σ

and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal cate- gory. A spinor representation as a left C`(V, σ, F )-module is identified with an exterior algebra over F -isotropic C-subspace of V. We give a sufficient condition on braid σ that the spinor representation is faithful and irreducible.

1. Introduction. Clifford and Weyl algebra for a Hecke braid was considered among other in [Hayashi 1990, Oziewicz 1995, Bautista et al. 1996]. The aim of this paper is to extend Clifford algebra and spinors to braided geometry and we do not assume that a braid need to be a Hecke braid.

Let V be a finite dimensional C-space, σ ∈ End(V

) be a pre-braid operator and let F ∈ lin(V

, C ). This paper offers a sufficient condition on (σ, F ) that there exists a Clifford algebra C`(V, σ, F ) as the Chevalley F -dependent deformation of a braided exterior algebra C`(V, σ, 0) ≡ V

(σ),

(1) (F ⊗ id

)(id

⊗ σ) = (id

⊗ F )(σ ⊗ id

).

1991 Mathematics Subject Classification: Primary 15A66, 15A75, 16S80; Secondary 18D10.

Z. Oziewicz is a member of Sistema Nacional de Investigadores, M´ exico. Partially supported by KBN grant 2 P302 023 07.

Resubmitted May 17, 1996.

The paper is in final form and no version of it will be published elsewhere.

[315]

The condition (1) should be contrasted with a condition that a form F is a σ-morphism in a σ-braided monoidal category. The condition (1) is valid also for σ = 0 in contrast to the σ-morphism which for F 6= 0 do not holds for σ = 0. For involutive braids σ = σ

condition (1) coincides with a σ-morphism for F . The condition (1) is invariant with respect to rescaling, for λ ∈ C, σ → λ · σ, whereas a condition for F to be a σ-morphism is not rescaling invariant.

From the point of view of the deformation theory one can ask if exists an F -dependent deformation of exterior Hopf algebra V

(σ)?

Let sh be a shuffle comultiplication on a C-space V

. Then, from one point of view, the answer to the above question is positive in the framework of generalized braided quantum groups [Durdevi´ c 1996],

V

(σ) ,→ {C`(V, σ, F ), sh}.

From another point of view the answer is negative because for F 6= 0, {C`(V, σ, F ), sh} is not a braided Hopf algebra as in [Majid 1991-1993].

An open question is whether exists an F -dependent deformation, in the framework of a braided category, of the braided exterior bialgebra V

(σ) which need not to be bi-unital (i.e. not unital or not co-unital)?

In the last section the Cartan approach to algebraic spinors is applied in the braided geometry. This last section is a variation and a generalization of a construction given in [Bautista at al. 1996]. A spinor space S is defined as an exterior Hopf C-algebra over F -isotropic C-subspace of V. A spinor space S is a left Clifford module and in contrast to the standard formulation, we shall not require the existence of a hermitian form (a spinor norm), see e. g. [Crumeyrolle 1990]. We found sufficient conditions on a braid σ that the spinor representation {C`(V, σ, F ) → End S} is irreducible and faithful.

Throughout this paper algebra, cogebra, Hopf algebra means C-algebra, C-cogebra and Hopf C-algebra and ⊗ ≡ ⊗

. All maps are C-linear, lin ≡ lin

and End ≡ End

.

2. Deformation of shuffle comutiplication. In what follows V is a C-space, T V is a tensor algebra universal on V with a realization T V ≡ {V

, ⊗}, and CV is a tensor cogebra co-universal on V with a shuffle co-universal commultiplication sh, i.e. CV ≡ {V

, sh}, see [Sweedler 1969, chapter XII, page 247].

Henceforth we define

s

∈ lin(V

, V

⊗ V

),

s

(v

⊗ . . . ⊗ v

) ≡ (v

⊗ . . . ⊗ v

) ⊗ (v

⊗ . . . ⊗ v

), s

≡ 1 ⊗ id, s

≡ id ⊗ 1,

sh: V

→ V

⊗ V

, sh|V

≡

n

X

i=0

s

.

In this section we consider a pre-braid - dependent coassociative deformation of co- universal shuffle comultiplication sh.

A pre-braid on C-space V is a map σ ∈ End(V

) for which the braid equation holds

(2) (σ ⊗ id

)(id

⊗ σ)(σ ⊗ id

) = (id

⊗ σ)(σ ⊗ id

)(id

⊗ σ).

A pre-braid σ extends to pre-braiding σ

∈ End(V

⊗ V

) on V

, e. g.

(σ

)

≡ σ

|{V ⊗ V

} = (id

⊗ σ)(σ ⊗ id

), (σ

)

≡ σ

|{V

⊗ V } = (σ ⊗ id

)(id

⊗ σ).

Let V

≡ lin(V, C) be a dual C-space. The evaluation V

⊗ V → C extends to a pairing T V

⊗ T V → C according to the following convention

for α, β ∈ T V

, t, u ∈ T V and |α| = |u|, |β| = |t|, (α ⊗ β)(t ⊗ u) ≡ (βt)(αu) ∈ C.

The dual space V

≡ lin(V, C ) possess the transposed pre-braid σ

∈ End(V

) with respect to the above pairing.

Let V σV denote an associative unital algebra and a coassociative counital cogebra with a C-space V

⊗ V

and with σ-dependent structures,

a multiplication: [(⊗) ⊗ (⊗)] ◦ (id

⊗ σ

⊗ id

), a unit: 1 ⊗ 1, 1 ∈ lin(C, V

),

a comultiplication: (id

⊗ σ

⊗ id

) ◦ [sh ⊗ sh], a counit: ε ⊗ ε, ε ∈ lin(V

, C).

It is an interesting question whether the above structure can be extended to some braided bialgebra. This question is not investigated in this paper.

For a pre-braid σ ∈ End(V

) we shall define a σ-dependent coassociative comulti- plication C(σ) and associative multiplication Q(σ) in a C-space V

,

V

−→ V

⊗ V

, V

←− V

⊗ V

, (3) Q(σ) ≡ [C(σ

)]

, the graded dual.

Definition 1 (Comultiplication). Let 1 ∈ T V. An algebra map C(σ) ∈ alg(T V, V σV ) is determined by value on generating C-space V,

C(σ)1 ≡ 1 ⊗ 1, C(σ)|V ≡ 1 ⊗ id

+ id

⊗ 1.

Let S

be the permutation group on n elements, π ∈ S

and let σ

∈ End(V

) be a map obtained by replacing transpositions in a minimal decomposition of π by a pre-braid σ ∈ End(V

).

Let sh

⊆ S

be a set of riffle shuffles with a cut n from S

[Sweedler 1969, chapter XII; Sternberg 1993, p. 43], i.e. a set of permutations preserving an order of sub-sets {1, . . . , n} and {n + 1, . . . , n + k}.

Proposition 2 (Deformation of shuffle comultiplication). An algebra map C(σ) is a coassociative σ-deformation of co-universal tensor shuffle comultiplication,

C(σ)|V

=

n

X

i=0

s

◦ C

(σ) ,

C

(σ) = C

(σ) = id

,

C

(σ) = X

π∈sh_n,k

σ

∈ End(V

),

C

(0) = id

, C(0) = sh.

P r o o f. The above formulas are proved by induction on n. The coassociativity of the coproduct C(σ) is due to braid equation (2).

The operators C

(σ) for n = 1 or k = 1 are the same as braided integers in [Majid 1993] or in [Majid 1995, Definition 10.4.8],

 1 + n n ; σ



= C

(σ), [1 + k; σ] = C

(σ).

In particular

C

(σ) = id

+ σ,

C

(σ) = id

+ σ ⊗ id

+ (id

⊗ σ)(σ ⊗ id

), C

(σ) = id

+ id

⊗ σ + (σ ⊗ id

)(id

⊗ σ).

3. Braided exterior Hopf algebra. Hopf algebras in a braided monoidal category, i.e. a braided Hopf algebras, has been introduced by Majid in a series of papers in years 1991-1993, we refer to monograph by Majid [1995]. In this section we generalize this to pre-braided case when a braid needs not to be invertible and we consider two important examples of pre-braid - dependent families: pre-braided universal Hopf algebra and pre- braided co-universal Hopf algebra as the deformation of bi-universal (i.e. universal and couniversal) Hopf algebra.

A bi-universal σ-braided Hopf algebra exists if and only if σ|V

= 0 [Oziewicz, Paal and R´ o˙za´ nski 1995, section 8]. A realization of the bi-universal 0-braided Hopf algebra is given by {V

, ⊗, sh} with the antipode

S|C = id, S|V = −id, S|V

= 0.

Proposition 3 (Braided Hopf algebras).

(i) Q(0) = ⊗, i.e. Q(σ) is an associative σ-deformation of a tensor multiplication (con- catenation) and Q(σ) ∈ cog(V σV, CV ).

(ii) CV (σ) ≡ {V

, ⊗, C(σ)} is a σ-braided universal biassociative and biunital Hopf algebra which is a σ-deformation of bi-universal Hopf algebra CV (0).

(iii) QV (σ) ≡ {V

, Q(σ), sh} is a σ-braided co-universal biassociative and biunital Hopf algebra which is σ-deformation of bi-universal Hopf algebra QV (0) = CV (0).

(iv) There exists a unique homomorphism W (σ) ∈ hopf[CV (σ), QV (σ)] of σ-deformed universal Hopf algebra into σ-deformed co-universal Hopf algebra such that W (σ)|(C ⊕ V ) = id. An operator W (σ) commutes with antipod and W (σ) is σ- deformation of identity , i.e. W (0) = id

.

P r o o f. See [Oziewicz, Paal and R´ o˙za´ nski 1995, section 8]. In the proof we need to use braid equation (1).

From the graded duality in definition (3) and from Proposition 2 we get Q

(σ) ≡ Q(σ)|(V

⊗ V

) = X

π∈sh_n,k

[(σ

)

]

: V

⊗ V

−→ V

.

Moreover

[(σ

)

]

= σ

and this follows from the equality of the tangles corresponding to the left and the right sides of the above equation.

By definition an operator W (σ) must be an algebra and cogebra map,

(4) W (σ) ∈ alg [T V, (V

, Q(σ))],

V

⊗ V

−→

V

↓ W ⊗ W ↓ W

V

⊗ V

−→ V

(5) W (σ) ∈ cog [(V

, C(σ)), CV ],

V

−→ V

⊗ V

↓ W ↓ W ⊗ W

V

sh

−→ V

⊗ V

An image im W (σ) is a Hopf sub-algebra of QV (σ).

In particular a Hopf algebra map W (σ) coincide with a braided symmetrizer/alter- nator introduced by Woronowicz [1989, pp. 153-155]

End(V

) 3 W

(σ) ≡ X

π∈Sn

σ

. Moreover

for λ ∈ C, (λ · σ)

= λ

· σ

.

A subspace ker W (σ) < CV (σ) is a two-sided biideal in a universal tensor Hopf algebra. A factor Hopf algebra V

(σ) ≡ CV (σ)/ker W (σ) is said to be an exterior Hopf algebra for a pre-braid σ. Let π

∈ hopf [CV (σ), V

(σ)] be an epimorphism of Hopf algebras and ker π

≡ ker W (σ),

∧ ≡ ∧

≡ ⊗ mod ker W (σ) : V

⊗ V

→ V

.

One can show that a pre-braiding σ

factors to a pre-braiding σ

on a factor algebra V

⊗ V

and that an exterior Hopf algebra V

(σ) is σ

-braided, i. e. all structure maps are σ

-morphisms and in particular a factor multiplication ∧

is a σ

-morphism,

σ

(∧ ⊗ id) =(id ⊗ ∧)(σ

⊗ id)(id ⊗ σ

), σ

(id ⊗ ∧) =(∧ ⊗ id)(id ⊗ σ

)(σ

⊗ id).

A Hopf algebra im W (σ) is isomorphic as Hopf algebra to the exterior algebra. The following algebra map is invertible,

π

|im W (σ) ∈ alg [(im W (σ), Q(σ)), (V

(σ), ∧

)],

V

(σ) 3 π

ψ ≡ ψ

≡ [ψ mod ker W (σ)] ←→ W (σ)ψ ∈ im W (σ) < V

. We have a pairings

for α, β ∈ T V

, t, u ∈ T V and |α| = |u|, |β| = |t|, (α ⊗ β)(t ⊗ u) ≡ (βt)(αu) ∈ C,

V

(σ) ⊗ V

(σ) → C,

(6) C 3 α

t

≡ αW (σ)t or 0 if |α| 6= |t|.

4. Inner product. An inner product is a graded derivation of degree −1. In this sec- tion an inner product is generalized to a braided geometry. A general theory of derivations of arbitrary degree in braided geometry was presented in [Oziewicz, Paal and R´ o˙za´ nski 1995].

For f ∈ V

and c

≡ f ⊗ id let c

|V

∈ lin(V

, V

) be contraction in a tensor algebra V

. Then exists k

(σ) ∈ End(V

) such that

c

◦ W (σ) = W (σ) ◦ k

(σ).

Therefore k

(σ) factors to a map k(σ) ∈ lin(V

⊗ V

, V

).

Lemma 4 (Braided derivation). The Leibniz rule holds,

k

(σ) ◦ ∧ = ∧ ◦ {k

(σ) ⊗ id + σ

◦ [k

(σ) ⊗ id] ◦ σ

}.

P r o o f. The Leibniz rule for k

(σ) in a tensor algebra factors to the above Leibniz rule in a factor algebra.

An operator k(σ) extends to an algebra map

t ∈ alg [T V

, End(V

)], t|V

≡ k(σ).

Lemma 5. If f ∈ kerW

(σ) then t

= 0.

P r o o f. The statement is the consequence of (3) because W (σ) ∈ alg [T V, QV (σ)].

Hence, we can pass from V

to V

in the first argument of t and we obtain a map t: V

⊗ V

→ V

.

Definition 6. The map t: V

⊗ V

→ V

is said to be the inner product.

The inner product t can be defined equivalently as the transposed exterior multipli- cation. For f ∈ V

, let ∧

∈ End(V

) be a linear map given by ∧

φ ≡ φ ∧ f.

Proposition 7. Let f, g ∈ T V

, ψ ∈ T V and |f | + |g| = |ψ|. Then the inner product t

and an exterior product ∧

are mutually transposed ,

(π

g)(t

π

ψ) = (∧

π

g)(π

ψ).

P r o o f. We have

(g

∧ f

)ψ

= (∧

g

)ψ

= [W

(g ⊗ f )]ψ

= (g ⊗ f )W ψ = g

(f

t ψ

) = g

(t

ψ

).

5. Clifford Algebra. Let F ∈ lin(V

, C ) be a scalar product and `

∈ lin(V, V

) be an associated correlation,

ev(`

⊗ id

) ≡ F and T `

∈ alg (T V, T V

).

Clifford and Weyl algebras for Hecke braids were considered among others in [Hayashi 1990, Oziewicz 1995, Bautista et al. 1996]. In this section a braid σ ∈ End(V

) needs not to be a Hecke braid.

In what follows we shall assume

(7) T `

◦ W (σ) = W

(σ) ◦ T `

, W

(σ) = W (σ

).

This is equivalent that

(`

⊗ `

) ◦ σ = σ

◦ (`

⊗ `

).

Factorizing T `

through ideals ker {W, W

} we obtain an algebra homomorphism

∧`

∈ alg (V

, V

).

A sufficient condition that (7) holds is

(8) (F ⊗ id

)(id

⊗ σ) = (id

⊗ F )(σ ⊗ id

).

Let ι

be a contraction map multliplicative on the first factor with values in braided derivations (Lemma 4),

(9) ι

: V

⊗ V

→ V

, ι

≡ t[(∧`

) ⊗ id], for x ∈ V and σ(x ⊗ ϑ) = X

k

ϑ

⊗ x

, ι

◦ ∧

= ∧

+ X

∧

◦ ι

.

We shall introduce contraction operators h , i

in V

which we need for construction of a Clifford algebra C`(V, σ, F ) as the Chevalley F -deformation of a σ-braided exterior algebra C`(V, σ, 0) ≡ V

(σ). We define

for ψ ∈ V

, ψ

∈ V

, ψ

∈ V

, [π

◦ C

(σ)]ψ = X

ψ

∧ ψ

, h , i

: V

⊗ V

→ V

, hψ, ·i

≡ X

∧

◦ ι

, if n < k then h , i

≡ 0.

Consistency of this definition follows from bialgebra map (4-5).

The Chevalley F -dependent deformed product ∨ ≡ ∨

on V

(σ) is defined as follows,

∨ ≡ ∧ + X

k≥1

h·, ·i

,

in particular for x ∈ V, ∨

= ∧

+ ι

.

Proposition 8. An F -deformed algebra {V

(σ), ∨

} is an associative algebra with the unity 1 ∈ V

(σ).

P r o o f. The proof can be performed diagramatically, using tangle and braid diagrams, and using condition (1 − 8).

Definition 9 (Clifford algebra as a deformation). An algebra C`(V, σ, F ) ≡ {V

(σ), ∨

}

is said to be a Clifford algebra as the Chevalley F -deformation of an exterior algebra V

(σ).

The graded algebra associated to a filtered algebra C`(V, σ, F ) is isomorphic to V

(σ).

Let π

∈ alg [T V, C`(V, σ, F )] be an algebra epimorphism extending the identity map on V and let I

be two-sided ideal in a tensor algebra,

I

≡ ker π

/ T V.

The Clifford algebra C`(V, σ, F ) can be presented as a factor algebra C`(V, σ, F ) = T V /I

.

6. The Bourbaki bijection in a tensor algebra. The ideal I

/ T V can be described in another way, using a bijection λ

∈ End V

introduced by Bourbaki [1959].

Let for x ∈ V, ι

∈ σder T V be a braided derivation on T V , Lemma 4. Bourbaki [1959] introduced the following map λ

∈ EndV

,

for x ∈ V and ψ ∈ V

,

λ

|C = id, λ

(x ⊗ ψ) = x ⊗ (λ

ψ) + (ι

◦ λ

)ψ.

Then λ

|V = id and the Bourbaki map λ

is bijective.

Lemma 10. We have

(10) ker W (σ) = λ

(I

).

P r o o f. The statement follows from π

◦ λ

= π

.

The Bourbaki bijection λ

allows to define a new product ∨

in C-space V

, (11) ∨

≡ λ

◦ λ

⊗ λ

.

With respect to product ∨

(11) the space ker W (σ) is a left ideal in {T V, ∨

}. The condition (1 − 8) ensures that ker W (σ) is also a right ∨

-ideal.

An algebra epimorphism π

∈ alg [T V, V

(σ)] by construction is also an algebra epimorphism of F -deformed algebras.

If the braid operator σ is such that ker W (σ) is quadratic, then the ideal I

is generated by elements of the form

ψ − F (ψ)1 ⊗ 1, where ψ ∈ V

is σ-invariant, σψ = ψ.

This covers Clifford and Weyl algebras for a Hecke braid [Hayashi 1990, Oziewicz 1995].

To define a Clifford algebra as the Chevalley F -deformation of braided exterior algebra V

(σ), it is neccessary and sufficient that ker W (σ) is also a right-ideal in (V

, ∨

). This assumption is weaker then (1 − 8). If (8) does not hold, then the symmetry between left and right is broken.

If σ = σ

then the braided Hopf algebra V

(σ) can be deformed to a generalized braided quantum group {C`(V, σ, F ), sh} [Durdevi´ c 1994, 1996]. A generalized braided quantum group {C`(V, σ, F ), sh} is not a braided Hopf algebra as in [Majid 1991-1993].

However, the axiom for the antipode is the same as for Hopf algebra. The antipode is F -dependent. The shuffle coproduct sh in {C`(V, σ, F ), sh} is the same as in V

(σ), however the intrinsic braid determined by the Clifford product, shuffle coproduct and the antipode will be F -dependent. The generalized braided quantum group is not includable in the framework of braided categories, because the coproduct map does not obey the functoriality properties relative to the mentioned braiding.

7. Spinors. This section is devoted to braided generalization of the Cartan theory of spinors [3], [1]. Consider the Witt F -isotropic splitting

(12) V = V

⊕ V

, V

are F -isotropic.

The Witt splitting is said to be compatible with the braid σ ∈ End(V

) if for i 6= j, σ(V

⊗ V

) = V

⊗ V

,

σ

|{(V

⊗ V

) ⊕ (V

⊗ V

)} = id.

Let F | V

⊗ V

be nondegenerate. In this case V

' V

. Let for f ∈ V

and x ∈ V

a form F be given by F (f ⊗ x) ≡ f (x).

Exterior algebras (V

)

and (V

)

are subalgebras of C`(V, σ, F ).

Lemma 11 (The Cartan map). The following Cartan map µ is bijective, (13) µ: (V

)

⊗ (V

)

−→ C`(V, σ, F ), µ ≡ ∨

.

P r o o f. Let u ∈ V

, v ∈ V

and σ(v ⊗ u) ≡ X

k

u

⊗ v

. Then vu + X

k

u

v

− F (v ⊗ u) · 1 = 0.

This implies that µ is surjective.

Let p

: V

→ (V

)

⊗ (V

)

be a projection. We shall prove that the map

∧: (V

)

⊗ (V

)

→ V

, which is the grade-preserving component of µ, is injective.

Indeed, we have p

◦ ∧ = id and hence ∧ is injective.

Denote K = (V

)

and let κ ∈ alg (K, C) be a character, κ(1) = 1 such that κ(V

) = 0.

This gives a left K-module structure on C. On the other hand C`(V, σ, F ) is a right K- module.

Definition 12. A left C`(V, σ, F )-module S ≡ C`(V, σ, F ) ⊗

C is called the spinor module associated to the F -isotropic splitting (12).

According to Lemma 11, the space S is isomorphic to the exterior algebra (V

)

and for x = x

+ x

∈ V, x

∈ V

, xξ = x

∧ ξ + x

t ξ.

Proposition 13. A C`(V, σ, F )-module S is faithful and irreducible.

P r o o f. We prove that each vector ψ ∈ S \ {0} is cyclic which implies that the module is simple. The unit element 1

∈ S is cyclic, by construction. The duality between spaces V

and V

extends to the duality between exterior algebras (V

)

and (V

)

, as explained in Section 4. In terms of this duality, the contraction between elements of the same degree becomes the pairing map. It follows that there exists ϕ ∈ K such that ϕ t ψ = 1. Hence, ψ is cyclic.

Let X

u

⊗v

6= 0 be a component consisting of summands having the minimal second degree n. We can assume that v

are linearly independent vectors. Let x ∈ C`(V, σ, F ) \ {0}. We have

µ

(x) = X

u

⊗ v

+ ψ.

There exist spinors ξ

∈ S satisfying ψξ

= 0 and v

ξ

= δ

. This gives xξ

= u

which implies that the representation is faithful.

A left C`(V, σ, F )-module S is completely characterized by the existence of a cyclic

vector 1

∈ S anihilated by the subspace V

.

In other words let V be an arbitrary left C`(V, σ, F )-module, possesing a vector v satisfying V

v = 0. Then exists the unique module map %: S → V satisfying % 1

= v.

The map % is injective because of the simplicity of S. In particular, if v is cyclic then % is a module isomorphism.

Clifford and Weyl algebra for a Hecke braid considered by Hayashi [1990] and by Oziewicz [1995] and spinors for a Hecke braid introduced in [Bautista et al. 1996] are included in the theory presented here. Clifford algebra of [1] is defined for Hecke braid σ ∈ End(V

) (where V ≡ W ⊕ W

is a finite-dimensional space) admitting extensions to braidings between W and W

, so that the contraction map is a σ-morphism. If V ≡ W ⊕ W

then a form F and a braid σ on V are expressible in terms of the extended braiding τ and the contraction map.

In the classical theory spinors can be equivalently viewed as elements of the left C`(V, σ, F )-ideal, generated by a volume element of V

< V ≡ V

⊕ V

. A similar descrip- tion is possible in the braided geometry if an external algebra (V

)

admits a volume ele- ment. A form ω ∈ V

(σ) is said to be a volume on V

(σ) if V

ω = 0. Let ω be a volume on V

. Then a left C`(V, σ, F )-ideal gen ω is isomorphic to S as a left C`(V, σ, F )-module.

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