Clifford Algebra Anti-self-dual Form

Clifford Algebra

Note 3

Anti-self-dual Form

TANAKA Akio

1

Oriented Euclid vector space     V

Complexification of V     V ⊗C

Non-negatively definitive value inner product of V     Q

Tensor algebra of V     T ( V )

Ideal of T ( V )     I_Q

Clifford algebra     C ( V, Q ) = T ( V ) / I_Q

Clifford algebra of mod.2     C ( V ) = C ⁺ ( V ) ⊕ C ^- ( V )

Oriented orthonormal basis of V   (e_i )ⁿ_i=1

Charity operator    C(V) ⊗_RC ∋ : = i^n/2e₁…e_n

Exterior product of V     ΛV

Clifford module   ΛV ⊗_RC

Oriented n-dimensional Riemann manifold     M

Tangent vector bundle of M     TM

Bundle of exterior differential of TM     ΛT *M

Complexificated exterior product bundle of M       ΛT *M ⊗_RC

Hodge star operator     * : Λ^kT *M ⊗_RC →Λ^n-kT *M ⊗_RC

2

Cross section space    Γ( M, ΛT *M ) = Ω( M )

Exterior product of Ω( M )     Ωⁱ( M ) =Γ( M, ΛⁱT *M )

Differential form space     Ω( M ) = _I Ωⁱ( M )

Exterior differential     d : Ω^●( M ) →Ω^●+1( M )

Adjoint operator of exterior differential d     d* : Ω^●( M ) →Ω^●-1( M )

de Rham complex    ( Ω( M ), d )

de Rham Cohomology group of M     Hⁱ ( M ) = Hⁱ ( Ω( M ), d ) = ker (d : Ωⁱ( M ) →Ωⁱ⁺¹( M ) ) / Im ( d : Ω^i-1( M ) →Ωⁱ( M ) )

Vector space Hⁱ ( M )

dα= 0  closed form of α

α = dβ exact form of α

Family of forms     [α]

Product of vector space is algebra.     [α₁]∧[α₂] = [α₁∧α₂]

*    Dirac operator      d + d*

Differential form that satisfies *α = α     Self-dual form    

Differential form that satisfies *α = - α     Anti-self-dual form

3

4-multiple dimensional oriented compact Riemann manifold     M

Signature operator     Operator d + d* over Clifford algebra ΛT *M ⊗_RC

[Note]

Exterior differential d and adjoint operator of exterior differential d* are corresponded with the concept of <orbit table> that is described in the paper, Quantum Theory for Language.

More details, refer to <24> and <#10> in the next paper.

Quantum Theory for Language Synopsis / Tokyo January 15, 2004

On history of Quantum Theory for Language, refer to the next.

Invitation to Quantum Theory for Language / Data arranged at Tokyo January 24, 2005

Tokyo January 15, 2008

Sekinan Research Field of Language

Clifford Algebra Anti-automorphism

Clifford Algebra

Note 2

Anti-automorphism

TANAKA Akio

1

n-dimensional real vector space     V

Inner product of V     Q

Tensor algebra of V     T ( V )

Ideal of T ( V )     I_Q that is generated from all the elements of v⊗w + w⊗v + 2Q(v, w)     (v, w∈V)

Clifford algebra     C ( V, Q ) = T ( V ) / I_Q

2

C ( V ) Clifford module    E

Super module     E = E⁺ ⊕ E^-

C⁺(V)·⊂

C^-(V)·⊂

3

Automorphism of V     v → v* = -v   ( v∈V )

4

Anti-automorphism of T ( V )     (vw)* = w*v*   ( v, w∈T ( V ) )

5

Anti-automorphism of C ( V ) : = a → a*

[Reference]

World and anti-world is considerable from Clifford algebra’s anti-automorphism.

Refer to the next.

Duplicability Theory / Antiworld / Tokyo December 26, 2005

Also refer to the next.

Mirror Theory / Tokyo June 5, 2004

Mirror Language / Tokyo June 10, 2004

Guarantee of Language / Tokyo June 12, 2004

Actual Language and Imaginary Language / Tokyo September 23, 2004

Tokyo January 12, 2008

Sekinan Research Field of Language

Clifford Algebra From Super Space to Quantization

Clifford Algebra

Note 1

From Super Space to Quantization

TANAKA Akio

1

Super space   Vector space that has Z₂ grading  E = E⁺ ⊕ E^-

Elements of E⁺ are called even.

Elements of E⁺ are called odd.

2

Super algebra   Aⁱ ·A^j = A^i+ji, j ∈ Z₂

+, - of elements of Z₂ are expressed by 0, 1.

3

Element of super algebra A = A⁺ ⊕ A^-      a

|a| = 0  a ∈ A⁺

|a| = 1  a ∈ A^-

Two elements of super algebra   a, b

a, b ∈ 

Super commutator    [ a, b ] = ab – ( -1 )^|a||b|ba

Supper commutator satisfies super Lie algebra’s next axioms.

[ a, b ]  + ( -1 )^|a||b|[ b, a ] = 0

[ a, [ b,c] ] = [ [a, b ], c ] + ( -1 )^|a||b|[ b, [ a, c] ]

When [ a, b ] =0, a, b are called super commutative.

3

Super spaces     E = E⁺ ⊕ E^-   F = F⁺ ⊕ F^-

Tensor product E⊗F becomes super space by the next.

(E⊗F) ⁺ = E⁺⊗F⁺ ⊕ E^-⊗F^-

(E⊗F) ^- = E⁺⊗F^- ⊕ E^-⊗F⁺

4

Super algebra   A, B

Super space   A ⊗ B

Tensor product of super algebra     (a₁⊗b₁) ·(a₂⊗b₂) = (-1)^|b1||a2|(a₁a₂⊗b₁b₂)

5

Hermitian super space is complex vector space in which E⁺ and E^- are both Hermitian metric.

6

n-dimensional real vector space     V

Inner product of V     Q

Tensor algebra of V     T ( V ) = T^k ( V )

Ideal of T ( V )     I _Q

Clifford algebra    C ( V, Q ) = T ( V ) / I _Q

C ( V, Q ) satisfies relation  vw + wv = -2Q (v, w ) ( v, w ∈ V )

7

Super module     E = E⁺ ⊕ E^-

Clifford module    C⁺(V)·⊂

                 C^-(V)·⊂

8

Exterior product space’s n-dimensional vector space V over field has direct sum and product e_i  e_j = - e_j  e_i.  ( e₁, …e_n    basis of V )

Exterior algebra of V     V

9

Family of V’s inner product     Q

Family of Clifford algebra     C ( V, Q )

10

When  is seemed to be Planck constant, C ( V ) is represented as quantization of V.

[Reference]

Quantization of language and property of quantum are considerable from Clifford algebra’s quantization.

Refer to the next.

Property of Language / Tokyo May 21, 2004

Tokyo January 10, 2008

Sekinan Research Field of Language

www.sekinan.org

Kac-Moody Lie Algebra Conjecture 1 Finiteness in Infinity on Language

Kac-Moody Lie Algebra

Conjecture 1

Finiteness in Infinity on Language

TANAKA Akio

1 Total of words is finite at a certain point of time.

2 Total of finite words combination, phrase, is probably finite.

3 Sentence is free combination of words. So total of sentences is seemed to be infinite.

4 Now cognition of sentence is probably seemed to be based on finite condition of language.

5 If sentences be really infinite, cognition of them is at last seemed to be impossible.

6 Sentence’s appearance of infinity is possibly finite, or finite in infinite.  

7 From freeness on generation of sentence, sentence is radically infinite and from cognition of that, sentence is seemed to be a certain type of finite in infinity.

[References]

Subdivision / Tokyo April 10, 2005

Algebraic Linguistics /Linguistic Result /Deep Fissure between Word and Sentence /Tokyo September 10, 2007

Tokyo February 10, 2008

Sekinan Research Field of Language

[Basis February 26, 2008]

Operator Algebra / Note 1 / Differential Operator and Symbol / Tokyo February 24, 2008

Kac-Moody Lie Algebra Quantum Group

Kac-Moody Lie Algebra

Note 2

Quantum Group

TANAKA Akio

1 <Cartan matrix>

Base field     K

Finite index set     I

Square matrix that has elements by integer     A = ( a_ij )_{i, j ∈ I}

Matrix that satisfies the next is called Cartan matrix.

i, j ∈ I

(1) a_ii = 2

(2) a_ij ≤ 0  ( i ≠j )

(3) a_ij = 0 ⇔ a_ji = 0

2 <Symmetrizable>

Cartan matrix     A = (a_ij)_i, j ∈I

Family of positive rational number    {d_i}_i∈I

Arbitrary i, j∈I    d_ia_ij = d_ja_ji

A is called symmetrizable.

3 <Fundamental root data>

Finite dimension vector space     h

Linearly independent subset of h     {h_i}_i∈I

Dual space of h     h*= Hom_K (h, K )

Linearly independent subset of h*     {α_i} _i∈I

Φ = {h, {h_i}_i∈I, {α_i} _i∈I }

Cartan matrix A = {α_i(h_i)} _{I, j∈I}

Φis called fundamental root data of A that is Cartan matrix.

4 <Standard form>

Symmetrizable Cartan matrix    A = (a_ij)_i, j ∈I

Fundamental root data     {h, {h_i}_i∈I, {α_i} _i∈I }

E = α_i ⊂h*

Family of positive rational number     {d_i}_i∈I

d_ia_ij = d_ja_ji

Symmetry bilinear form over E     ( , ) : E×E → K     ( (α_i ,α_j ) = d_ia_ij )

The form is called standard form.

5 <Lattice>

n-dimensional Euclid space    Rⁿ

Linear independent vector     v₁, …, v_n

Lattice of Rⁿ     m₁v₁+ … +m_nv_n     ( m₁, …, m_n ∈ Z )

Lattice of h     h_Z

6 <Integer fundamental root data>

From the upperv3, 4 and 5, the next three components are defined.

(Φ, ( , ), h_Z )

When the components satisfy the next, they are called integer fundamental root data.

 i ∈ I

(1)  ∈ Z

(2) α_i ( h_z ) ⊂ Z

(3) t_i :=  h_i ∈ h_z

7 <Associative algebra>

Vector space over K     A

Bilinear product over K     A×A → A

When A is ring, it is called associative algebra.

8 <Similarity>

Integer     m

t similarity of m    [m]_t

[m]_t = t^m-t^-m / t- t^-1

Integer   m, n   m≧n≧0

Binomial coefficient     (^m_n)

t similarity of m!     [m]_t! = [m]_t! [m-1]_t!...[1]_t

t similarity of (^m_n)    [^m_n]_t = [m]_t! / [n]_t! [m-n]_t!

[^m₀] = [^m_m]_t = 1

8 <Quantum group>

Integer fundamental root data that has Cartan matrix A = ( a_ij )_{i, j ∈ I}

      Ψ = ((h, {h_i}_i∈I, {α_i} _i∈I ), ( , ), h_z )

Generating set     {K^h}_h∈hz ∪{E_i, F_i}_i∈I

Associative algebra U over K (q), that is defined the next relations, is called quantum group associated with Ψ.

(1) k^hk^h’ = k^h+h’     ( h, h’∈h_Z )

(2) k⁰ = 1

(3) K^hE_iK^-h = q^αi(h)E_i    ( h∈h_Z , i∈I )

(4) K^hF_iK^-h = q^αi(h)F_i   ( h∈h_Z , i∈I )

(5) E_i F_j – F_jE_i = _ij  K_i - K_i^-1 / q_i – q_i^-1     ( i , j∈I )

(6) ^p [^1-aij_p]_qiE_i^1-aij-pE_jE_i^p = 0     ( i , j∈I , i ≠j )

(7) ^p [^1-aij_p]_qiF_i^1-aij-pF_jF_i^p = 0     ( i , j∈I , i ≠j )

[Note]

Parameter q in K is thinkable in connection with the concept of <jump> at the paper Place where Quantum of Language exists / 27 /.

Refer to the next.

Place where Quantum of Language exists / Tokyo July 18, 2004

Tokyo February 9, 2008

Sekinan Research Field of Language