Definition 3.21 .

Given n β‰₯ 1 𝑛 1 n\geq 1 italic_n β‰₯ 1 , denote by G n subscript 𝐺 𝑛 G_{n} italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT the groupoid n Γ— G Γ— n 𝑛 𝐺 𝑛 n\times G\times n italic_n Γ— italic_G Γ— italic_n endowed with the product topology, with set of objects G 0 Γ— n superscript 𝐺 0 𝑛 G^{0}\times n italic_G start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT Γ— italic_n , and operations defined by

s ⁒ ( i , Ξ³ , j ) = ( s ⁒ ( Ξ³ ) , j ) , r ⁒ ( i , Ξ³ , j ) = ( r ⁒ ( Ξ³ ) , i ) and ( i , Ξ³ , j ) ⁒ ( j , ρ , k ) = ( i , Ξ³ ⁒ ρ , k ) . formulae-sequence 𝑠 𝑖 𝛾 𝑗 𝑠 𝛾 𝑗 formulae-sequence π‘Ÿ 𝑖 𝛾 𝑗 π‘Ÿ 𝛾 𝑖 and 𝑖 𝛾 𝑗 𝑗 𝜌 π‘˜ 𝑖 𝛾 𝜌 π‘˜ s(i,\gamma,j)=(s(\gamma),j)\ ,\ \ r(i,\gamma,j)=(r(\gamma),i)\ \ \mbox{ and }\ \ (i,\gamma,j)(j,\rho,k)=(i,\gamma\rho,k). italic_s ( italic_i , italic_Ξ³ , italic_j ) = ( italic_s ( italic_Ξ³ ) , italic_j ) , italic_r ( italic_i , italic_Ξ³ , italic_j ) = ( italic_r ( italic_Ξ³ ) , italic_i ) and ( italic_i , italic_Ξ³ , italic_j ) ( italic_j , italic_ρ , italic_k ) = ( italic_i , italic_Ξ³ italic_ρ , italic_k ) .

Denote by 𝒡 ( n ) superscript 𝒡 𝑛 \mathcal{Z}^{(n)} caligraphic_Z start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT the Borel Banach bundle over G 0 Γ— n superscript 𝐺 0 𝑛 G^{0}\times n italic_G start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT Γ— italic_n such that 𝒡 ( x , j ) ( n ) = 𝒡 x superscript subscript 𝒡 π‘₯ 𝑗 𝑛 subscript 𝒡 π‘₯ \mathcal{Z}_{(x,j)}^{(n)}=\mathcal{Z}_{x} caligraphic_Z start_POSTSUBSCRIPT ( italic_x , italic_j ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT = caligraphic_Z start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , with basic sequence ( Οƒ k ( n ) ) k ∈ Ο‰ subscript superscript subscript 𝜎 π‘˜ 𝑛 π‘˜ πœ” (\sigma_{k}^{(n)})_{k\in\omega} ( italic_Οƒ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ italic_Ο‰ end_POSTSUBSCRIPT defined by

Οƒ k , ( x , j ) ( n ) = Οƒ k , x superscript subscript 𝜎 π‘˜ π‘₯ 𝑗 𝑛 subscript 𝜎 π‘˜ π‘₯ \sigma_{k,(x,j)}^{(n)}=\sigma_{k,x} italic_Οƒ start_POSTSUBSCRIPT italic_k , ( italic_x , italic_j ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT = italic_Οƒ start_POSTSUBSCRIPT italic_k , italic_x end_POSTSUBSCRIPT

for ( x , j ) ∈ G 0 Γ— n π‘₯ 𝑗 superscript 𝐺 0 𝑛 (x,j)\in G^{0}\times n ( italic_x , italic_j ) ∈ italic_G start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT Γ— italic_n . Endow G 0 Γ— n superscript 𝐺 0 𝑛 G^{0}\times n italic_G start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT Γ— italic_n with the measure ΞΌ ( n ) = ΞΌ Γ— c n superscript πœ‡ 𝑛 πœ‡ subscript 𝑐 𝑛 \mu^{(n)}=\mu\times c_{n} italic_ΞΌ start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT = italic_ΞΌ Γ— italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , and define the amplification T ( n ) : G n β†’ Iso ⁒ ( 𝒡 ( n ) ) : superscript 𝑇 𝑛 β†’ subscript 𝐺 𝑛 Iso superscript 𝒡 𝑛 T^{(n)}\colon G_{n}\rightarrow\mathrm{Iso}(\mathcal{Z}^{(n)}) italic_T start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT : italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT β†’ roman_Iso ( caligraphic_Z start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ) of T 𝑇 T italic_T by T ( i , Ξ³ , j ) ( n ) = T Ξ³ superscript subscript 𝑇 𝑖 𝛾 𝑗 𝑛 subscript 𝑇 𝛾 T_{(i,\gamma,j)}^{(n)}=T_{\gamma} italic_T start_POSTSUBSCRIPT ( italic_i , italic_Ξ³ , italic_j ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT = italic_T start_POSTSUBSCRIPT italic_Ξ³ end_POSTSUBSCRIPT for ( i , Ξ³ , j ) ∈ G n 𝑖 𝛾 𝑗 subscript 𝐺 𝑛 (i,\gamma,j)\in G_{n} ( italic_i , italic_Ξ³ , italic_j ) ∈ italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .

Definition 8 .

Let Q 𝑄 Q italic_Q be a quantity space over β„› β„› \mathcal{R} caligraphic_R . We set

Ξ» ⁒ p + Ξ» β€² ⁒ p = ( Ξ» + Ξ» β€² ) ⁒ p πœ† 𝑝 superscript πœ† β€² 𝑝 πœ† superscript πœ† β€² 𝑝 \lambda p+\lambda^{\prime}p=\left(\lambda+\lambda^{\prime}\right)p italic_Ξ» italic_p + italic_Ξ» start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT italic_p = ( italic_Ξ» + italic_Ξ» start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) italic_p

for any invertible p ∈ Q 𝑝 𝑄 p\in Q italic_p ∈ italic_Q and Ξ» , Ξ» β€² ∈ β„› πœ† superscript πœ† β€² β„› \lambda,\lambda^{\prime}\in\mathcal{R} italic_Ξ» , italic_Ξ» start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∈ caligraphic_R . β–‘ β–‘ \hphantom{\square}\square β–‘

As a trivial consequence of this definition, addition of quantities is commutative.

Definition 2

A probability density function f ⁒ ( x ) 𝑓 π‘₯ f(x) italic_f ( italic_x ) has heavy tails if for any Ξ· > 0 πœ‚ 0 \eta>0 italic_Ξ· > 0 holds that

f ⁒ ( x ) = β„“ ⁒ ( x ) ⁒ x - Ξ· , 𝑓 π‘₯ β„“ π‘₯ superscript π‘₯ πœ‚ f(x)=\ell(x)x^{-\eta}, italic_f ( italic_x ) = roman_β„“ ( italic_x ) italic_x start_POSTSUPERSCRIPT - italic_Ξ· end_POSTSUPERSCRIPT ,

where β„“ normal-β„“ \ell roman_β„“ is a slow-varying function in the infinite, and Ξ· πœ‚ \eta italic_Ξ· is the tail index.

Definition 10.37 .

For a ring ( R , + , β‹… ) 𝑅 β‹… (R,+,\cdot) ( italic_R , + , β‹… ) and a commutative group ( M , + ) 𝑀 (M,+) ( italic_M , + ) a function ( r , m ) ↦ r ⁒ m maps-to π‘Ÿ π‘š π‘Ÿ π‘š (r,m)\mapsto rm ( italic_r , italic_m ) ↦ italic_r italic_m from R Γ— M 𝑅 𝑀 R\times M italic_R Γ— italic_M to M 𝑀 M italic_M is called a module over R 𝑅 R italic_R or an R 𝑅 R italic_R - module (symbolised by M 𝑀 M italic_M ) when it holds

( r + s ) ⁒ m = r ⁒ m + s ⁒ m , r ⁒ ( m + n ) = r ⁒ m + r ⁒ n , r ⁒ ( s ⁒ m ) = ( r ⁒ s ) ⁒ m , 1 ⁒ m = m formulae-sequence π‘Ÿ 𝑠 π‘š π‘Ÿ π‘š 𝑠 π‘š formulae-sequence π‘Ÿ π‘š 𝑛 π‘Ÿ π‘š π‘Ÿ 𝑛 formulae-sequence π‘Ÿ 𝑠 π‘š π‘Ÿ 𝑠 π‘š 1 π‘š π‘š (r+s)m=rm+sm,r(m+n)=rm+rn,r(sm)=(rs)m,1m=m ( italic_r + italic_s ) italic_m = italic_r italic_m + italic_s italic_m , italic_r ( italic_m + italic_n ) = italic_r italic_m + italic_r italic_n , italic_r ( italic_s italic_m ) = ( italic_r italic_s ) italic_m , 1 italic_m = italic_m

for all r , s ∈ R , m , n ∈ M formulae-sequence π‘Ÿ 𝑠 𝑅 π‘š 𝑛 𝑀 r,s\in R,m,n\in M italic_r , italic_s ∈ italic_R , italic_m , italic_n ∈ italic_M . 63 63 63 When M 𝑀 M italic_M is even a ring with additional property r ⁒ ( m ⁒ n ) = ( r ⁒ m ) ⁒ n π‘Ÿ π‘š 𝑛 π‘Ÿ π‘š 𝑛 r(mn)=(rm)n italic_r ( italic_m italic_n ) = ( italic_r italic_m ) italic_n for all r ∈ R , m , n ∈ M formulae-sequence π‘Ÿ 𝑅 π‘š 𝑛 𝑀 r\in R,m,n\in M italic_r ∈ italic_R , italic_m , italic_n ∈ italic_M then it is called an algebra . In case R 𝑅 R italic_R is a field M 𝑀 M italic_M is called an R 𝑅 R italic_R - vectorspace . A group homomorphism l : M β†’ N : 𝑙 β†’ 𝑀 𝑁 l:M\to N italic_l : italic_M β†’ italic_N between R 𝑅 R italic_R -modules M , N 𝑀 𝑁 M,N italic_M , italic_N is called linear when l ⁒ ( Ξ» ⁒ m ) = Ξ» ⁒ l ⁒ ( m ) 𝑙 πœ† π‘š πœ† 𝑙 π‘š l(\lambda m)=\lambda l(m) italic_l ( italic_Ξ» italic_m ) = italic_Ξ» italic_l ( italic_m ) for all m ∈ M , Ξ» ∈ R formulae-sequence π‘š 𝑀 πœ† 𝑅 m\in M,\lambda\in R italic_m ∈ italic_M , italic_Ξ» ∈ italic_R . For R 𝑅 R italic_R -modules M , N 𝑀 𝑁 M,N italic_M , italic_N a map M Γ— … Γ— M β†’ N β†’ 𝑀 … 𝑀 𝑁 M\times...\times M\to N italic_M Γ— … Γ— italic_M β†’ italic_N is called multilinear when it is linear in each variable. A multlinear function M Γ— M β†’ N β†’ 𝑀 𝑀 𝑁 M\times M\to N italic_M Γ— italic_M β†’ italic_N is also called bilinear . In case of N = R 𝑁 𝑅 N=R italic_N = italic_R a linear/bilinear/multilinear function is called a linear/bilinear/multilinear form . A subset of an R-module M 𝑀 M italic_M is called an R- submodule of M 𝑀 M italic_M when it is an R-module. When the ring R 𝑅 R italic_R is clear from context we just say submodule . Analogously a subspace of a vectorspace is defined.

Definition 2.3 .

A 2-cocycle on 𝒒 𝒒 \mathcal{G} caligraphic_G is a continuous circle-valued function Ο‰ : 𝒒 ( 2 ) β†’ 𝕋 : πœ” β†’ superscript 𝒒 2 𝕋 \omega:\mathcal{G}^{(2)}\to\mathbb{T} italic_Ο‰ : caligraphic_G start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT β†’ blackboard_T such that whenever ( x , y ) , ( y , z ) ∈ 𝒒 ( 2 ) π‘₯ 𝑦 𝑦 𝑧 superscript 𝒒 2 (x,y),\,(y,z)\in\mathcal{G}^{(2)} ( italic_x , italic_y ) , ( italic_y , italic_z ) ∈ caligraphic_G start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , the cocycle condition holds:

Ο‰ ⁒ ( x ⁒ y , z ) ⁒ Ο‰ ⁒ ( x , y ) = Ο‰ ⁒ ( x , y ⁒ z ) ⁒ Ο‰ ⁒ ( y , z ) . πœ” π‘₯ 𝑦 𝑧 πœ” π‘₯ 𝑦 πœ” π‘₯ 𝑦 𝑧 πœ” 𝑦 𝑧 \omega(xy,z)\omega(x,y)=\omega(x,yz)\omega(y,z). italic_Ο‰ ( italic_x italic_y , italic_z ) italic_Ο‰ ( italic_x , italic_y ) = italic_Ο‰ ( italic_x , italic_y italic_z ) italic_Ο‰ ( italic_y , italic_z ) . (2)
Definition 21.1 .

[ 4 , LemmaΒ 3.2] Let 𝔰 ⁒ 𝔩 2 𝔰 subscript 𝔩 2 \mathfrak{sl}_{2} fraktur_s fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT denote the Lie algebra over 𝔽 𝔽 \mathbb{F} blackboard_F with basis x , y , z π‘₯ 𝑦 𝑧 x,y,z italic_x , italic_y , italic_z and Lie bracket

[ x , y ] = 2 ⁒ x + 2 ⁒ y , [ y , z ] = 2 ⁒ y + 2 ⁒ z , [ z , x ] = 2 ⁒ z + 2 ⁒ x . formulae-sequence π‘₯ 𝑦 2 π‘₯ 2 𝑦 formulae-sequence 𝑦 𝑧 2 𝑦 2 𝑧 𝑧 π‘₯ 2 𝑧 2 π‘₯ [{x,y}]=2{x}+2{y},\qquad[{y,z}]=2{y}+2{z},\qquad[{z,x}]=2{z}+2{x}. [ italic_x , italic_y ] = 2 italic_x + 2 italic_y , [ italic_y , italic_z ] = 2 italic_y + 2 italic_z , [ italic_z , italic_x ] = 2 italic_z + 2 italic_x . (41)
Definition 2.3 .

A gyrogroup ( G , βŠ• ) 𝐺 direct-sum (G,\oplus) ( italic_G , βŠ• ) having the additional property that

a βŠ• b = gyr ⁒ [ a , b ] ⁒ ( b βŠ• a ) direct-sum π‘Ž 𝑏 gyr π‘Ž 𝑏 direct-sum 𝑏 π‘Ž a\oplus b={\mathrm{gyr}[{a,b}]}{(}{b\oplus a}) italic_a βŠ• italic_b = roman_gyr [ italic_a , italic_b ] ( italic_b βŠ• italic_a ) (gyrocommutative law)

for all a , b ∈ G π‘Ž 𝑏 𝐺 a,b\in G italic_a , italic_b ∈ italic_G is called a gyrocommutative gyrogroup .

Definition 5.3 .

Let K 𝐾 K italic_K be a commutative ring. Given a cobordism category and a fixed parameter Ξ΄ ∈ K 𝛿 𝐾 \delta\in K italic_Ξ΄ ∈ italic_K , the diagram category 𝖣 𝖣 \mathsf{D} sansserif_D has objects β„• β„• \mathbb{N} blackboard_N . For r , s ∈ β„• π‘Ÿ 𝑠 β„• r,s\in\mathbb{N} italic_r , italic_s ∈ blackboard_N , the set of morphisms Hom 𝖣 ⁒ ( r , s ) subscript Hom 𝖣 π‘Ÿ 𝑠 \mathrm{Hom}_{\mathsf{D}}(r,s) roman_Hom start_POSTSUBSCRIPT sansserif_D end_POSTSUBSCRIPT ( italic_r , italic_s ) is the free K 𝐾 K italic_K -module with basis D ⁒ ( r , s ) 𝐷 π‘Ÿ 𝑠 D(r,s) italic_D ( italic_r , italic_s ) . The composition in 𝖣 𝖣 \mathsf{D} sansserif_D , which we will refer to as multiplication henceforth, is defined as

(10) x β‹… y = Ξ΄ c ⁒ ( x , y ) ⁒ x ∘ y , β‹… π‘₯ 𝑦 superscript 𝛿 𝑐 π‘₯ 𝑦 π‘₯ 𝑦 \displaystyle x\cdot y=\delta^{c(x,y)}x\circ y, italic_x β‹… italic_y = italic_Ξ΄ start_POSTSUPERSCRIPT italic_c ( italic_x , italic_y ) end_POSTSUPERSCRIPT italic_x ∘ italic_y ,

for basis elements x ∈ D ⁒ ( r , s ) , y ∈ D ⁒ ( s , t ) formulae-sequence π‘₯ 𝐷 π‘Ÿ 𝑠 𝑦 𝐷 𝑠 𝑑 x\in D(r,s),y\in D(s,t) italic_x ∈ italic_D ( italic_r , italic_s ) , italic_y ∈ italic_D ( italic_s , italic_t ) , and extended K 𝐾 K italic_K -bilinearly.

Thus, 𝖣 𝖣 \mathsf{D} sansserif_D is a K 𝐾 K italic_K -linear (or pre-additive) category. The tensor product from 𝖒 𝖒 \mathsf{C} sansserif_C carries over to 𝖣 𝖣 \mathsf{D} sansserif_D and makes the diagram category into a strict monoidal category.

Definition 2 .

Because Ο‰ πœ” \omega italic_Ο‰ is a primitive 8 t ⁒ h superscript 8 𝑑 β„Ž 8^{th} 8 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT root of unity, β„€ ⁒ [ Ο‰ ] β„€ delimited-[] πœ” \mathbb{Z}[\omega] blackboard_Z [ italic_Ο‰ ] has Ο• ⁒ ( 8 ) = 4 italic-Ο• 8 4 \phi(8)=4 italic_Ο• ( 8 ) = 4 automorphisms. One such automorphism is the usual complex conjugation which maps i 𝑖 i italic_i to - i 𝑖 -i - italic_i and 2 2 \sqrt{2} square-root start_ARG 2 end_ARG to itself. We will denote complex conjugation by ( - ) † superscript † (-)^{\dagger} ( - ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT . For any element in β„€ ⁒ [ Ο‰ ] β„€ delimited-[] πœ” \mathbb{Z}[\omega] blackboard_Z [ italic_Ο‰ ] , we have

( a ⁒ Ο‰ 3 + b ⁒ Ο‰ 2 + c ⁒ Ο‰ + d ) † = - c ⁒ Ο‰ 3 + b ⁒ Ο‰ 2 - a ⁒ Ο‰ + d . superscript π‘Ž superscript πœ” 3 𝑏 superscript πœ” 2 𝑐 πœ” 𝑑 † 𝑐 superscript πœ” 3 𝑏 superscript πœ” 2 π‘Ž πœ” 𝑑 (a\omega^{3}+b\omega^{2}+c\omega+d)^{\dagger}=-c\omega^{3}+b\omega^{2}-a\omega% +d. ( italic_a italic_Ο‰ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_b italic_Ο‰ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c italic_Ο‰ + italic_d ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = - italic_c italic_Ο‰ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_b italic_Ο‰ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_a italic_Ο‰ + italic_d .

Another automorphism is 2 2 \sqrt{2} square-root start_ARG 2 end_ARG -conjugation, which maps i 𝑖 i italic_i to itself and 2 2 \sqrt{2} square-root start_ARG 2 end_ARG to - 2 2 -\sqrt{2} - square-root start_ARG 2 end_ARG . We will denote 2 2 \sqrt{2} square-root start_ARG 2 end_ARG -conjugation by ( - ) βˆ™ superscript βˆ™ (-)^{\bullet} ( - ) start_POSTSUPERSCRIPT βˆ™ end_POSTSUPERSCRIPT . For any element in β„€ ⁒ [ Ο‰ ] β„€ delimited-[] πœ” \mathbb{Z}[\omega] blackboard_Z [ italic_Ο‰ ] , we have

( a ⁒ Ο‰ 3 + b ⁒ Ο‰ 2 + c ⁒ Ο‰ + d ) † = - a ⁒ Ο‰ 3 + b ⁒ Ο‰ 2 - c ⁒ Ο‰ + d . superscript π‘Ž superscript πœ” 3 𝑏 superscript πœ” 2 𝑐 πœ” 𝑑 † π‘Ž superscript πœ” 3 𝑏 superscript πœ” 2 𝑐 πœ” 𝑑 (a\omega^{3}+b\omega^{2}+c\omega+d)^{\dagger}=-a\omega^{3}+b\omega^{2}-c\omega% +d. ( italic_a italic_Ο‰ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_b italic_Ο‰ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c italic_Ο‰ + italic_d ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = - italic_a italic_Ο‰ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_b italic_Ο‰ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_c italic_Ο‰ + italic_d .

The remaining two automorphisms are obviously the identity function and ( - ) † βˆ™ = ( - ) βˆ™ † superscript † absent βˆ™ superscript βˆ™ absent † (-)^{\dagger\bullet}=(-)^{\bullet\dagger} ( - ) start_POSTSUPERSCRIPT † βˆ™ end_POSTSUPERSCRIPT = ( - ) start_POSTSUPERSCRIPT βˆ™ † end_POSTSUPERSCRIPT .

Definition 1 .

A finite forest algebra ( H , V ) 𝐻 𝑉 (H,V) ( italic_H , italic_V ) is an 𝖀π–₯ 𝖀π–₯ \mathsf{EF} sansserif_EF -algebra if it satisfies the identities

h + h β€² = h β€² + h , v ⁒ h + h = v ⁒ h formulae-sequence β„Ž superscript β„Ž β€² superscript β„Ž β€² β„Ž 𝑣 β„Ž β„Ž 𝑣 β„Ž h+h^{\prime}=h^{\prime}+h,vh+h=vh italic_h + italic_h start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT = italic_h start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT + italic_h , italic_v italic_h + italic_h = italic_v italic_h

for all h , h β€² ∈ H β„Ž superscript β„Ž normal-β€² 𝐻 h,h^{\prime}\in H italic_h , italic_h start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∈ italic_H and v ∈ V . 𝑣 𝑉 v\in V. italic_v ∈ italic_V . The second identity with v = 1 𝑣 1 v=1 italic_v = 1 gives h + h = h . β„Ž β„Ž β„Ž h+h=h. italic_h + italic_h = italic_h . Thus every 𝖀π–₯ 𝖀π–₯ \mathsf{EF} sansserif_EF -algebra is horizontally idempotent and commutative.

Definition 1 .

A team is any set of assignments for a fixed set of variables. A team X 𝑋 X italic_X is said to satisfy the dependence atom

= ( x , y ) , absent π‘₯ 𝑦 {=\mskip-1.2mu }(x,y), = ( italic_x , italic_y ) , (1)

where x π‘₯ x italic_x and y 𝑦 y italic_y are finite sequences of variables, if any two assignments s 𝑠 s italic_s and s β€² superscript 𝑠 normal-β€² s^{\prime} italic_s start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT in X 𝑋 X italic_X satisfy

s ⁒ ( x ) = s β€² ⁒ ( x ) β†’ s ⁒ ( y ) = s β€² ⁒ ( y ) . 𝑠 π‘₯ superscript 𝑠 β€² π‘₯ β†’ 𝑠 𝑦 superscript 𝑠 β€² 𝑦 s(x)=s^{\prime}(x)\rightarrow s(y)=s^{\prime}(y). italic_s ( italic_x ) = italic_s start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ( italic_x ) β†’ italic_s ( italic_y ) = italic_s start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ( italic_y ) . (2)
Definition 1.11 (Invariant measure) .

Let 𝒒 𝒒 \mathcal{G} caligraphic_G be a measurable groupoid with a transversal function Ξ½ 𝜈 \nu italic_Ξ½ . A measure m π‘š m italic_m on the base space ( Ξ© , ℬ Ξ© ) Ξ© subscript ℬ Ξ© (\Omega,\mathcal{B}_{\Omega}) ( roman_Ξ© , caligraphic_B start_POSTSUBSCRIPT roman_Ξ© end_POSTSUBSCRIPT ) of units is called Ξ½ 𝜈 \nu italic_Ξ½ -invariant (or simply invariant , if there is no ambiguity in the choice of Ξ½ 𝜈 \nu italic_Ξ½ ) if

m ∘ Ξ½ = ( m ∘ Ξ½ ) ∼ , π‘š 𝜈 superscript π‘š 𝜈 similar-to m\circ\nu=(m\circ\nu)^{\sim}, italic_m ∘ italic_Ξ½ = ( italic_m ∘ italic_Ξ½ ) start_POSTSUPERSCRIPT ∼ end_POSTSUPERSCRIPT ,

where ( m ∘ Ξ½ ) ⁒ ( f ) = ∫ Ξ© Ξ½ Ο‰ ⁒ ( f ) ⁒ 𝑑 m ⁒ ( Ο‰ ) π‘š 𝜈 𝑓 subscript Ξ© superscript 𝜈 πœ” 𝑓 differential-d π‘š πœ” (m\circ\nu)(f)=\int_{\Omega}\nu^{\omega}(f)dm(\omega) ( italic_m ∘ italic_Ξ½ ) ( italic_f ) = ∫ start_POSTSUBSCRIPT roman_Ξ© end_POSTSUBSCRIPT italic_Ξ½ start_POSTSUPERSCRIPT italic_Ο‰ end_POSTSUPERSCRIPT ( italic_f ) italic_d italic_m ( italic_Ο‰ ) and ( m ∘ Ξ½ ) ∼ ⁒ ( f ) = ( m ∘ Ξ½ ) ⁒ ( f ~ ) superscript π‘š 𝜈 similar-to 𝑓 π‘š 𝜈 ~ 𝑓 (m\circ\nu)^{\sim}(f)=(m\circ\nu)(\tilde{f}) ( italic_m ∘ italic_Ξ½ ) start_POSTSUPERSCRIPT ∼ end_POSTSUPERSCRIPT ( italic_f ) = ( italic_m ∘ italic_Ξ½ ) ( ~ start_ARG italic_f end_ARG ) with f ~ ⁒ ( g ) = f ⁒ ( g - 1 ) ~ 𝑓 𝑔 𝑓 superscript 𝑔 1 \tilde{f}(g)=f(g^{-1}) ~ start_ARG italic_f end_ARG ( italic_g ) = italic_f ( italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) .

Definition 1.2 .

We say that E = { E ⁒ [ β‹… ] ⁒ ( Ξ» ) : Ξ» ∈ Ξ“ } 𝐸 conditional-set 𝐸 delimited-[] normal-β‹… πœ† πœ† normal-Ξ“ E=\{E[\cdot](\lambda):\lambda\in\Gamma\} italic_E = { italic_E [ β‹… ] ( italic_Ξ» ) : italic_Ξ» ∈ roman_Ξ“ } is a complete family of functionals E ⁒ [ β‹… ] ⁒ ( Ξ» ) ∈ C β€² 𝐸 delimited-[] normal-β‹… πœ† superscript 𝐢 normal-β€² E[\cdot](\lambda)\in C^{\prime} italic_E [ β‹… ] ( italic_Ξ» ) ∈ italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT if

Ο• ∈ Ξ¦ ⁒ and ⁒ E ⁒ [ Ο• ] ⁒ ( Ξ» ) = 0 ⁒ βˆ€ Ξ» ∈ Ξ“ β‡’ Ο• = 0 . formulae-sequence italic-Ο• Ξ¦ and 𝐸 delimited-[] italic-Ο• πœ† 0 for-all πœ† Ξ“ β‡’ italic-Ο• 0 \phi\in\Phi\mbox{ and }E[\phi](\lambda)=0\hskip 2.845276pt\forall\hskip 2.8452% 76pt\lambda\in\Gamma\quad\Rightarrow\quad\phi=0. italic_Ο• ∈ roman_Ξ¦ and italic_E [ italic_Ο• ] ( italic_Ξ» ) = 0 βˆ€ italic_Ξ» ∈ roman_Ξ“ β‡’ italic_Ο• = 0 . (1.22)
Definition \thedefinition .

Let u , v 𝑒 𝑣 u,v italic_u , italic_v be operators in G ⁒ ( 𝔄 ) G 𝔄 \text{G}(\mathfrak{A}) G ( fraktur_A ) such that u βŸ‚ πš› ⁒ ( v ) perpendicular-to 𝑒 πš› v u\perp\tt r\rm(v) italic_u βŸ‚ typewriter_r ( roman_v ) . The execution of u 𝑒 u italic_u and πš› ⁒ ( v ) πš› v \tt r\rm(v) typewriter_r ( roman_v ) , denoted by u ⁒ : : πš› ⁒ ( v ) 𝑒 fragments : : πš› v u\mathop{\mathopen{:}\mathclose{:}}\tt r\rm(v) italic_u start_BIGOP : : end_BIGOP typewriter_r ( roman_v ) , is defined as:

u : : πš› ( v ) = ( 1 - πš›πš› ) βˆ— ( 1 - u πš› ( v ) ) - 1 ( 1 - πš›πš› ) βˆ— fragments u fragments : : r fragments ( v ) fragments ( 1 rr superscript ) βˆ— superscript fragments ( 1 u r fragments ( v ) ) 1 fragments ( 1 rr superscript ) βˆ— u\mathop{\mathopen{:}\mathclose{:}}\tt r\rm(v)=(1-\tt r\rm\tt r{}^{\ast})(1-u% \tt r\rm(v))^{-1}(1-\tt r\rm\tt r{}^{\ast}) italic_u start_BIGOP : : end_BIGOP typewriter_r ( roman_v ) = ( 1 - typewriter_rr start_FLOATSUPERSCRIPT βˆ— end_FLOATSUPERSCRIPT ) ( 1 - roman_u typewriter_r ( roman_v ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - typewriter_rr start_FLOATSUPERSCRIPT βˆ— end_FLOATSUPERSCRIPT )

Since u βŸ‚ πš› ⁒ ( v ) perpendicular-to 𝑒 πš› v u\perp\tt r\rm(v) italic_u βŸ‚ typewriter_r ( roman_v ) , the inverse of 1 - u ⁒ πš› ⁒ ( v ) 1 𝑒 πš› v 1-u\tt r\rm(v) 1 - italic_u typewriter_r ( roman_v ) always exists and can be computed as the series βˆ‘ i = 0 ∞ ( u ⁒ πš› ⁒ ( v ) ) i superscript subscript 𝑖 0 superscript 𝑒 πš› v i \sum_{i=0}^{\infty}(u\tt r\rm(v))^{i} βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_u typewriter_r ( roman_v ) ) start_POSTSUPERSCRIPT roman_i end_POSTSUPERSCRIPT . One can show that 6 6 6 We will not present a proof here, but it would be similar to the proof of subsectionΒ 4.4 . if u 𝑒 u italic_u and πš› ⁒ ( v ) πš› v \tt r\rm(v) typewriter_r ( roman_v ) are in G β„’ ⁒ ( ℍ ) ⁒ ( 𝔄 ) subscript G β„’ ℍ 𝔄 \text{G}_{\mathcal{L}(\mathbb{H})}(\mathfrak{A}) G start_POSTSUBSCRIPT caligraphic_L ( blackboard_H ) end_POSTSUBSCRIPT ( fraktur_A ) , then u ⁒ : : d ⁒ ( v ) ∈ G β„’ ⁒ ( ℍ ) ⁒ ( 𝔄 ) 𝑒 fragments : : 𝑑 𝑣 subscript G β„’ ℍ 𝔄 u\mathop{\mathopen{:}\mathclose{:}}d(v)\in\text{G}_{\mathcal{L}(\mathbb{H})}(% \mathfrak{A}) italic_u start_BIGOP : : end_BIGOP italic_d ( italic_v ) ∈ G start_POSTSUBSCRIPT caligraphic_L ( blackboard_H ) end_POSTSUBSCRIPT ( fraktur_A ) .

Definition \thedefinition .

Let u , v 𝑒 𝑣 u,v italic_u , italic_v be GoI operators. We define

u ⁒ \with ⁒ v = ( 1 βŠ— p ) ⁒ u ⁒ ( 1 βŠ— p ) βˆ— + ( 1 βŠ— q ) ⁒ v ⁒ ( 1 βŠ— q ) βˆ— 𝑒 \with 𝑣 tensor-product 1 𝑝 𝑒 superscript tensor-product 1 𝑝 βˆ— tensor-product 1 π‘ž 𝑣 superscript tensor-product 1 π‘ž βˆ— u\with v=(1\otimes p)u(1\otimes p)^{\ast}+(1\otimes q)v(1\otimes q)^{\ast} italic_u italic_v = ( 1 βŠ— italic_p ) italic_u ( 1 βŠ— italic_p ) start_POSTSUPERSCRIPT βˆ— end_POSTSUPERSCRIPT + ( 1 βŠ— italic_q ) italic_v ( 1 βŠ— italic_q ) start_POSTSUPERSCRIPT βˆ— end_POSTSUPERSCRIPT

If A , B 𝐴 𝐡 A,B italic_A , italic_B are (weak) types, we define

A ⁒ \with ⁒ B = { u ⁒ \with ⁒ v | u ∈ A , v ∈ B } βŠ₯ βŠ₯ 𝐴 \with 𝐡 superscript conditional-set 𝑒 \with 𝑣 formulae-sequence 𝑒 𝐴 𝑣 𝐡 bottom absent bottom A\with B=\{u\with v\leavevmode\nobreak\ |\leavevmode\nobreak\ u\in A,v\in B\}^% {\bot\bot} italic_A italic_B = { italic_u italic_v | italic_u ∈ italic_A , italic_v ∈ italic_B } start_POSTSUPERSCRIPT βŠ₯ βŠ₯ end_POSTSUPERSCRIPT
Definition 10 .

An m π‘š m italic_m -admissible Segre family is a 2-parameter antiholomorphic family of planar holomorphic curves in a polydisc Ξ” Ξ΄ Γ— Ξ” Ξ΅ subscript Ξ” 𝛿 subscript Ξ” πœ€ \Delta_{\delta}\times\Delta_{\varepsilon} roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT Γ— roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΅ end_POSTSUBSCRIPT which can be parameterized in the form

w = Ξ· Β― ⁒ e Β± i ⁒ Ξ· Β― m - 1 ⁒ Ο† ⁒ ( z , ΞΎ Β― , Ξ· Β― ) , 𝑀 Β― πœ‚ superscript 𝑒 plus-or-minus 𝑖 superscript Β― πœ‚ π‘š 1 πœ‘ 𝑧 Β― πœ‰ Β― πœ‚ w=\bar{\eta}e^{\pm i\bar{\eta}^{m-1}\varphi(z,\bar{\xi},\bar{\eta})}, italic_w = Β― start_ARG italic_Ξ· end_ARG italic_e start_POSTSUPERSCRIPT Β± italic_i Β― start_ARG italic_Ξ· end_ARG start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT italic_Ο† ( italic_z , Β― start_ARG italic_ΞΎ end_ARG , Β― start_ARG italic_Ξ· end_ARG ) end_POSTSUPERSCRIPT , (3.6)

where m β‰₯ 1 π‘š 1 m\geq 1 italic_m β‰₯ 1 is an integer, ΞΎ ∈ Ξ” Ξ΄ , Ξ· ∈ Ξ” Ξ΅ formulae-sequence πœ‰ subscript Ξ” 𝛿 πœ‚ subscript Ξ” πœ€ \xi\in\Delta_{\delta},\eta\in\Delta_{\varepsilon} italic_ΞΎ ∈ roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT , italic_Ξ· ∈ roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΅ end_POSTSUBSCRIPT are holomorphic parameters, and the function Ο† ⁒ ( x , y , u ) πœ‘ π‘₯ 𝑦 𝑒 \varphi(x,y,u) italic_Ο† ( italic_x , italic_y , italic_u ) is holomorphic in the polydisc Ξ” Ξ΄ Γ— Ξ” Ξ΄ Γ— Ξ” Ξ΅ subscript Ξ” 𝛿 subscript Ξ” 𝛿 subscript Ξ” πœ€ \Delta_{\delta}\times\Delta_{\delta}\times\Delta_{\varepsilon} roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT Γ— roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT Γ— roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΅ end_POSTSUBSCRIPT and has there an expansion

Ο† ⁒ ( x , y , u ) = x ⁒ y + βˆ‘ k , l β‰₯ 2 Ο† k ⁒ l ⁒ ( u ) ⁒ x k ⁒ y l , Ο† k ⁒ l ⁒ ( u ) ∈ π’ͺ ⁒ ( Ξ” Ξ΅ ) . formulae-sequence πœ‘ π‘₯ 𝑦 𝑒 π‘₯ 𝑦 subscript π‘˜ 𝑙 2 subscript πœ‘ π‘˜ 𝑙 𝑒 superscript π‘₯ π‘˜ superscript 𝑦 𝑙 subscript πœ‘ π‘˜ 𝑙 𝑒 π’ͺ subscript Ξ” πœ€ \varphi(x,y,u)=xy+\sum\limits_{k,l\geq 2}\varphi_{kl}(u)x^{k}y^{l},\ \ \varphi% _{kl}(u)\in\mathcal{O}(\Delta_{\varepsilon}). italic_Ο† ( italic_x , italic_y , italic_u ) = italic_x italic_y + βˆ‘ start_POSTSUBSCRIPT italic_k , italic_l β‰₯ 2 end_POSTSUBSCRIPT italic_Ο† start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ( italic_u ) italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT , italic_Ο† start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ( italic_u ) ∈ caligraphic_O ( roman_Ξ” start_POSTSUBSCRIPT italic_Ξ΅ end_POSTSUBSCRIPT ) .
Definition 4.3 ( Basic Forms ) .

Let G 𝐺 G italic_G be a Lie group acting on a manifold M 𝑀 M italic_M . A differential form Ξ± 𝛼 \alpha italic_Ξ± on M 𝑀 M italic_M is horizontal if for any x ∈ M π‘₯ 𝑀 x\in M italic_x ∈ italic_M and v ∈ T x ⁒ ( G β‹… x ) 𝑣 subscript 𝑇 π‘₯ β‹… 𝐺 π‘₯ v\in T_{x}(G\cdot x) italic_v ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_G β‹… italic_x ) we have

v ⁒ ⌟ ⁒ Ξ± = 0 . 𝑣 ⌟ 𝛼 0 v{\lrcorner\,}\alpha=0. italic_v ⌟ italic_Ξ± = 0 .

A form that is both horizontal and G 𝐺 G italic_G -invariant is called basic . When the G 𝐺 G italic_G action is understood, we denote the set of basic k π‘˜ k italic_k -forms on M 𝑀 M italic_M by Ξ© basic k ⁒ ( M ) subscript superscript Ξ© π‘˜ basic 𝑀 \Omega^{k}_{{\text{basic}}}(M) roman_Ξ© start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT basic end_POSTSUBSCRIPT ( italic_M ) .