Definition 4.8 .

Let ( 𝔤 , [ ⋅ , ⋯ , ⋅ ] 𝔤 , α ) 𝔤 subscript normal-⋅ normal-⋯ normal-⋅ 𝔤 𝛼 (\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}},\alpha) ( fraktur_g , [ ⋅ , ⋯ , ⋅ ] start_POSTSUBSCRIPT fraktur_g end_POSTSUBSCRIPT , italic_α ) and ( 𝔤 ′ , [ ⋅ , ⋯ , ⋅ ] 𝔤 ′ , β ) superscript 𝔤 normal-′ subscript normal-⋅ normal-⋯ normal-⋅ superscript 𝔤 normal-′ 𝛽 (\mathfrak{g}^{{}^{\prime}},[\cdot,\cdots,\cdot]_{\mathfrak{g}^{{}^{\prime}}},\beta) ( fraktur_g start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT , [ ⋅ , ⋯ , ⋅ ] start_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_β ) be two n 𝑛 n italic_n -ary Hom-Nambu-Lie superalgebras. A linear isomorphism map ϕ : 𝔤 → 𝔤 ′ normal-: italic-ϕ normal-→ 𝔤 superscript 𝔤 normal-′ \phi:\mathfrak{g}\rightarrow\mathfrak{g}^{{}^{\prime}} italic_ϕ : fraktur_g → fraktur_g start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT is called an isomorphism of n 𝑛 n italic_n -ary Hom-Nambu-Lie superalgebras, if

ϕ ∘ α = β ∘ ϕ ; italic-ϕ 𝛼 𝛽 italic-ϕ \phi\circ\alpha=\beta\circ\phi; italic_ϕ ∘ italic_α = italic_β ∘ italic_ϕ ;
ϕ ⁢ [ x 1 , ⋯ , x n ] 𝔤 = [ ϕ ⁢ ( x 1 ) , ⋯ , ϕ ⁢ ( x n ) ] 𝔤 ′ , ∀ x 1 , x 2 , ⋯ , x n ∈ 𝔤 . formulae-sequence italic-ϕ subscript subscript 𝑥 1 ⋯ subscript 𝑥 𝑛 𝔤 subscript italic-ϕ subscript 𝑥 1 ⋯ italic-ϕ subscript 𝑥 𝑛 superscript 𝔤 ′ for-all subscript 𝑥 1 subscript 𝑥 2 ⋯ subscript 𝑥 𝑛 𝔤 \phi[x_{1},\cdots,x_{n}]_{\mathfrak{g}}=[\phi(x_{1}),\cdots,\phi(x_{n})]_{% \mathfrak{g}^{{}^{\prime}}},\forall x_{1},x_{2},\cdots,x_{n}\in\mathfrak{g}. italic_ϕ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT fraktur_g end_POSTSUBSCRIPT = [ italic_ϕ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ⋯ , italic_ϕ ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , ∀ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ fraktur_g .
Definition 4.21 .

Let A 𝐴 A italic_A and B 𝐵 B italic_B be algebras with anti-involution. A Morita context ( M , N ) 𝑀 𝑁 (M,N) ( italic_M , italic_N ) between A 𝐴 A italic_A and B 𝐵 B italic_B is said to be involutive if there is an isomorphism M ≃ N * similar-to-or-equals 𝑀 superscript 𝑁 M\simeq N^{*} italic_M ≃ italic_N start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT of ( A , B ) 𝐴 𝐵 (A,B) ( italic_A , italic_B ) -bimodules (so M * ≃ N similar-to-or-equals superscript 𝑀 𝑁 M^{*}\simeq N italic_M start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ≃ italic_N ) which satisfies

η ⁢ ( x ⊗ y ) * = ( - 1 ) | x | ⁢ | y | ⁢ η ⁢ ( y * ⊗ x * ) , ρ ⁢ ( y ⊗ x ) * = ( - 1 ) | x | ⁢ | y | ⁢ ρ ⁢ ( x * ⊗ y * ) formulae-sequence 𝜂 superscript tensor-product 𝑥 𝑦 superscript 1 𝑥 𝑦 𝜂 tensor-product superscript 𝑦 superscript 𝑥 𝜌 superscript tensor-product 𝑦 𝑥 superscript 1 𝑥 𝑦 𝜌 tensor-product superscript 𝑥 superscript 𝑦 \eta(x\otimes y)^{*}=(-1)^{\lvert x\rvert\lvert y\rvert}\eta(y^{*}\otimes x^{*% }),\quad\rho(y\otimes x)^{*}=(-1)^{\lvert x\rvert\lvert y\rvert}\rho(x^{*}% \otimes y^{*}) italic_η ( italic_x ⊗ italic_y ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = ( - 1 ) start_POSTSUPERSCRIPT | italic_x | | italic_y | end_POSTSUPERSCRIPT italic_η ( italic_y start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⊗ italic_x start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) , italic_ρ ( italic_y ⊗ italic_x ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = ( - 1 ) start_POSTSUPERSCRIPT | italic_x | | italic_y | end_POSTSUPERSCRIPT italic_ρ ( italic_x start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⊗ italic_y start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT )

for every x ∈ M 𝑥 𝑀 x\in M italic_x ∈ italic_M , y ∈ N 𝑦 𝑁 y\in N italic_y ∈ italic_N .

Definition 2.5 .

Let M i , M j subscript 𝑀 𝑖 subscript 𝑀 𝑗 M_{i},M_{j} italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT be modules over rings R i , R j subscript 𝑅 𝑖 subscript 𝑅 𝑗 R_{i},R_{j} italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT respectively. A module homomorphism f : M i → M j normal-: 𝑓 normal-→ subscript 𝑀 𝑖 subscript 𝑀 𝑗 f:M_{i}\to M_{j} italic_f : italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over a ring homomorphism f ¯ : R i ⟶ R j normal-: normal-¯ 𝑓 normal-⟶ subscript 𝑅 𝑖 subscript 𝑅 𝑗 \bar{f}:R_{i}\longrightarrow R_{j} ¯ start_ARG italic_f end_ARG : italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟶ italic_R start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is an abelian group homomorphism which is equivariant with respect to f ¯ normal-¯ 𝑓 \bar{f} ¯ start_ARG italic_f end_ARG . That is, for all r ∈ R i 𝑟 subscript 𝑅 𝑖 r\in R_{i} italic_r ∈ italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and m ∈ M i 𝑚 subscript 𝑀 𝑖 m\in M_{i} italic_m ∈ italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,

f ⁢ ( r ⁢ m ) = f ¯ ⁢ ( r ) ⁢ f ⁢ ( m ) . 𝑓 𝑟 𝑚 ¯ 𝑓 𝑟 𝑓 𝑚 f(rm)=\bar{f}(r)\;f(m). italic_f ( italic_r italic_m ) = ¯ start_ARG italic_f end_ARG ( italic_r ) italic_f ( italic_m ) .
Definition 2.1

A Novikov algebra is a vector space 𝒜 𝒜 \mathcal{A} caligraphic_A equipped with an operation ∘ \circ ∘ such that for x , y , z ∈ 𝒜 𝑥 𝑦 𝑧 𝒜 \ x,y,z\in\mathcal{A} italic_x , italic_y , italic_z ∈ caligraphic_A :

( x ∘ y ) ∘ z = ( x ∘ z ) ∘ y , ( x , y , z ) = ( y , x , z ) , formulae-sequence 𝑥 𝑦 𝑧 𝑥 𝑧 𝑦 𝑥 𝑦 𝑧 𝑦 𝑥 𝑧 \displaystyle(x\circ y)\circ z=(x\circ z)\circ y,\ \ (x,y,z)=(y,x,z), ( italic_x ∘ italic_y ) ∘ italic_z = ( italic_x ∘ italic_z ) ∘ italic_y , ( italic_x , italic_y , italic_z ) = ( italic_y , italic_x , italic_z ) , (2.1)

where the associator ( x , y , z ) = ( x ∘ y ) ∘ z - x ∘ ( y ∘ z ) . 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 (x,y,z)=(x\circ y)\circ z-x\circ(y\circ z). ( italic_x , italic_y , italic_z ) = ( italic_x ∘ italic_y ) ∘ italic_z - italic_x ∘ ( italic_y ∘ italic_z ) .

Definition 2.3

A Hom-Lie algebra is a vector space L 𝐿 L italic_L with a bilinear map [ ⋅ , ⋅ ] : L × L ⟶ L : ⋅ ⋅ ⟶ 𝐿 𝐿 𝐿 [\cdot,\cdot]:L\times L\longrightarrow L [ ⋅ , ⋅ ] : italic_L × italic_L ⟶ italic_L and a linear map α : L ⟶ L : 𝛼 ⟶ 𝐿 𝐿 \alpha:L\longrightarrow L italic_α : italic_L ⟶ italic_L , such that the following relations hold for x , y , z ∈ L 𝑥 𝑦 𝑧 𝐿 x,y,z\in L italic_x , italic_y , italic_z ∈ italic_L :

[ x , y ] = - [ y , x ] , (skew-symmetry) 𝑥 𝑦 𝑦 𝑥 (skew-symmetry) \displaystyle[x,y]=-[y,x],\hskip 14.226378pt\mbox{(skew-symmetry)} [ italic_x , italic_y ] = - [ italic_y , italic_x ] , (skew-symmetry) (2.4)
[ [ x , y ] , α ( z ) ] + [ [ y , z ] , α ( x ) ] + [ [ z , x ] , α ( y ) ] = 0 . (Hom-Jacobi   identity) fragments fragments [ fragments [ x , y ] , α fragments ( z ) ] fragments [ fragments [ y , z ] , α fragments ( x ) ] fragments [ fragments [ z , x ] , α fragments ( y ) ] 0 . italic- (Hom-Jacobi   identity) \displaystyle[[x,y],\alpha(z)]+[[y,z],\alpha(x)]+[[z,x],\alpha(y)]=0.\hskip 14% .226378pt\mbox{(Hom-Jacobi \ identity)} [ [ italic_x , italic_y ] , italic_α ( italic_z ) ] + [ [ italic_y , italic_z ] , italic_α ( italic_x ) ] + [ [ italic_z , italic_x ] , italic_α ( italic_y ) ] = 0 . (Hom-Jacobi identity) (2.5)
Definition 2.4

A Gel’fand-Dorfman bialgebra is a vector space 𝒜 𝒜 \mathcal{A} caligraphic_A , equipped with two operations [ ⋅ , ⋅ ] ⋅ ⋅ [\cdot,\cdot] [ ⋅ , ⋅ ] and ∘ \circ ∘ such that ( 𝒜 , [ ⋅ , ⋅ ] ) 𝒜 ⋅ ⋅ (\mathcal{A},[\cdot,\cdot]) ( caligraphic_A , [ ⋅ , ⋅ ] ) forms a Lie algebra, ( 𝒜 , ∘ ) 𝒜 (\mathcal{A},\circ) ( caligraphic_A , ∘ ) forms a Novikov algebra and the compatibility condition holds for x , y , z ∈ 𝒜 𝑥 𝑦 𝑧 𝒜 x,y,z\in\mathcal{A} italic_x , italic_y , italic_z ∈ caligraphic_A :

[ x ∘ y , z ] - [ x ∘ z , y ] + [ x , y ] ∘ z - [ x , z ] ∘ y - x ∘ [ y , z ] = 0 . 𝑥 𝑦 𝑧 𝑥 𝑧 𝑦 𝑥 𝑦 𝑧 𝑥 𝑧 𝑦 𝑥 𝑦 𝑧 0 \displaystyle[x\circ y,z]-[x\circ z,y]+[x,y]\circ z-[x,z]\circ y-x\circ[y,z]=0. [ italic_x ∘ italic_y , italic_z ] - [ italic_x ∘ italic_z , italic_y ] + [ italic_x , italic_y ] ∘ italic_z - [ italic_x , italic_z ] ∘ italic_y - italic_x ∘ [ italic_y , italic_z ] = 0 . (2.6)
Definition 3.1

A Hom Gel’fand-Dorfman bialgebra is a vector space 𝒜 𝒜 \mathcal{A} caligraphic_A equipped with a linear endomorphism α 𝛼 \alpha italic_α and two operations [ ⋅ , ⋅ ] ⋅ ⋅ [\cdot,\cdot] [ ⋅ , ⋅ ] and ∘ \circ ∘ , such that ( 𝒜 , [ ⋅ , ⋅ ] , α ) 𝒜 ⋅ ⋅ 𝛼 (\mathcal{A},[\cdot,\cdot],\alpha) ( caligraphic_A , [ ⋅ , ⋅ ] , italic_α ) is a Hom-Lie algebra, ( 𝒜 , ∘ , α ) 𝒜 𝛼 (\mathcal{A},\circ,\alpha) ( caligraphic_A , ∘ , italic_α ) is a Hom-Novikov algebra and the following compatibility condition holds for x , y , z ∈ 𝒜 𝑥 𝑦 𝑧 𝒜 x,y,z\in\mathcal{A} italic_x , italic_y , italic_z ∈ caligraphic_A :

[ x ∘ y , α ⁢ ( z ) ] - [ x ∘ z , α ⁢ ( y ) ] + [ x , y ] ∘ α ⁢ ( z ) - [ x , z ] ∘ α ⁢ ( y ) - α ⁢ ( x ) ∘ [ y , z ] = 0 . 𝑥 𝑦 𝛼 𝑧 𝑥 𝑧 𝛼 𝑦 𝑥 𝑦 𝛼 𝑧 𝑥 𝑧 𝛼 𝑦 𝛼 𝑥 𝑦 𝑧 0 \displaystyle[x\circ y,\alpha(z)]-[x\circ z,\alpha(y)]+[x,y]\circ\alpha(z)-[x,% z]\circ\alpha(y)-\alpha(x)\circ[y,z]=0. [ italic_x ∘ italic_y , italic_α ( italic_z ) ] - [ italic_x ∘ italic_z , italic_α ( italic_y ) ] + [ italic_x , italic_y ] ∘ italic_α ( italic_z ) - [ italic_x , italic_z ] ∘ italic_α ( italic_y ) - italic_α ( italic_x ) ∘ [ italic_y , italic_z ] = 0 . (3.1)
Definition 3.14

A Hom-Poisson algebra is a vector space 𝒜 𝒜 \mathcal{A} caligraphic_A equipped with two operations ⋅ ⋅ \cdot ⋅ and [ ⋅ , ⋅ ] ⋅ ⋅ [\cdot,\cdot] [ ⋅ , ⋅ ] , and a linear endomorphism α 𝛼 \alpha italic_α , such that ( 𝒜 , ⋅ , α ) 𝒜 ⋅ 𝛼 (\mathcal{A},\cdot,\alpha) ( caligraphic_A , ⋅ , italic_α ) is a commutative Hom-associative algebra, ( 𝒜 , [ ⋅ , ⋅ ] , α ) 𝒜 ⋅ ⋅ 𝛼 (\mathcal{A},[\cdot,\cdot],\alpha) ( caligraphic_A , [ ⋅ , ⋅ ] , italic_α ) is a Hom-Lie algebra, and the following relation holds for x , y , z ∈ 𝒜 𝑥 𝑦 𝑧 𝒜 x,y,z\in\mathcal{A} italic_x , italic_y , italic_z ∈ caligraphic_A :

[ α ⁢ ( x ) , y ⋅ z ] = α ⁢ ( y ) ⋅ [ x , z ] + α ⁢ ( z ) ⋅ [ x , y ] . 𝛼 𝑥 ⋅ 𝑦 𝑧 ⋅ 𝛼 𝑦 𝑥 𝑧 ⋅ 𝛼 𝑧 𝑥 𝑦 \displaystyle[\alpha(x),y\cdot z]=\alpha(y)\cdot[x,z]+\alpha(z)\cdot[x,y]. [ italic_α ( italic_x ) , italic_y ⋅ italic_z ] = italic_α ( italic_y ) ⋅ [ italic_x , italic_z ] + italic_α ( italic_z ) ⋅ [ italic_x , italic_y ] . (3.18)
Definition 2.13

An averaging operator f 𝑓 f italic_f on an R 𝑅 R italic_R -algebra A 𝐴 A italic_A is called a Reynolds-averaging operator and ( A , f ) 𝐴 𝑓 (A,f) ( italic_A , italic_f ) is called a Reynolds-averaging algebra if f also satisfies the Reynolds identity :

f ⁢ ( x ) ⁢ f ⁢ ( y ) + f ⁢ ( f ⁢ ( x ) ⁢ f ⁢ ( y ) ) = f ⁢ ( x ⁢ f ⁢ ( y ) ) + f ⁢ ( y ⁢ f ⁢ ( x ) ) 𝑓 𝑥 𝑓 𝑦 𝑓 𝑓 𝑥 𝑓 𝑦 𝑓 𝑥 𝑓 𝑦 𝑓 𝑦 𝑓 𝑥 f(x)f(y)+f(f(x)f(y))=f(xf(y))+f(yf(x)) italic_f ( italic_x ) italic_f ( italic_y ) + italic_f ( italic_f ( italic_x ) italic_f ( italic_y ) ) = italic_f ( italic_x italic_f ( italic_y ) ) + italic_f ( italic_y italic_f ( italic_x ) )

for all x , y ∈ A 𝑥 𝑦 𝐴 x,y\in A italic_x , italic_y ∈ italic_A .

Definition 3.1 (Isothermal Coordinates) .

On a surface S 𝑆 S italic_S with a Riemannian metric 𝐠 𝐠 \mathbf{g} bold_g , a local coordinates system ( u , v ) 𝑢 𝑣 (u,v) ( italic_u , italic_v ) is an isothermal coordinate system, if

(3.1) 𝐠 ⁢ ( u , v ) = e 2 ⁢ λ ⁢ ( u , v ) ⁢ ( d ⁢ u 2 + d ⁢ v 2 ) , 𝐠 𝑢 𝑣 superscript 𝑒 2 𝜆 𝑢 𝑣 𝑑 superscript 𝑢 2 𝑑 superscript 𝑣 2 \mathbf{g}(u,v)=e^{2\lambda(u,v)}(du^{2}+dv^{2}), bold_g ( italic_u , italic_v ) = italic_e start_POSTSUPERSCRIPT 2 italic_λ ( italic_u , italic_v ) end_POSTSUPERSCRIPT ( italic_d italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ,

where λ : S → ℝ : 𝜆 → 𝑆 ℝ \lambda:S\to\mathbb{R} italic_λ : italic_S → blackboard_R is a function defined on the surface, and called conformal factor.

Definition 6.3 .

Let 𝒴 ⁢ ( 𝔰 ⁢ 𝔩 3 , 𝔰 ⁢ 𝔬 3 ) t ⁢ w 𝒴 superscript 𝔰 subscript 𝔩 3 𝔰 subscript 𝔬 3 𝑡 𝑤 {\cal Y}(\mathfrak{sl}_{3},\mathfrak{so}_{3})^{tw} caligraphic_Y ( fraktur_s fraktur_l start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , fraktur_s fraktur_o start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_t italic_w end_POSTSUPERSCRIPT denote the associative unital algebra with a basis of eight generators 𝗁 , 𝖾 , 𝖿 , 𝖧 , 𝖤 , 𝖥 , 𝖤 2 , 𝖥 2 𝗁 𝖾 𝖿 𝖧 𝖤 𝖥 subscript 𝖤 2 subscript 𝖥 2 \mathsf{h},\mathsf{e},\mathsf{f},\mathsf{H},\mathsf{E},\mathsf{F},\mathsf{E}_{% 2},\mathsf{F}_{2} sansserif_h , sansserif_e , sansserif_f , sansserif_H , sansserif_E , sansserif_F , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the defining level-0 relations (of the 𝔰 ⁢ 𝔬 3 𝔰 subscript 𝔬 3 \mathfrak{so}_{3} fraktur_s fraktur_o start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Lie algebra)

(6.6) [ 𝖾 , 𝖿 ] = 𝗁 , [ 𝗁 , 𝖾 ] = 𝖾 , [ 𝗁 , 𝖿 ] = - 𝖿 , formulae-sequence 𝖾 𝖿 𝗁 formulae-sequence 𝗁 𝖾 𝖾 𝗁 𝖿 𝖿 [\mathsf{e},\mathsf{f}]=\mathsf{h},\quad[\mathsf{h},\mathsf{e}]=\mathsf{e},% \quad[\mathsf{h},\mathsf{f}]=-\mathsf{f}, [ sansserif_e , sansserif_f ] = sansserif_h , [ sansserif_h , sansserif_e ] = sansserif_e , [ sansserif_h , sansserif_f ] = - sansserif_f ,

level-1 Lie relations

(6.7) [ 𝖾 , 𝖥 ] = [ 𝖤 , 𝖿 ] = 𝖧 , [ 𝗁 , 𝖤 ] = 𝖤 , [ 𝗁 , 𝖥 ] = - 𝖥 , formulae-sequence 𝖾 𝖥 𝖤 𝖿 𝖧 formulae-sequence 𝗁 𝖤 𝖤 𝗁 𝖥 𝖥 \displaystyle[\mathsf{e},\mathsf{F}]=[\mathsf{E},\mathsf{f}]=\mathsf{H},\quad[% \mathsf{h},\mathsf{E}]=\mathsf{E},\quad[\mathsf{h},\mathsf{F}]=-\mathsf{F}, [ sansserif_e , sansserif_F ] = [ sansserif_E , sansserif_f ] = sansserif_H , [ sansserif_h , sansserif_E ] = sansserif_E , [ sansserif_h , sansserif_F ] = - sansserif_F ,
[ 𝖾 , 𝖤 ] = 2 ⁢ 𝖤 2 , [ 𝖿 , 𝖥 ] = 2 ⁢ 𝖥 2 , [ 𝖾 , 𝖤 2 ] = [ 𝖿 , 𝖥 2 ] = 0 , formulae-sequence 𝖾 𝖤 2 subscript 𝖤 2 formulae-sequence 𝖿 𝖥 2 subscript 𝖥 2 𝖾 subscript 𝖤 2 𝖿 subscript 𝖥 2 0 \displaystyle[\mathsf{e},\mathsf{E}]=2\mathsf{E}_{2},\quad[\mathsf{f},\mathsf{% F}]=2\mathsf{F}_{2},\quad[\mathsf{e},\mathsf{E}_{2}]=[\mathsf{f},\mathsf{F}_{2% }]=0, [ sansserif_e , sansserif_E ] = 2 sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_f , sansserif_F ] = 2 sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_e , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = [ sansserif_f , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = 0 ,
[ 𝖾 , 𝖥 2 ] = 𝖥 , [ 𝖿 , 𝖤 2 ] = 𝖤 , [ 𝗁 , 𝖥 2 ] = - 2 ⁢ 𝖥 2 , [ 𝗁 , 𝖤 2 ] = 2 ⁢ 𝖤 2 , formulae-sequence 𝖾 subscript 𝖥 2 𝖥 formulae-sequence 𝖿 subscript 𝖤 2 𝖤 formulae-sequence 𝗁 subscript 𝖥 2 2 subscript 𝖥 2 𝗁 subscript 𝖤 2 2 subscript 𝖤 2 \displaystyle[\mathsf{e},\mathsf{F}_{2}]=\mathsf{F},\quad[\mathsf{f},\mathsf{E% }_{2}]=\mathsf{E},\quad[\mathsf{h},\mathsf{F}_{2}]=-2\mathsf{F}_{2},\quad[% \mathsf{h},\mathsf{E}_{2}]=2\mathsf{E}_{2}, [ sansserif_e , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = sansserif_F , [ sansserif_f , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = sansserif_E , [ sansserif_h , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = - 2 sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_h , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = 2 sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,
[ 𝖧 , 𝖾 ] = 3 ⁢ 𝖤 , [ 𝖧 , 𝖿 ] = - 3 ⁢ 𝖥 , [ 𝖧 , 𝗁 ] = 0 , formulae-sequence 𝖧 𝖾 3 𝖤 formulae-sequence 𝖧 𝖿 3 𝖥 𝖧 𝗁 0 \displaystyle[\mathsf{H},\mathsf{e}]=3\mathsf{E},\quad[\mathsf{H},\mathsf{f}]=% -3\mathsf{F},\quad[\mathsf{H},\mathsf{h}]=0, [ sansserif_H , sansserif_e ] = 3 sansserif_E , [ sansserif_H , sansserif_f ] = - 3 sansserif_F , [ sansserif_H , sansserif_h ] = 0 ,

level-2 horrific relation

(6.8) [ 𝖤 , 𝖥 ] + [ 𝖤 2 , 𝖥 2 ] = 1 4 ⁢ ℏ 2 ⁢ ( { 𝗁 , 𝗁 , 𝗁 } - 3 ⁢ { 𝖾 , 𝖿 , 𝗁 } ) , 𝖤 𝖥 subscript 𝖤 2 subscript 𝖥 2 1 4 superscript Planck-constant-over-2-pi 2 𝗁 𝗁 𝗁 3 𝖾 𝖿 𝗁 \displaystyle[\mathsf{E},\mathsf{F}]+[\mathsf{E}_{2},\mathsf{F}_{2}]=\tfrac{1}% {4}\hbar^{2}\big{(}\{\mathsf{h},\mathsf{h},\mathsf{h}\}-3\{\mathsf{e},\mathsf{% f},\mathsf{h}\}\big{)}, [ sansserif_E , sansserif_F ] + [ sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( { sansserif_h , sansserif_h , sansserif_h } - 3 { sansserif_e , sansserif_f , sansserif_h } ) ,

level-3 horrific relation

(6.9) [ [ 𝖤 , 𝖥 ] , 𝖧 ] = 3 2 ⁢ ℏ 2 ⁢ ( { 𝖤 2 , 𝖿 , 𝖿 } + { 𝖥 2 , 𝖾 , 𝖾 } ) + 15 4 ⁢ ℏ 2 ⁢ ( { 𝖤 , 𝖿 , 𝗁 } - { 𝖥 , 𝖾 , 𝗁 } ) . 𝖤 𝖥 𝖧 3 2 superscript Planck-constant-over-2-pi 2 subscript 𝖤 2 𝖿 𝖿 subscript 𝖥 2 𝖾 𝖾 15 4 superscript Planck-constant-over-2-pi 2 𝖤 𝖿 𝗁 𝖥 𝖾 𝗁 \displaystyle[[\mathsf{E},\mathsf{F}],\mathsf{H}]=\tfrac{3}{2}\hbar^{2}\big{(}% \{\mathsf{E}_{2},\mathsf{f},\mathsf{f}\}+\{\mathsf{F}_{2},\mathsf{e},\mathsf{e% }\}\big{)}+\tfrac{15}{4}\hbar^{2}\big{(}\{\mathsf{E},\mathsf{f},\mathsf{h}\}-% \{\mathsf{F},\mathsf{e},\mathsf{h}\}\big{)}. [ [ sansserif_E , sansserif_F ] , sansserif_H ] = divide start_ARG 3 end_ARG start_ARG 2 end_ARG roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( { sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_f , sansserif_f } + { sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_e , sansserif_e } ) + divide start_ARG 15 end_ARG start_ARG 4 end_ARG roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( { sansserif_E , sansserif_f , sansserif_h } - { sansserif_F , sansserif_e , sansserif_h } ) .
Definition 6.5 .

Let 𝒴 ⁢ ( 𝔰 ⁢ 𝔩 3 , 𝔤 ⁢ 𝔩 2 ) t ⁢ w 𝒴 superscript 𝔰 subscript 𝔩 3 𝔤 subscript 𝔩 2 𝑡 𝑤 {\cal Y}(\mathfrak{sl}_{3},\mathfrak{gl}_{2})^{tw} caligraphic_Y ( fraktur_s fraktur_l start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , fraktur_g fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_t italic_w end_POSTSUPERSCRIPT denote the associative unital algebra with a basis of eight generators 𝗁 , 𝖾 , 𝖿 , 𝗄 𝗁 𝖾 𝖿 𝗄 \mathsf{h},\mathsf{e},\mathsf{f},\mathsf{k} sansserif_h , sansserif_e , sansserif_f , sansserif_k and 𝖤 2 , 𝖥 2 , 𝖤 3 , 𝖥 3 subscript 𝖤 2 subscript 𝖥 2 subscript 𝖤 3 subscript 𝖥 3 \mathsf{E}_{2},\mathsf{F}_{2},\mathsf{E}_{3},\mathsf{F}_{3} sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and the defining level-0 relations (of the 𝔤 ⁢ 𝔩 2 𝔤 subscript 𝔩 2 \mathfrak{gl}_{2} fraktur_g fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Lie algebra) 3 3 3 We have the standard 𝔤 ⁢ 𝔩 2 𝔤 subscript 𝔩 2 \mathfrak{gl}_{2} fraktur_g fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT basis { e i ⁢ j } subscript 𝑒 𝑖 𝑗 \{e_{ij}\} { italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT } with the defining relations [ e i ⁢ j , e k ⁢ l ] = δ k ⁢ j ⁢ e i ⁢ l - δ i ⁢ l ⁢ e k ⁢ j subscript 𝑒 𝑖 𝑗 subscript 𝑒 𝑘 𝑙 subscript 𝛿 𝑘 𝑗 subscript 𝑒 𝑖 𝑙 subscript 𝛿 𝑖 𝑙 subscript 𝑒 𝑘 𝑗 [e_{ij},e_{kl}]=\delta_{kj}e_{il}-\delta_{il}e_{kj} [ italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ] = italic_δ start_POSTSUBSCRIPT italic_k italic_j end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_k italic_j end_POSTSUBSCRIPT that follow from the identification e 11 = - ( 𝗁 + 𝗄 ) / 2 subscript 𝑒 11 𝗁 𝗄 2 e_{11}=-(\mathsf{h}+\mathsf{k})/2 italic_e start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = - ( sansserif_h + sansserif_k ) / 2 , e 22 = ( 𝗁 - 𝗄 ) / 2 subscript 𝑒 22 𝗁 𝗄 2 e_{22}=(\mathsf{h}-\mathsf{k})/2 italic_e start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT = ( sansserif_h - sansserif_k ) / 2 , e 12 = 𝖿 subscript 𝑒 12 𝖿 e_{12}=\mathsf{f} italic_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT = sansserif_f and e 21 = 𝖾 subscript 𝑒 21 𝖾 e_{21}=\mathsf{e} italic_e start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT = sansserif_e .

(6.16) [ 𝖾 , 𝖿 ] = 𝗁 , [ 𝗁 , 𝖾 ] = 2 ⁢ 𝖾 , [ 𝗁 , 𝖿 ] = - 2 ⁢ 𝖿 , [ 𝖾 , 𝗄 ] = [ 𝖿 , 𝗄 ] = [ 𝗁 , 𝗄 ] = 0 , formulae-sequence 𝖾 𝖿 𝗁 formulae-sequence 𝗁 𝖾 2 𝖾 formulae-sequence 𝗁 𝖿 2 𝖿 𝖾 𝗄 𝖿 𝗄 𝗁 𝗄 0 \displaystyle[\mathsf{e},\mathsf{f}]=\mathsf{h},\quad[\mathsf{h},\mathsf{e}]=2% \mathsf{e},\quad[\mathsf{h},\mathsf{f}]=-2\mathsf{f},\quad[\mathsf{e},\mathsf{% k}]=[\mathsf{f},\mathsf{k}]=[\mathsf{h},\mathsf{k}]=0, [ sansserif_e , sansserif_f ] = sansserif_h , [ sansserif_h , sansserif_e ] = 2 sansserif_e , [ sansserif_h , sansserif_f ] = - 2 sansserif_f , [ sansserif_e , sansserif_k ] = [ sansserif_f , sansserif_k ] = [ sansserif_h , sansserif_k ] = 0 ,

level-1 Lie relations

(6.17) [ 𝖾 , 𝖤 2 ] = 𝖤 3 , [ 𝖿 , 𝖥 2 ] = 𝖥 3 , [ 𝖾 , 𝖥 2 ] = [ 𝖿 , 𝖤 2 ] = 0 , formulae-sequence 𝖾 subscript 𝖤 2 subscript 𝖤 3 formulae-sequence 𝖿 subscript 𝖥 2 subscript 𝖥 3 𝖾 subscript 𝖥 2 𝖿 subscript 𝖤 2 0 \displaystyle[\mathsf{e},\mathsf{E}_{2}]=\mathsf{E}_{3},\qquad[\mathsf{f},% \mathsf{F}_{2}]=\mathsf{F}_{3},\qquad[\mathsf{e},\mathsf{F}_{2}]=[\mathsf{f},% \mathsf{E}_{2}]=0, [ sansserif_e , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , [ sansserif_f , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , [ sansserif_e , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = [ sansserif_f , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = 0 ,
[ 𝖾 , 𝖥 3 ] = 𝖥 2 , [ 𝖿 , 𝖤 3 ] = 𝖤 2 , [ 𝖾 , 𝖤 3 ] = [ 𝖿 , 𝖥 3 ] = 0 , formulae-sequence 𝖾 subscript 𝖥 3 subscript 𝖥 2 formulae-sequence 𝖿 subscript 𝖤 3 subscript 𝖤 2 𝖾 subscript 𝖤 3 𝖿 subscript 𝖥 3 0 \displaystyle[\mathsf{e},\mathsf{F}_{3}]=\mathsf{F}_{2},\qquad[\mathsf{f},% \mathsf{E}_{3}]=\mathsf{E}_{2},\qquad[\mathsf{e},\mathsf{E}_{3}]=[\mathsf{f},% \mathsf{F}_{3}]=0, [ sansserif_e , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_f , sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_e , sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = [ sansserif_f , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = 0 ,
[ 𝗁 , 𝖤 2 ] = - 𝖤 2 , [ 𝗁 , 𝖥 2 ] = 𝖥 2 , [ 𝗄 , 𝖤 i ] = 3 ⁢ 𝖤 i , formulae-sequence 𝗁 subscript 𝖤 2 subscript 𝖤 2 formulae-sequence 𝗁 subscript 𝖥 2 subscript 𝖥 2 𝗄 subscript 𝖤 𝑖 3 subscript 𝖤 𝑖 \displaystyle[\mathsf{h},\mathsf{E}_{2}]=-\mathsf{E}_{2},\quad[\mathsf{h},% \mathsf{F}_{2}]=\mathsf{F}_{2},\qquad[\mathsf{k},\mathsf{E}_{i}]=3\mathsf{E}_{% i}, [ sansserif_h , sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = - sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_h , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_k , sansserif_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] = 3 sansserif_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,
[ 𝗁 , 𝖤 3 ] = 𝖤 3 , [ 𝗁 , 𝖥 3 ] = - 𝖥 3 , [ 𝗄 , 𝖥 i ] = - 3 ⁢ 𝖥 i , formulae-sequence 𝗁 subscript 𝖤 3 subscript 𝖤 3 formulae-sequence 𝗁 subscript 𝖥 3 subscript 𝖥 3 𝗄 subscript 𝖥 𝑖 3 subscript 𝖥 𝑖 \displaystyle[\mathsf{h},\mathsf{E}_{3}]=\mathsf{E}_{3},\quad\;\;\;[\mathsf{h}% ,\mathsf{F}_{3}]=-\mathsf{F}_{3},\quad\;[\mathsf{k},\mathsf{F}_{i}]=-3\mathsf{% F}_{i}, [ sansserif_h , sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , [ sansserif_h , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = - sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , [ sansserif_k , sansserif_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] = - 3 sansserif_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,

level-2 horrific relations

(6.18) [ 𝖤 2 , 𝖤 3 ] = 0 , [ 𝖥 2 , 𝖥 3 ] = 0 , formulae-sequence subscript 𝖤 2 subscript 𝖤 3 0 subscript 𝖥 2 subscript 𝖥 3 0 \displaystyle[\mathsf{E}_{2},\mathsf{E}_{3}]=0,\qquad[\mathsf{F}_{2},\mathsf{F% }_{3}]=0, [ sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = 0 , [ sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] = 0 ,

level-3 horrific relations

(6.19) [ 𝖤 2 , [ 𝖤 2 , 𝖥 3 ] ] = - 2 ⁢ ℏ 2 ⁢ { 𝖤 2 , 𝖿 , 𝗄 } , [ 𝖥 2 , [ 𝖤 3 , 𝖥 2 ] ] = - 2 ⁢ ℏ 2 ⁢ { 𝖥 2 , 𝖿 , 𝗄 } . formulae-sequence subscript 𝖤 2 subscript 𝖤 2 subscript 𝖥 3 2 superscript Planck-constant-over-2-pi 2 subscript 𝖤 2 𝖿 𝗄 subscript 𝖥 2 subscript 𝖤 3 subscript 𝖥 2 2 superscript Planck-constant-over-2-pi 2 subscript 𝖥 2 𝖿 𝗄 \displaystyle[\mathsf{E}_{2},[\mathsf{E}_{2},\mathsf{F}_{3}]]=-2\hbar^{2}\{% \mathsf{E}_{2},\mathsf{f},\mathsf{k}\},\qquad[\mathsf{F}_{2},[\mathsf{E}_{3},% \mathsf{F}_{2}]]=-2\hbar^{2}\{\mathsf{F}_{2},\mathsf{f},\mathsf{k}\}. [ sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] ] = - 2 roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT { sansserif_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_f , sansserif_k } , [ sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ sansserif_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] ] = - 2 roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT { sansserif_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_f , sansserif_k } .
Definition 5 .

We call Shanks cubic polynomials (SCPs) the polynomials

ρ ⁢ ( h , - 1 , x ) = x 3 - h ⁢ x 2 - ( h + 3 ) ⁢ x - 1 , 𝜌 ℎ 1 𝑥 superscript 𝑥 3 ℎ superscript 𝑥 2 ℎ 3 𝑥 1 \rho(h,-1,x)=x^{3}-hx^{2}-(h+3)x-1, italic_ρ ( italic_h , - 1 , italic_x ) = italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_h italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_h + 3 ) italic_x - 1 , (3)

since Shanks [ 5 ] deeply studied the cyclic cubic fields generated by the polynomials ( 3 ).

Definition 3.3 .

Let X 𝑋 X italic_X be a complex manifold and let f ∈ 𝖧𝗈𝗅 ⁢ ( X , X ) 𝑓 𝖧𝗈𝗅 𝑋 𝑋 f\in{\sf Hol}(X,X) italic_f ∈ sansserif_Hol ( italic_X , italic_X ) . A semi-model for f 𝑓 f italic_f is a triple ( Ω , h , ψ ) Ω ℎ 𝜓 (\Omega,h,\psi) ( roman_Ω , italic_h , italic_ψ ) where Ω Ω \Omega roman_Ω is a complex manifold, h : X → Ω : ℎ → 𝑋 Ω h\colon X\to\Omega italic_h : italic_X → roman_Ω is a holomorphic mapping, ψ : Ω → Ω : 𝜓 → Ω Ω \psi\colon\Omega\to\Omega italic_ψ : roman_Ω → roman_Ω is an automorphism such that

h ∘ f = ψ ∘ h , ℎ 𝑓 𝜓 ℎ h\circ f=\psi\circ h, italic_h ∘ italic_f = italic_ψ ∘ italic_h , (3.1)

and

⋃ n ≥ 0 ψ - n ⁢ ( h ⁢ ( X ) ) = Ω . subscript 𝑛 0 superscript 𝜓 𝑛 ℎ 𝑋 Ω \bigcup_{n\geq 0}\psi^{-n}(h(X))=\Omega. ⋃ start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT italic_ψ start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( italic_h ( italic_X ) ) = roman_Ω . (3.2)

We call the manifold Ω Ω \Omega roman_Ω the base space and the mapping h ℎ h italic_h the intertwining mapping .

If there exists an f 𝑓 f italic_f -absorbing domain A ⊂ X 𝐴 𝑋 A\subset X italic_A ⊂ italic_X such that h | A : A → Ω : evaluated-at ℎ 𝐴 → 𝐴 Ω h|_{A}\colon A\to\Omega italic_h | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT : italic_A → roman_Ω is univalent, we call the triple ( Ω , h , ψ ) Ω ℎ 𝜓 (\Omega,h,\psi) ( roman_Ω , italic_h , italic_ψ ) a model for f 𝑓 f italic_f .

Let ( Ω , h , ψ ) Ω ℎ 𝜓 (\Omega,h,\psi) ( roman_Ω , italic_h , italic_ψ ) and ( Λ , k , φ ) Λ 𝑘 𝜑 (\Lambda,k,\varphi) ( roman_Λ , italic_k , italic_φ ) be two semi-models for f 𝑓 f italic_f . A morphism of models η ^ : ( Ω , h , ψ ) → ( Λ , k , φ ) : ^ 𝜂 → Ω ℎ 𝜓 Λ 𝑘 𝜑 \hat{\eta}\colon(\Omega,h,\psi)\to(\Lambda,k,\varphi) ^ start_ARG italic_η end_ARG : ( roman_Ω , italic_h , italic_ψ ) → ( roman_Λ , italic_k , italic_φ ) is given by a holomorphic mapping η : Ω → Λ : 𝜂 → Ω Λ \eta\colon\Omega\to\Lambda italic_η : roman_Ω → roman_Λ such that

η ∘ h = k , 𝜂 ℎ 𝑘 \eta\circ h=k, italic_η ∘ italic_h = italic_k ,

and

φ ∘ η = η ∘ ψ . 𝜑 𝜂 𝜂 𝜓 \varphi\circ\eta=\eta\circ\psi. italic_φ ∘ italic_η = italic_η ∘ italic_ψ .

An isomorphism of models is a morphism of models which admits an inverse.

Definition 3.2.1 .

A Poisson structure on a smooth manifold M 𝑀 M italic_M is the structure determined by a bilinear, skew-symmetric composition law on the space of smooth functions, called the Poisson bracket and denoted by ( f , g ) ↦ { f , g } maps-to 𝑓 𝑔 𝑓 𝑔 (f,g)\mapsto\{f,g\} ( italic_f , italic_g ) ↦ { italic_f , italic_g } , satisfying the Leibniz identity

{ f , g ⁢ h } = { f , g } ⁢ h + g ⁢ { f , h } 𝑓 𝑔 ℎ 𝑓 𝑔 ℎ 𝑔 𝑓 ℎ \{f,gh\}=\{f,g\}h+g\{f,h\} { italic_f , italic_g italic_h } = { italic_f , italic_g } italic_h + italic_g { italic_f , italic_h }

and the Jacobi identity

{ { f , g } , h } + { { g , h } , f } + { { h , f } , g } = 0 . 𝑓 𝑔 ℎ 𝑔 ℎ 𝑓 ℎ 𝑓 𝑔 0 \bigl{\{}\{f,g\},h\bigr{\}}+\bigl{\{}\{g,h\},f\bigr{\}}+\bigl{\{}\{h,f\},g% \bigr{\}}=0\,. { { italic_f , italic_g } , italic_h } + { { italic_g , italic_h } , italic_f } + { { italic_h , italic_f } , italic_g } = 0 .

A manifold endowed with a Poisson structure is called a Poisson manifold .

Definition II.1

A translocation φ ⁢ ( i , j ) 𝜑 𝑖 𝑗 \varphi{(i,j)} italic_φ ( italic_i , italic_j ) is a permutation defined as follows: If i ≤ j 𝑖 𝑗 i\leq j italic_i ≤ italic_j , we have

φ ⁢ ( i , j ) = ( 1 , ⋯ , i - 1 , i + 1 , ⋯ , j - 1 , j , i , j + 1 , ⋯ , n ) , 𝜑 𝑖 𝑗 1 ⋯ 𝑖 1 𝑖 1 ⋯ 𝑗 1 𝑗 𝑖 𝑗 1 ⋯ 𝑛 \varphi{(i,j)}=(1,\cdots,i-1,i+1,\cdots,j-1,j,i,j+1,\cdots,n), italic_φ ( italic_i , italic_j ) = ( 1 , ⋯ , italic_i - 1 , italic_i + 1 , ⋯ , italic_j - 1 , italic_j , italic_i , italic_j + 1 , ⋯ , italic_n ) ,

and if i > j 𝑖 𝑗 i>j italic_i > italic_j , we have

φ ⁢ ( i , j ) = ( 1 , ⋯ , j - 1 , i , j , j + 1 , ⋯ , i - 1 , i + 1 , ⋯ , n ) . 𝜑 𝑖 𝑗 1 ⋯ 𝑗 1 𝑖 𝑗 𝑗 1 ⋯ 𝑖 1 𝑖 1 ⋯ 𝑛 \varphi{(i,j)}=(1,\cdots,j-1,i,j,j+1,\cdots,i-1,i+1,\cdots,n)\;. italic_φ ( italic_i , italic_j ) = ( 1 , ⋯ , italic_j - 1 , italic_i , italic_j , italic_j + 1 , ⋯ , italic_i - 1 , italic_i + 1 , ⋯ , italic_n ) .

For i ≤ j 𝑖 𝑗 i\leq j italic_i ≤ italic_j , the permutation φ ⁢ ( i , j ) 𝜑 𝑖 𝑗 \varphi{(i,j)} italic_φ ( italic_i , italic_j ) is called a right translocation while the permutation φ ⁢ ( j , i ) 𝜑 𝑗 𝑖 \varphi{(j,i)} italic_φ ( italic_j , italic_i ) is called a left translocation. Translocations arise due to independent falls and rises of elements in a ranking.

Definition 2.1 .

The pair { L , α } 𝐿 𝛼 \{L,\alpha\} { italic_L , italic_α } is a Lie-Rinehart algebra if the following equation holds for all x , y ∈ L 𝑥 𝑦 𝐿 x,y\in L italic_x , italic_y ∈ italic_L and a ∈ B 𝑎 𝐵 a\in B italic_a ∈ italic_B :

[ x , a ⁢ y ] = a ⁢ [ x , y ] + α ⁢ ( x ) ⁢ ( a ) ⁢ y . 𝑥 𝑎 𝑦 𝑎 𝑥 𝑦 𝛼 𝑥 𝑎 𝑦 [x,ay]=a[x,y]+\alpha(x)(a)y. [ italic_x , italic_a italic_y ] = italic_a [ italic_x , italic_y ] + italic_α ( italic_x ) ( italic_a ) italic_y .

The map α 𝛼 \alpha italic_α is usually called the anchor map .

Definition 2.2 .

The map ∇ ∇ \nabla ∇ is an L-connection if the following equation holds for all x ∈ L , a ∈ B formulae-sequence 𝑥 𝐿 𝑎 𝐵 x\in L,a\in B italic_x ∈ italic_L , italic_a ∈ italic_B and w ∈ W 𝑤 𝑊 w\in W italic_w ∈ italic_W :

∇ ⁡ ( x ) ⁡ ( a ⁢ w ) = a ⁢ ∇ ⁡ ( x ) ⁡ ( w ) + α ⁢ ( x ) ⁢ ( a ) ⁢ w . ∇ 𝑥 𝑎 𝑤 𝑎 ∇ 𝑥 𝑤 𝛼 𝑥 𝑎 𝑤 \nabla(x)(aw)=a\nabla(x)(w)+\alpha(x)(a)w. ∇ ( italic_x ) ( italic_a italic_w ) = italic_a ∇ ( italic_x ) ( italic_w ) + italic_α ( italic_x ) ( italic_a ) italic_w .
Definition 2.2 .

A function h : M → N normal-: ℎ normal-→ 𝑀 𝑁 h\colon M\to N italic_h : italic_M → italic_N is a homomorphism between monoids M , N 𝑀 𝑁 M,N italic_M , italic_N if it preserves the product:

h ⁢ ( s ⋅ t ) = h ⁢ ( s ) ⋅ h ⁢ ( t ) . ℎ ⋅ 𝑠 𝑡 ⋅ ℎ 𝑠 ℎ 𝑡 h(s\cdot t)=h(s)\cdot h(t). italic_h ( italic_s ⋅ italic_t ) = italic_h ( italic_s ) ⋅ italic_h ( italic_t ) .
Definition 1 (Injective Function)

A function ψ : D → 𝒱 normal-: 𝜓 normal-→ 𝐷 𝒱 \psi:D\rightarrow{\cal V} italic_ψ : italic_D → caligraphic_V is injective if

∀ v , v ′ ∈ D : ψ ⁢ ( v ) = ψ ⁢ ( v ′ ) ⇒ v = v ′ . : for-all 𝑣 superscript 𝑣 ′ 𝐷 𝜓 𝑣 𝜓 superscript 𝑣 ′ ⇒ 𝑣 superscript 𝑣 ′ \forall v,v^{\prime}\in D:\psi(v)=\psi(v^{\prime})\Rightarrow v=v^{\prime}. ∀ italic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_D : italic_ψ ( italic_v ) = italic_ψ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⇒ italic_v = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

The inverse ψ - 1 : 𝒱 → D ⊥ normal-: superscript 𝜓 1 normal-→ 𝒱 subscript 𝐷 bottom \psi^{-1}:{\cal V}\rightarrow D_{\bot} italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : caligraphic_V → italic_D start_POSTSUBSCRIPT ⊥ end_POSTSUBSCRIPT of injective function ψ 𝜓 \psi italic_ψ is defined as

ψ - 1 ⁢ ( w ) = { v if ⁢ v ∈ D ∧ ψ ⁢ ( v ) = w ⊥ otherwise superscript 𝜓 1 𝑤 cases 𝑣 if 𝑣 𝐷 𝜓 𝑣 𝑤 bottom otherwise \psi^{-1}(w)=\left\{\begin{array}[]{ll}v&\mbox{ if }v\in D\ \wedge\ \psi(v)=w% \\ \bot&\mbox{ otherwise}\end{array}\right. italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w ) = { start_ARRAY start_ROW start_CELL italic_v end_CELL start_CELL if italic_v ∈ italic_D ∧ italic_ψ ( italic_v ) = italic_w end_CELL end_ROW start_ROW start_CELL ⊥ end_CELL start_CELL otherwise end_CELL end_ROW end_ARRAY

where D ⊥ = D ∪ { ⊥ } . subscript 𝐷 bottom 𝐷 bottom D_{\bot}=D\cup\{\bot\}. italic_D start_POSTSUBSCRIPT ⊥ end_POSTSUBSCRIPT = italic_D ∪ { ⊥ } .

Definition 5.1 .

Fix t 0 ≥ 0 subscript 𝑡 0 0 t_{0}\geq 0 italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ 0 , and let f : [ t 0 , ∞ ) → [ 0 , ∞ ) : 𝑓 → subscript 𝑡 0 0 f:[t_{0},\infty)\to[0,\infty) italic_f : [ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ∞ ) → [ 0 , ∞ ) be a continuous function. We say that f 𝑓 f italic_f is recursively integrable if for some t 1 ≥ t 0 subscript 𝑡 1 subscript 𝑡 0 t_{1}\geq t_{0} italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT the differential equation

(5.1) - g ′ ⁢ ( x ) = g 2 ⁢ ( x ) + f ⁢ ( x ) superscript 𝑔 ′ 𝑥 superscript 𝑔 2 𝑥 𝑓 𝑥 -g^{\prime}(x)=g^{2}(x)+f(x) - italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) = italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) + italic_f ( italic_x )

has a solution g : [ t 1 , ∞ ) → [ 0 , ∞ ) : 𝑔 → subscript 𝑡 1 0 g:[t_{1},\infty)\to[0,\infty) italic_g : [ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ∞ ) → [ 0 , ∞ ) . The class of recursively integrable functions will be denoted ℛ ℛ \mathcal{R} caligraphic_R . A solution g 𝑔 g italic_g of ( 5.1 ) will be called a recursive antiderivative of f 𝑓 f italic_f (regardless of its domain and range).

Definition 5.1 .

A QSO V 𝑉 V italic_V is called associative if the corresponding multiplication given by ( 5.1 ) is associative, i.e

(5.2) ( 𝐱 ∘ 𝐲 ) ∘ 𝐳 = 𝐱 ∘ ( 𝐲 ∘ 𝐳 ) 𝐱 𝐲 𝐳 𝐱 𝐲 𝐳 (\mathbf{x}\circ\mathbf{y})\circ\mathbf{z}=\mathbf{x}\circ(\mathbf{y}\circ% \mathbf{z}) ( bold_x ∘ bold_y ) ∘ bold_z = bold_x ∘ ( bold_y ∘ bold_z )

hold for all 𝐱 , 𝐲 , 𝐳 ∈ ℝ n 𝐱 𝐲 𝐳 superscript ℝ 𝑛 \mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{R}^{n} bold_x , bold_y , bold_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

Definition 1.3 .

Let a = a ⁢ ( x ) 𝑎 𝑎 𝑥 a=a(x) italic_a = italic_a ( italic_x ) denote the (unique) solution of the cubic equation

1 + 2 ⁢ x ⁢ a 2 - 4 ⁢ a 3 = 0 1 2 𝑥 superscript 𝑎 2 4 superscript 𝑎 3 0 1+2xa^{2}-4a^{3}=0 1 + 2 italic_x italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = 0 (1.16)

subject to boundary condition

a = x 2 + 𝒪 ⁢ ( x - 2 ) , x → ∞ . formulae-sequence 𝑎 𝑥 2 𝒪 superscript 𝑥 2 → 𝑥 a=\frac{x}{2}+\mathcal{O}\left(x^{-2}\right),\hskip 14.226378ptx\rightarrow\infty. italic_a = divide start_ARG italic_x end_ARG start_ARG 2 end_ARG + caligraphic_O ( italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ) , italic_x → ∞ .

The three branch points x k = - 3 2 3 ⁢ e 2 ⁢ π ⁢ i 3 ⁢ k , k = 0 , 1 , 2 formulae-sequence subscript 𝑥 𝑘 3 3 2 superscript 𝑒 2 𝜋 𝑖 3 𝑘 𝑘 0 1 2 x_{k}=-\frac{3}{\sqrt[3]{2}}e^{\frac{2\pi i}{3}k},k=0,1,2 italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG nth-root start_ARG 3 end_ARG start_ARG 2 end_ARG end_ARG italic_e start_POSTSUPERSCRIPT divide start_ARG 2 italic_π italic_i end_ARG start_ARG 3 end_ARG italic_k end_POSTSUPERSCRIPT , italic_k = 0 , 1 , 2 of equation ( 1.16 ) form the vertices of the star shaped region Δ ¯ = Δ ∪ ∂ ⁡ Δ normal-¯ normal-Δ normal-Δ normal-Δ \overline{\Delta}=\Delta\cup\partial{\Delta} ¯ start_ARG roman_Δ end_ARG = roman_Δ ∪ ∂ roman_Δ depicted in Figure 1 below which contains the origin and whose boundary ∂ ⁡ Δ normal-Δ \partial\Delta ∂ roman_Δ consists of three edges defined implicitly via the requirement

ℜ ⁡ { - 2 ⁢ ln ⁡ ( 1 + 1 + 2 ⁢ a 3 i ⁢ a ⁢ 2 ⁢ a ) + 1 + 2 ⁢ a 3 ⁢ ( 4 ⁢ a 3 - 1 3 ⁢ a 3 ) } = 0 . 2 1 1 2 superscript 𝑎 3 𝑖 𝑎 2 𝑎 1 2 superscript 𝑎 3 4 superscript 𝑎 3 1 3 superscript 𝑎 3 0 \Re\left\{-2\ln\left(\frac{1+\sqrt{1+2a^{3}}}{ia\sqrt{2a}}\right)+\sqrt{1+2a^{% 3}}\left(\frac{4a^{3}-1}{3a^{3}}\right)\right\}=0. roman_ℜ { - 2 roman_ln ( divide start_ARG 1 + square-root start_ARG 1 + 2 italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG italic_i italic_a square-root start_ARG 2 italic_a end_ARG end_ARG ) + square-root start_ARG 1 + 2 italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ( divide start_ARG 4 italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - 1 end_ARG start_ARG 3 italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ) } = 0 . (1.17)

Here, all branches of fractional exponents and logarithms are chosen to be principal ones.

Definition 1.1 .

[ References ] delimited-[] References {}^{[\ref{ref04}]} start_FLOATSUPERSCRIPT [ ] end_FLOATSUPERSCRIPT Let k 𝑘 k italic_k be a field of characteristic zero. A Lie-Yamaguti algebra(LYA for short) is a vector space T 𝑇 T italic_T over k 𝑘 k italic_k with a bilinear composition a ⁢ b 𝑎 𝑏 ab italic_a italic_b and a trilinear composition [ a , b , c ] 𝑎 𝑏 𝑐 [a,b,c] [ italic_a , italic_b , italic_c ] satisfying:

a 2 = 0 , superscript 𝑎 2 0 a^{2}=0, italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 , (1.1)
[ a , a , b ] = 0 , 𝑎 𝑎 𝑏 0 [a,a,b]=0, [ italic_a , italic_a , italic_b ] = 0 , (1.2)
[ a , b , c ] + [ b , c , a ] + [ c , a , b ] + ( a ⁢ b ) ⁢ c + ( b ⁢ c ) ⁢ a + ( c ⁢ a ) ⁢ b = 0 , 𝑎 𝑏 𝑐 𝑏 𝑐 𝑎 𝑐 𝑎 𝑏 𝑎 𝑏 𝑐 𝑏 𝑐 𝑎 𝑐 𝑎 𝑏 0 [a,b,c]+[b,c,a]+[c,a,b]+(ab)c+(bc)a+(ca)b=0, [ italic_a , italic_b , italic_c ] + [ italic_b , italic_c , italic_a ] + [ italic_c , italic_a , italic_b ] + ( italic_a italic_b ) italic_c + ( italic_b italic_c ) italic_a + ( italic_c italic_a ) italic_b = 0 , (1.3)
[ a ⁢ b , c , d ] + [ b ⁢ c , a , d ] + [ c ⁢ a , b , d ] = 0 , 𝑎 𝑏 𝑐 𝑑 𝑏 𝑐 𝑎 𝑑 𝑐 𝑎 𝑏 𝑑 0 [ab,c,d]+[bc,a,d]+[ca,b,d]=0, [ italic_a italic_b , italic_c , italic_d ] + [ italic_b italic_c , italic_a , italic_d ] + [ italic_c italic_a , italic_b , italic_d ] = 0 , (1.4)
[ a , b , c ⁢ d ] = [ a , b , c ] ⁢ d + c ⁢ [ a , b , d ] , 𝑎 𝑏 𝑐 𝑑 𝑎 𝑏 𝑐 𝑑 𝑐 𝑎 𝑏 𝑑 [a,b,cd]=[a,b,c]d+c[a,b,d], [ italic_a , italic_b , italic_c italic_d ] = [ italic_a , italic_b , italic_c ] italic_d + italic_c [ italic_a , italic_b , italic_d ] , (1.5)
[ a , b , [ c , d , e ] ] = [ [ a , b , c ] , d , e ] + [ c , [ a , b , d ] , e ] + [ c , d , [ a , b , e ] ] , 𝑎 𝑏 𝑐 𝑑 𝑒 𝑎 𝑏 𝑐 𝑑 𝑒 𝑐 𝑎 𝑏 𝑑 𝑒 𝑐 𝑑 𝑎 𝑏 𝑒 [a,b,[c,d,e]]=[[a,b,c],d,e]+[c,[a,b,d],e]+[c,d,[a,b,e]], [ italic_a , italic_b , [ italic_c , italic_d , italic_e ] ] = [ [ italic_a , italic_b , italic_c ] , italic_d , italic_e ] + [ italic_c , [ italic_a , italic_b , italic_d ] , italic_e ] + [ italic_c , italic_d , [ italic_a , italic_b , italic_e ] ] , (1.6)

for any a , b , c , d , e ∈ T 𝑎 𝑏 𝑐 𝑑 𝑒 𝑇 a,b,c,d,e\in T italic_a , italic_b , italic_c , italic_d , italic_e ∈ italic_T .

Definition 2.19 .

Let 𝔇 𝔇 {\mathfrak{D}} fraktur_D be the Lie algebra with generators d α , r superscript 𝑑 𝛼 𝑟 d^{\alpha,r} italic_d start_POSTSUPERSCRIPT italic_α , italic_r end_POSTSUPERSCRIPT for α , r ∈ ℤ 𝛼 𝑟 ℤ \alpha,r\in\mathbb{Z} italic_α , italic_r ∈ blackboard_Z , subject to relations

⁢ d - α , r = - d α , r , superscript 𝑑 𝛼 𝑟 superscript 𝑑 𝛼 𝑟 \displaystyle                 d^{-\alpha,r}=-d^{\alpha,r}, italic_d start_POSTSUPERSCRIPT - italic_α , italic_r end_POSTSUPERSCRIPT = - italic_d start_POSTSUPERSCRIPT italic_α , italic_r end_POSTSUPERSCRIPT , (2.22)
[ d α , r , d β , s ] = δ α + β , s - r ⁢ d α + β , - α + s - δ α + β , r - s ⁢ d α + β , α + s superscript 𝑑 𝛼 𝑟 superscript 𝑑 𝛽 𝑠 subscript 𝛿 𝛼 𝛽 𝑠 𝑟 superscript 𝑑 𝛼 𝛽 𝛼 𝑠 subscript 𝛿 𝛼 𝛽 𝑟 𝑠 superscript 𝑑 𝛼 𝛽 𝛼 𝑠 \displaystyle[d^{\alpha,r},d^{\beta,s}]=\delta_{\alpha+\beta,s-r}d^{\alpha+% \beta,-\alpha+s}-\delta_{\alpha+\beta,r-s}d^{\alpha+\beta,\alpha+s} [ italic_d start_POSTSUPERSCRIPT italic_α , italic_r end_POSTSUPERSCRIPT , italic_d start_POSTSUPERSCRIPT italic_β , italic_s end_POSTSUPERSCRIPT ] = italic_δ start_POSTSUBSCRIPT italic_α + italic_β , italic_s - italic_r end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT italic_α + italic_β , - italic_α + italic_s end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT italic_α + italic_β , italic_r - italic_s end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT italic_α + italic_β , italic_α + italic_s end_POSTSUPERSCRIPT
- δ α - β , s - r ⁢ d α - β , - α + s + δ α - β , r - s ⁢ d α - β , α + s subscript 𝛿 𝛼 𝛽 𝑠 𝑟 superscript 𝑑 𝛼 𝛽 𝛼 𝑠 subscript 𝛿 𝛼 𝛽 𝑟 𝑠 superscript 𝑑 𝛼 𝛽 𝛼 𝑠 \displaystyle                   -\delta_{\alpha-\beta,s-r}d^{\alpha-\beta,-% \alpha+s}+\delta_{\alpha-\beta,r-s}d^{\alpha-\beta,\alpha+s} - italic_δ start_POSTSUBSCRIPT italic_α - italic_β , italic_s - italic_r end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT italic_α - italic_β , - italic_α + italic_s end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_α - italic_β , italic_r - italic_s end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT italic_α - italic_β , italic_α + italic_s end_POSTSUPERSCRIPT (2.23)

for α , β , r , s ∈ ℤ 𝛼 𝛽 𝑟 𝑠 ℤ \alpha,\beta,r,s\in{\mathbb{Z}} italic_α , italic_β , italic_r , italic_s ∈ blackboard_Z .

Definition 2.4 .

We say that a set A ⊂ ∗ X superscript ∗ 𝐴 𝑋 A\subset\,^{\ast}X italic_A ⊂ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_X is intra-convex (or ∗ ∗ \ast ∗ -convex) if

∀ α , β ∈ ∗ [ 0 , 1 ] ∀ x , y ∈ A ( α + β = 1 ⇒ α x + β y ∈ A ) . fragments for-all α , β superscript ∗ fragments [ 0 , 1 ] for-all x , y A fragments ( α β 1 ⇒ α x β y A ) . \forall\alpha,\beta\in\,^{\ast}[0,1]\ \mathbb{\forall}x,y\in A\ (\alpha+\beta=% 1\Rightarrow\alpha x+\beta y\in A). ∀ italic_α , italic_β ∈ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT [ 0 , 1 ] ∀ italic_x , italic_y ∈ italic_A ( italic_α + italic_β = 1 ⇒ italic_α italic_x + italic_β italic_y ∈ italic_A ) .