Definition 2.16 .

[ 40 ] Consider the autonomous system

𝐱 ˙ = 𝐟 ⁢ ( 𝐱 ) ˙ 𝐱 𝐟 𝐱 \mathbf{\dot{x}=f(x)} ˙ start_ARG bold_x end_ARG = bold_f ( bold_x ) (4)

with 𝐟 ⁢ ( 𝐱 ) ∈ 𝒞 1 ⁢ ( E ) 𝐟 𝐱 superscript 𝒞 1 𝐸 \mathbf{f(x)}\in\mathcal{C}^{1}(E) bold_f ( bold_x ) ∈ caligraphic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_E ) where E 𝐸 E italic_E is an open subset of ℝ 2 superscript ℝ 2 \mathbb{R}^{2} blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . A point 𝐩 ~ ∈ E normal-~ 𝐩 𝐸 \mathbf{\tilde{p}}\in E ~ start_ARG bold_p end_ARG ∈ italic_E is an ω 𝜔 \omega italic_ω - limit point of the trajectory ϕ ⁢ ( ⋅ , 𝐱 ) italic-ϕ normal-⋅ 𝐱 \phi(\cdot,{\mathbf{x}}) italic_ϕ ( ⋅ , bold_x ) (which is a function from ℝ ℝ \mathbb{R} blackboard_R to E 𝐸 E italic_E ) of the system ( 4 ) if there is a sequence t n → ∞ normal-→ subscript 𝑡 𝑛 t_{n}\rightarrow\infty italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → ∞ such that lim n → ∞ ⁡ ϕ ⁢ ( t n , 𝐱 ) = 𝐩 ~ subscript normal-→ 𝑛 italic-ϕ subscript 𝑡 𝑛 𝐱 normal-~ 𝐩 \displaystyle{\lim_{n\rightarrow\infty}{\phi(t_{n},{\mathbf{x}})}=\mathbf{% \tilde{p}}} roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_ϕ ( italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_x ) = ~ start_ARG bold_p end_ARG . Similarly, if there is a sequence t n → - ∞ normal-→ subscript 𝑡 𝑛 t_{n}\rightarrow-\infty italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → - ∞ such that lim n → ∞ ⁡ ϕ ⁢ ( t n , 𝐱 ) = 𝐪 ~ subscript normal-→ 𝑛 italic-ϕ subscript 𝑡 𝑛 𝐱 normal-~ 𝐪 \displaystyle{\lim_{n\rightarrow\infty}{\phi(t_{n},{\mathbf{x}})}=\mathbf{% \tilde{q}}} roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_ϕ ( italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_x ) = ~ start_ARG bold_q end_ARG , then the point 𝐪 ~ normal-~ 𝐪 \mathbf{\tilde{q}} ~ start_ARG bold_q end_ARG is called an α 𝛼 \alpha italic_α - limit point of the trajectory ϕ ⁢ ( ⋅ , 𝐱 ) italic-ϕ ⋅ 𝐱 \phi(\cdot,{\mathbf{x}}) italic_ϕ ( ⋅ , bold_x ) .

Definition 2.1 (Entropy flux triple) .

A triplet ( β , ζ , ν ) 𝛽 𝜁 𝜈 (\beta,\zeta,\nu) ( italic_β , italic_ζ , italic_ν ) is called an entropy flux triple if β ∈ C 2 ⁢ ( ℝ ) 𝛽 superscript 𝐶 2 ℝ \beta\in C^{2}(\mathbb{R}) italic_β ∈ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_R ) , Lipschitz and β ≥ 0 𝛽 0 \beta\geq 0 italic_β ≥ 0 , ζ = ( ζ 1 , ζ 2 , … . ζ d ) : ℝ ↦ ℝ d fragments ζ fragments ( subscript 𝜁 1 , subscript 𝜁 2 , … . subscript 𝜁 𝑑 ) : R maps-to superscript ℝ 𝑑 \zeta=(\zeta_{1},\zeta_{2},....\zeta_{d}):\mathbb{R}\mapsto\mathbb{R}^{d} italic_ζ = ( italic_ζ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ζ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … . italic_ζ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) : blackboard_R ↦ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is a vector valued function, and ν : ℝ ↦ ℝ : 𝜈 maps-to ℝ ℝ \nu:\mathbb{R}\mapsto\mathbb{R} italic_ν : blackboard_R ↦ blackboard_R is a scalar valued function such that

ζ ′ ⁢ ( r ) = β ′ ⁢ ( r ) ⁢ f ′ ⁢ ( r ) and ν ′ ⁢ ( r ) = β ′ ⁢ ( r ) ⁢ ϕ ′ ⁢ ( r ) . formulae-sequence superscript 𝜁 ′ 𝑟 superscript 𝛽 ′ 𝑟 superscript 𝑓 ′ 𝑟 and superscript 𝜈 ′ 𝑟 superscript 𝛽 ′ 𝑟 superscript italic-ϕ ′ 𝑟 \zeta^{\prime}(r)=\beta^{\prime}(r)f^{\prime}(r)\quad\text{and}\quad\nu^{% \prime}(r)=\beta^{\prime}(r)\phi^{\prime}(r). italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) = italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) and italic_ν start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) = italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) .

An entropy flux triple ( β , ζ , ν ) 𝛽 𝜁 𝜈 (\beta,\zeta,\nu) ( italic_β , italic_ζ , italic_ν ) is called convex if β ′′ ⁢ ( s ) ≥ 0 superscript 𝛽 ′′ 𝑠 0 \beta^{\prime\prime}(s)\geq 0 italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_s ) ≥ 0 .

Definition 2.1

The transition q 𝑞 q italic_q is commutative on X 𝑋 X italic_X if for all x ∈ X 𝑥 𝑋 x\in X italic_x ∈ italic_X , for all i , i ′ ∈ I 𝑖 superscript 𝑖 normal-′ 𝐼 i,i^{\prime}\in I italic_i , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_I and for all j , j ′ ∈ J 𝑗 superscript 𝑗 normal-′ 𝐽 j,j^{\prime}\in J italic_j , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_J ,

q ~ ⁢ ( q ⁢ ( x , i , j ) , i ′ , j ′ ) = q ~ ⁢ ( q ⁢ ( x , i ′ , j ′ ) , i , j ) . ~ 𝑞 𝑞 𝑥 𝑖 𝑗 superscript 𝑖 ′ superscript 𝑗 ′ ~ 𝑞 𝑞 𝑥 superscript 𝑖 ′ superscript 𝑗 ′ 𝑖 𝑗 \widetilde{q}(q(x,i,j),i^{\prime},j^{\prime})=\widetilde{q}(q(x,i^{\prime},j^{% \prime}),i,j). ~ start_ARG italic_q end_ARG ( italic_q ( italic_x , italic_i , italic_j ) , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ~ start_ARG italic_q end_ARG ( italic_q ( italic_x , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_i , italic_j ) .
Definition 2.5 .

Let U 𝑈 U italic_U be an open set in ℝ n superscript ℝ 𝑛 \mathbb{R}^{n} blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and ϕ : U → ℝ n : italic-ϕ → 𝑈 superscript ℝ 𝑛 \phi:U\to\mathbb{R}^{n} italic_ϕ : italic_U → blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be a real analytic mapping. Then we define the map ϕ ^ : C ω ⁢ ( ℝ n ) → C ω ⁢ ( U ) : ^ italic-ϕ → superscript 𝐶 𝜔 superscript ℝ 𝑛 superscript 𝐶 𝜔 𝑈 \widehat{\phi}:C^{\omega}(\mathbb{R}^{n})\to C^{\omega}(U) ^ start_ARG italic_ϕ end_ARG : italic_C start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) → italic_C start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( italic_U ) by

ϕ ^ ⁢ ( f ) = f ∘ ϕ . ^ italic-ϕ 𝑓 𝑓 italic-ϕ \widehat{\phi}(f)=f\raise 1.0pt\hbox{$\,\scriptstyle\circ\,$}\phi. ^ start_ARG italic_ϕ end_ARG ( italic_f ) = italic_f ∘ italic_ϕ .

It can be shown that ϕ ^ ^ italic-ϕ \widehat{\phi} ^ start_ARG italic_ϕ end_ARG is a unital algebra homomorphism between C ω ⁢ ( ℝ n ) superscript 𝐶 𝜔 superscript ℝ 𝑛 C^{\omega}(\mathbb{R}^{n}) italic_C start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and C ω ⁢ ( U ) superscript 𝐶 𝜔 𝑈 C^{\omega}(U) italic_C start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( italic_U ) .

Definition 1.1 .

A quasi-harmonic sphere is a harmonic map from ( ℝ m , exp ⁡ ( - x 2 / 2 ⁢ ( m - 2 ) ) ⁢ g 0 ) superscript ℝ 𝑚 superscript 𝑥 2 2 𝑚 2 subscript 𝑔 0 \left(\mathbb{R}^{m},\exp(-x^{2}/2(m-2))g_{0}\right) ( blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , roman_exp ( - italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ( italic_m - 2 ) ) italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) to a Riemannian manifold, where g 0 subscript 𝑔 0 g_{0} italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the Euclidean metric in ℝ m superscript ℝ 𝑚 \mathbb{R}^{m} blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( m > 2 𝑚 2 m>2 italic_m > 2 ), i.e.,

(1.1) τ ⁢ ( u ) = 1 2 ⁢ x ⋅ d ⁢ u , 𝜏 𝑢 ⋅ 1 2 𝑥 d 𝑢 \tau(u)=\dfrac{1}{2}x\cdot\mathrm{d}u, italic_τ ( italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x ⋅ roman_d italic_u ,

with finite energy

(1.2) ∫ ℝ m e ⁢ ( u ) ⁢ e - | x | 2 / 4 ⁢ d x < ∞ , subscript superscript ℝ 𝑚 𝑒 𝑢 superscript 𝑒 superscript 𝑥 2 4 differential-d 𝑥 \int_{\mathbb{R}^{m}}e(u)e^{-\left\lvert x\right\rvert^{2}/4}\mathrm{d}x<\infty, ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_e ( italic_u ) italic_e start_POSTSUPERSCRIPT - | italic_x | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 end_POSTSUPERSCRIPT roman_d italic_x < ∞ ,

where

e ⁢ ( u ) = 1 2 ⁢ | d ⁢ u | 2 . 𝑒 𝑢 1 2 superscript d 𝑢 2 e(u)=\dfrac{1}{2}\left\lvert\mathrm{d}u\right\rvert^{2}. italic_e ( italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | roman_d italic_u | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .
Definition 3 (Compatible signatures)

Let S 𝑆 S italic_S be a set of signatures. Then S 𝑆 S italic_S is compatible iff, for all s ⁢ i ⁢ g ∈ S 𝑠 𝑖 𝑔 𝑆 sig\in S italic_s italic_i italic_g ∈ italic_S , s ⁢ i ⁢ g ′ ∈ S 𝑠 𝑖 superscript 𝑔 normal-′ 𝑆 sig^{\prime}\in S italic_s italic_i italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_S , where s ⁢ i ⁢ g = ⟨ i ⁢ n , o ⁢ u ⁢ t , i ⁢ n ⁢ t ⟩ 𝑠 𝑖 𝑔 𝑖 𝑛 𝑜 𝑢 𝑡 𝑖 𝑛 𝑡 sig=\langle in,out,int\rangle italic_s italic_i italic_g = ⟨ italic_i italic_n , italic_o italic_u italic_t , italic_i italic_n italic_t ⟩ , s ⁢ i ⁢ g ′ = ⟨ i ⁢ n ′ , o ⁢ u ⁢ t ′ , i ⁢ n ⁢ t ′ ⟩ 𝑠 𝑖 superscript 𝑔 normal-′ 𝑖 superscript 𝑛 normal-′ 𝑜 𝑢 superscript 𝑡 normal-′ 𝑖 𝑛 superscript 𝑡 normal-′ sig^{\prime}=\langle in^{\prime},out^{\prime},int^{\prime}\rangle italic_s italic_i italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ⟨ italic_i italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_o italic_u italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_i italic_n italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ and s ⁢ i ⁢ g ≠ s ⁢ i ⁢ g ′ 𝑠 𝑖 𝑔 𝑠 𝑖 superscript 𝑔 normal-′ sig\neq sig^{\prime} italic_s italic_i italic_g ≠ italic_s italic_i italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , we have:

  1. 1.

    ( i ⁢ n ∪ o ⁢ u ⁢ t ∪ i ⁢ n ⁢ t ) ∩ i ⁢ n ⁢ t ′ = ∅ 𝑖 𝑛 𝑜 𝑢 𝑡 𝑖 𝑛 𝑡 𝑖 𝑛 superscript 𝑡 ′ (in\cup out\cup int)\cap int^{\prime}=\emptyset ( italic_i italic_n ∪ italic_o italic_u italic_t ∪ italic_i italic_n italic_t ) ∩ italic_i italic_n italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∅ , and

  2. 2.

    o ⁢ u ⁢ t ∩ o ⁢ u ⁢ t ′ = ∅ 𝑜 𝑢 𝑡 𝑜 𝑢 superscript 𝑡 ′ out\cap out^{\prime}=\emptyset italic_o italic_u italic_t ∩ italic_o italic_u italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∅ .

Definition 2.1 .

A special GDN-Poisson admissible algebra ( 𝒜 , ⋅ , ∗ , D ) 𝒜 normal-⋅ normal-∗ 𝐷 (\mathcal{A},\cdot,\ast,D) ( caligraphic_A , ⋅ , ∗ , italic_D ) is a vector space with three linear operations { ⋅ , ∗ , D } normal-⋅ normal-∗ 𝐷 \{\cdot,\ast,D\} { ⋅ , ∗ , italic_D } such that ( 𝒜 , ⋅ ) 𝒜 normal-⋅ (\mathcal{A},\cdot) ( caligraphic_A , ⋅ ) forms a commutative associative algebra with identity e 𝑒 e italic_e , ( 𝒜 , ∗ ) 𝒜 normal-∗ (\mathcal{A},\ast) ( caligraphic_A , ∗ ) forms a commutative associative algebra and ` ` ⋅ , ∗ , D " fragments ` ` normal-⋅ normal-, normal-∗ normal-, D " ``\cdot,\ast,D" ` ` ⋅ , ∗ , italic_D " are compatible in the sense that the following identities hold:

( x ⋅ y ) ∗ z = x ⋅ ( y ∗ z ) , ∗ ⋅ 𝑥 𝑦 𝑧 ⋅ 𝑥 ∗ 𝑦 𝑧 \displaystyle(x\cdot y)\ast z=x\cdot(y\ast z), ( italic_x ⋅ italic_y ) ∗ italic_z = italic_x ⋅ ( italic_y ∗ italic_z ) ,
D ⁢ ( x ∗ y ) = ( D ⁢ x ) ∗ y + x ∗ ( D ⁢ y ) , 𝐷 ∗ 𝑥 𝑦 ∗ 𝐷 𝑥 𝑦 ∗ 𝑥 𝐷 𝑦 \displaystyle D(x\ast y)=(Dx)\ast y+x\ast(Dy), italic_D ( italic_x ∗ italic_y ) = ( italic_D italic_x ) ∗ italic_y + italic_x ∗ ( italic_D italic_y ) ,
D ⁢ ( x ⋅ y ) = ( D ⁢ x ) ⋅ y + x ⋅ ( D ⁢ y ) - x ⋅ y ⋅ ( D ⁢ e ) . 𝐷 ⋅ 𝑥 𝑦 ⋅ 𝐷 𝑥 𝑦 ⋅ 𝑥 𝐷 𝑦 ⋅ 𝑥 𝑦 𝐷 𝑒 \displaystyle D(x\cdot y)=(Dx)\cdot y+x\cdot(Dy)-x\cdot y\cdot(De). italic_D ( italic_x ⋅ italic_y ) = ( italic_D italic_x ) ⋅ italic_y + italic_x ⋅ ( italic_D italic_y ) - italic_x ⋅ italic_y ⋅ ( italic_D italic_e ) .
Definition 4.1 .

A differential GDN-Poisson algebra ( 𝒜 , ⋅ , ∘ ) 𝒜 normal-⋅ (\mathcal{A},\cdot,\circ) ( caligraphic_A , ⋅ , ∘ ) is a GDN-Poisson algebra  satisfying the following identity:

x ∘ ( y ⋅ z ) = ( x ∘ y ) ⋅ z + ( x ∘ z ) ⋅ y ( ◆ ) 𝑥 ⋅ 𝑦 𝑧 ⋅ 𝑥 𝑦 𝑧 ⋅ 𝑥 𝑧 𝑦 ◆ x\circ(y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y\ \ \ \ \ \ \ \ \ (\lozenge) italic_x ∘ ( italic_y ⋅ italic_z ) = ( italic_x ∘ italic_y ) ⋅ italic_z + ( italic_x ∘ italic_z ) ⋅ italic_y ( ◆ )
Definition 13

Given a Radon measure ζ = ζ ⁢ ( t , r , v ) ∈ C ⁢ ( [ T 1 , T 2 ] , ℳ + ⁢ ( ℝ + × ℝ ) ) , 𝜁 𝜁 𝑡 𝑟 𝑣 𝐶 subscript 𝑇 1 subscript 𝑇 2 subscript ℳ subscript ℝ ℝ \zeta=\zeta\left(t,r,v\right)\in C\left(\left[T_{1},T_{2}\right],\mathcal{M}_{% +}\left(\mathbb{R}_{+}\times\mathbb{R}\right)\right), italic_ζ = italic_ζ ( italic_t , italic_r , italic_v ) ∈ italic_C ( [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] , caligraphic_M start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT × blackboard_R ) ) , - ∞ < T 1 < T 2 < ∞ , subscript 𝑇 1 subscript 𝑇 2 -\infty<T_{1}<T_{2}<\infty,\ - ∞ < italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < ∞ , suppose that ρ , p 𝜌 𝑝 \rho,\ p italic_ρ , italic_p defined by means of ( 2.17 ) are in C ⁢ ( [ T 1 , T 2 ] ; 𝒵 δ 0 ) 𝐶 subscript 𝑇 1 subscript 𝑇 2 subscript 𝒵 subscript 𝛿 0 C\left(\left[T_{1},T_{2}\right];\mathcal{Z}_{\delta_{0}}\right) italic_C ( [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] ; caligraphic_Z start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) for some δ 0 > 0 , subscript 𝛿 0 0 \delta_{0}>0, italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 , and that the functions λ , μ 𝜆 𝜇 \lambda,\ \mu italic_λ , italic_μ are given by ( 3.8 ), ( 3.9 ) for each t ∈ [ T 1 , T 2 ] . 𝑡 subscript 𝑇 1 subscript 𝑇 2 t\in\left[T_{1},T_{2}\right]. italic_t ∈ [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] . Let us denote as ζ ~ ⁢ ( t , r , v ¯ ) normal-~ 𝜁 𝑡 𝑟 normal-¯ 𝑣 \tilde{\zeta}\left(t,r,\bar{v}\right) ~ start_ARG italic_ζ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) the measure defined by means of:

ζ ~ ( t , r , v ¯ ) = ζ ( t , r , v ¯ e - λ ) , v = v ¯ e - λ fragments ~ 𝜁 fragments ( t , r , ¯ 𝑣 ) ζ fragments ( t , r , ¯ 𝑣 superscript 𝑒 𝜆 ) italic- , v ¯ 𝑣 superscript 𝑒 𝜆 \tilde{\zeta}\left(t,r,\bar{v}\right)=\zeta\left(t,r,\bar{v}e^{-\lambda}\right% )\ \ ,\ \ \ v=\bar{v}e^{-\lambda} ~ start_ARG italic_ζ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) = italic_ζ ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG italic_e start_POSTSUPERSCRIPT - italic_λ end_POSTSUPERSCRIPT ) , italic_v = ¯ start_ARG italic_v end_ARG italic_e start_POSTSUPERSCRIPT - italic_λ end_POSTSUPERSCRIPT (3.18)

Let us denote the support of ζ 𝜁 \zeta italic_ζ as S 𝑆 S italic_S . We will say that ζ 𝜁 \zeta italic_ζ is a solution of ( 2.5 )-( 2.8 ), ( 2.16 ), ( 2.17 ) in the sense of measures in the interval t ∈ [ T 1 , T 2 ] 𝑡 subscript 𝑇 1 subscript 𝑇 2 t\in\left[T_{1},T_{2}\right] italic_t ∈ [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] if the function Ψ ⁢ ( t , r , v ¯ ) normal-Ψ 𝑡 𝑟 normal-¯ 𝑣 \Psi\left(t,r,\bar{v}\right) roman_Ψ ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) defined in ( 3.11 ) is continuous in S ⊂ [ T 1 , T 2 ] × ( 0 , ∞ ) × ( - ∞ , ∞ ) 𝑆 subscript 𝑇 1 subscript 𝑇 2 0 S\subset\left[T_{1},T_{2}\right]\times\left(0,\infty\right)\times\left(-\infty% ,\infty\right) italic_S ⊂ [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] × ( 0 , ∞ ) × ( - ∞ , ∞ ) and for any test function φ ¯ = φ ¯ ⁢ ( t , r , v ¯ ) ∈ C 0 ⁢ ( [ T 1 , T 2 ] , [ 0 , ∞ ) × ( - ∞ , ∞ ) ) normal-¯ 𝜑 normal-¯ 𝜑 𝑡 𝑟 normal-¯ 𝑣 subscript 𝐶 0 subscript 𝑇 1 subscript 𝑇 2 0 \bar{\varphi}=\bar{\varphi}\left(t,r,\bar{v}\right)\in C_{0}\left(\left[T_{1},% T_{2}\right],\left[0,\infty\right)\times\left(-\infty,\infty\right)\right) ¯ start_ARG italic_φ end_ARG = ¯ start_ARG italic_φ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) ∈ italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( [ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] , [ 0 , ∞ ) × ( - ∞ , ∞ ) ) the following identity holds:

∫ ℝ + × ℝ ζ ~ ⁢ ( T 1 , r , v ¯ ) ⁢ φ ¯ ⁢ ( 0 , r , v ¯ ) ⁢ 𝑑 r ⁢ 𝑑 v ¯ + ∫ ∫ S ζ ~ ⁢ ( t , r , v ¯ ) ⁢ Δ ¯ ⁢ ( t , r , v ¯ ) ⁢ 𝑑 r ⁢ 𝑑 v ¯ ⁢ 𝑑 t subscript subscript ℝ ℝ ~ 𝜁 subscript 𝑇 1 𝑟 ¯ 𝑣 ¯ 𝜑 0 𝑟 ¯ 𝑣 differential-d 𝑟 differential-d ¯ 𝑣 subscript 𝑆 ~ 𝜁 𝑡 𝑟 ¯ 𝑣 ¯ Δ 𝑡 𝑟 ¯ 𝑣 differential-d 𝑟 differential-d ¯ 𝑣 differential-d 𝑡 \displaystyle\int_{\mathbb{R}_{+}\times\mathbb{R}}\tilde{\zeta}\left(T_{1},r,% \bar{v}\right)\bar{\varphi}\left(0,r,\bar{v}\right)drd\bar{v}+\int\int_{S}% \tilde{\zeta}\left(t,r,\bar{v}\right)\bar{\Delta}\left(t,r,\bar{v}\right)drd% \bar{v}dt ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT × blackboard_R end_POSTSUBSCRIPT ~ start_ARG italic_ζ end_ARG ( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r , ¯ start_ARG italic_v end_ARG ) ¯ start_ARG italic_φ end_ARG ( 0 , italic_r , ¯ start_ARG italic_v end_ARG ) italic_d italic_r italic_d ¯ start_ARG italic_v end_ARG + ∫ ∫ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ~ start_ARG italic_ζ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) ¯ start_ARG roman_Δ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) italic_d italic_r italic_d ¯ start_ARG italic_v end_ARG italic_d italic_t
= 0 absent 0 \displaystyle=0 = 0 (3.19)

where:

Δ ¯ ⁢ ( t , r , v ¯ ) = ∂ t ⁡ φ ¯ ⁢ ( t , r , v ¯ ) + ∂ r ⁡ ( e μ - 2 ⁢ λ ⁢ v ¯ E ~ ⁢ φ ¯ ) - ∂ v ¯ ⁡ ( ( - λ r ⁢ e μ - 2 ⁢ λ ⁢ v ¯ 2 E ~ + e μ ⁢ μ r ⁢ E ~ - e μ ⁢ 1 r 3 ⁢ E ~ ) ⁢ φ ¯ ) ¯ Δ 𝑡 𝑟 ¯ 𝑣 subscript 𝑡 ¯ 𝜑 𝑡 𝑟 ¯ 𝑣 subscript 𝑟 superscript 𝑒 𝜇 2 𝜆 ¯ 𝑣 ~ 𝐸 ¯ 𝜑 subscript ¯ 𝑣 subscript 𝜆 𝑟 superscript 𝑒 𝜇 2 𝜆 superscript ¯ 𝑣 2 ~ 𝐸 superscript 𝑒 𝜇 subscript 𝜇 𝑟 ~ 𝐸 superscript 𝑒 𝜇 1 superscript 𝑟 3 ~ 𝐸 ¯ 𝜑 \bar{\Delta}\left(t,r,\bar{v}\right)=\partial_{t}\bar{\varphi}\left(t,r,\bar{v% }\right)+\partial_{r}\left(e^{\mu-2\lambda}\frac{\bar{v}}{\tilde{E}}\bar{% \varphi}\right)-\partial_{\bar{v}}\left(\left(-\frac{\lambda_{r}e^{\mu-2% \lambda}\bar{v}^{2}}{\tilde{E}}+e^{\mu}\mu_{r}\tilde{E}-e^{\mu}\frac{1}{r^{3}% \tilde{E}}\right)\bar{\varphi}\right) ¯ start_ARG roman_Δ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) = ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ¯ start_ARG italic_φ end_ARG ( italic_t , italic_r , ¯ start_ARG italic_v end_ARG ) + ∂ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT italic_μ - 2 italic_λ end_POSTSUPERSCRIPT divide start_ARG ¯ start_ARG italic_v end_ARG end_ARG start_ARG ~ start_ARG italic_E end_ARG end_ARG ¯ start_ARG italic_φ end_ARG ) - ∂ start_POSTSUBSCRIPT ¯ start_ARG italic_v end_ARG end_POSTSUBSCRIPT ( ( - divide start_ARG italic_λ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_μ - 2 italic_λ end_POSTSUPERSCRIPT ¯ start_ARG italic_v end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ~ start_ARG italic_E end_ARG end_ARG + italic_e start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ~ start_ARG italic_E end_ARG - italic_e start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ~ start_ARG italic_E end_ARG end_ARG ) ¯ start_ARG italic_φ end_ARG ) (3.20)
Definition 3.1 .

Let B ⊂ M 𝐵 𝑀 B\subset M italic_B ⊂ italic_M be a * * * -subalgebra. A derivation is a map δ : B → M ⊗ M o ⁢ p : 𝛿 → 𝐵 tensor-product 𝑀 superscript 𝑀 𝑜 𝑝 \delta\colon B\to M\otimes M^{op} italic_δ : italic_B → italic_M ⊗ italic_M start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT satisfying the Leibniz rule:

δ ⁢ ( a ⁢ b ) = δ ⁢ ( a ) ⋅ b + a ⋅ δ ⁢ ( b ) . 𝛿 𝑎 𝑏 ⋅ 𝛿 𝑎 𝑏 ⋅ 𝑎 𝛿 𝑏 \delta(ab)=\delta(a)\cdot b+a\cdot\delta(b). italic_δ ( italic_a italic_b ) = italic_δ ( italic_a ) ⋅ italic_b + italic_a ⋅ italic_δ ( italic_b ) .

We call B 𝐵 B italic_B the domain of δ 𝛿 \delta italic_δ and write dom ⁢ ( δ ) = B dom 𝛿 𝐵 \text{dom}\,(\delta)=B dom ( italic_δ ) = italic_B . The conjugate derivation to δ 𝛿 \delta italic_δ is a derivation, denoted by δ ^ ^ 𝛿 \hat{\delta} ^ start_ARG italic_δ end_ARG and with dom ⁢ ( δ ^ ) = dom ⁢ ( δ ) dom ^ 𝛿 dom 𝛿 \text{dom}\,(\hat{\delta})=\text{dom}\,(\delta) dom ( ^ start_ARG italic_δ end_ARG ) = dom ( italic_δ ) , defined by δ ^ ⁢ ( x ) = δ ⁢ ( x * ) † ^ 𝛿 𝑥 𝛿 superscript superscript 𝑥 † \hat{\delta}(x)=\delta(x^{*})^{\dagger} ^ start_ARG italic_δ end_ARG ( italic_x ) = italic_δ ( italic_x start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT for x ∈ dom ⁢ ( δ ) 𝑥 dom 𝛿 x\in\text{dom}\,(\delta) italic_x ∈ dom ( italic_δ ) .

Definition 3.2 .

Let 𝚪 ⊆ 2 X 𝚪 superscript 2 𝑋 \mathbf{\Gamma}\subseteq 2^{X} bold_Γ ⊆ 2 start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT be nonempty and closed under subsets. An ideal 𝐈 𝐈 \mathbf{I} bold_I on X is 𝚪 𝚪 \mathbf{\Gamma} bold_Γ –maximal if 𝐈 ⊆ 𝚪 𝐈 𝚪 \mathbf{I}\subseteq\mathbf{\Gamma} bold_I ⊆ bold_Γ and the implication

(3.2) ( 𝐈 ⊆ 𝐉 ⊆ 𝚪 ) ⇒ ( 𝐈 = 𝐉 ) fragments fragments ( I J Γ ) ⇒ fragments ( I J ) (\mathbf{I}\subseteq\mathbf{J}\subseteq\mathbf{\Gamma})\Rightarrow(\mathbf{I}=% \mathbf{J}) ( bold_I ⊆ bold_J ⊆ bold_Γ ) ⇒ ( bold_I = bold_J )

holds for every ideal 𝐉 𝐉 \mathbf{J} bold_J on X 𝑋 X italic_X .

Definition 3.1 .

Let G 𝐺 G italic_G be a given non-abelian group. Define a binary operation ⋅ normal-⋅ \cdot ⋅ on G × G 𝐺 𝐺 G\times G italic_G × italic_G as

( a , x ) ⋅ ( b , y ) = ( a ⁢ b ⁢ [ b - 1 ⁢ x ⁢ b , y - 1 ] , b - 1 ⁢ x ⁢ b ⁢ o ⁢ y ) , ⋅ 𝑎 𝑥 𝑏 𝑦 𝑎 𝑏 superscript 𝑏 1 𝑥 𝑏 superscript 𝑦 1 superscript 𝑏 1 𝑥 𝑏 𝑜 𝑦 (a,x)\cdot(b,y)=(ab[b^{-1}xb,y^{-1}],b^{-1}xboy), ( italic_a , italic_x ) ⋅ ( italic_b , italic_y ) = ( italic_a italic_b [ italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_b , italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ] , italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_b italic_o italic_y ) ,

where x ⁢ o ⁢ y = y - 1 ⁢ x ⁢ y 2 𝑥 𝑜 𝑦 superscript 𝑦 1 𝑥 superscript 𝑦 2 xoy=y^{-1}xy^{2} italic_x italic_o italic_y = italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . Then ( G × G , ⋅ ) 𝐺 𝐺 normal-⋅ (G\times G,\cdot) ( italic_G × italic_G , ⋅ ) is a group. The identity of the group G × G 𝐺 𝐺 G\times G italic_G × italic_G is ( 1 , 1 ) 1 1 (1,1) ( 1 , 1 ) , where 1 1 1 1 denotes the identity of group G 𝐺 G italic_G and the inverse of ( a , x ) 𝑎 𝑥 (a,x) ( italic_a , italic_x ) is ( a - 1 , a - 1 ⁢ x - 1 ⁢ a ) superscript 𝑎 1 superscript 𝑎 1 superscript 𝑥 1 𝑎 (a^{-1},a^{-1}x^{-1}a) ( italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a ) , for each ( a , x ) 𝑎 𝑥 (a,x) ( italic_a , italic_x ) in G × G 𝐺 𝐺 G\times G italic_G × italic_G .

Definition 1 .

A relation algebra is an algebra 𝔄 𝔄 {\mathfrak{A}} fraktur_A in which the following axioms are satisfied for all elements r 𝑟 r italic_r , s 𝑠 s italic_s , and t 𝑡 t italic_t in 𝔄 𝔄 {\mathfrak{A}} fraktur_A .

  1. (R1)

    r + s = s + r 𝑟 𝑠 𝑠 𝑟 r+s=s+r italic_r + italic_s = italic_s + italic_r .

  2. (R2)

    r + ( s + t ) = ( r + s ) + t 𝑟 𝑠 𝑡 𝑟 𝑠 𝑡 r+(s+t)=(r+s)+t italic_r + ( italic_s + italic_t ) = ( italic_r + italic_s ) + italic_t .

  3. (R3)

    - ( - r + s ) + - ( - r + - s ) = r fragments fragments ( r s ) fragments ( r s ) r -(-r+s)+-(-r+-s)=r - ( - italic_r + italic_s ) + - ( - italic_r + - italic_s ) = italic_r .

  4. (R4)

    r ; ( s ; t ) = ( r ; s ) ; t formulae-sequence 𝑟 𝑠 𝑡 𝑟 𝑠 𝑡 r;(s;t)=(r;s);t italic_r ; ( italic_s ; italic_t ) = ( italic_r ; italic_s ) ; italic_t .

  5. (R5)

    r ; 1 ⁢ ’ = r 𝑟 1 ’ 𝑟 r;1{\mskip-1.4mu }\textnormal{'}=r italic_r ; 1 ’ = italic_r .

  6. (R6)

    r = ⌣ ⌣ r fragments r superscript superscript ⌣ ⌣ r r{}^{\scriptstyle\smallsmile}{}^{\scriptstyle\smallsmile}=r italic_r start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT = italic_r .

  7. (R7)

    ( r ; s ) = ⌣ s ; ⌣ r ⌣ fragments fragments ( r ; s ) superscript ⌣ s superscript ; ⌣ r ⌣ (r;s){}^{\scriptstyle\smallsmile}=s{}^{\scriptstyle\smallsmile};r{}^{% \scriptstyle\smallsmile} ( italic_r ; italic_s ) start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT = italic_s start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT ; italic_r start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT .

  8. (R8)

    ( r + s ) ; t = r ; t + s ; t formulae-sequence 𝑟 𝑠 𝑡 𝑟 𝑡 𝑠 𝑡 (r+s);t=r;t+s;t ( italic_r + italic_s ) ; italic_t = italic_r ; italic_t + italic_s ; italic_t .

  9. (R9)

    ( r + s ) = ⌣ r + ⌣ s ⌣ fragments fragments ( r s ) superscript ⌣ r superscript ⌣ s ⌣ (r+s){}^{\scriptstyle\smallsmile}=r{}^{\scriptstyle\smallsmile}+s{}^{% \scriptstyle\smallsmile} ( italic_r + italic_s ) start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT = italic_r start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT + italic_s start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT .

  10. (R10)

    r ; ⌣ - ( r ; s ) + - s = - s fragments r superscript ; ⌣ fragments ( r ; s ) s s r{}^{\scriptstyle\smallsmile};-(r;s)+-s=-s italic_r start_FLOATSUPERSCRIPT ⌣ end_FLOATSUPERSCRIPT ; - ( italic_r ; italic_s ) + - italic_s = - italic_s .∎

Definition 3.11 .

Let A 𝐴 A italic_A be a 3 3 3 3 -Lie algebra and r ∈ A ⊗ A 𝑟 tensor-product 𝐴 𝐴 r\in A\otimes A italic_r ∈ italic_A ⊗ italic_A . The equation

[ [ r , r , r ] ] = 0 delimited-[] 𝑟 𝑟 𝑟 0 [[r,r,r]]=0 [ [ italic_r , italic_r , italic_r ] ] = 0

is called the 3 3 3 3 -Lie classical Yang-Baxter equation (3-Lie CYBE) .

Definition 1 .

The vector field commutator is the skew-symmetric bi-linear form [ ⋅ , ⋅ ] : 𝔛 ⁢ ( Ω ) × 𝔛 ⁢ ( Ω ) → 𝔛 ⁢ ( Ω ) : ⋅ ⋅ → 𝔛 Ω 𝔛 Ω 𝔛 Ω [\cdot,\cdot]\colon\mathfrak{X}(\Omega)\times\mathfrak{X}(\Omega)\to\mathfrak{% X}(\Omega) [ ⋅ , ⋅ ] : fraktur_X ( roman_Ω ) × fraktur_X ( roman_Ω ) → fraktur_X ( roman_Ω ) given by

[ 𝐯 , 𝐮 ] = ∇ ⁡ 𝐮 ⋅ 𝐯 - ∇ ⁡ 𝐯 ⋅ 𝐮 , 𝐯 𝐮 ∇ ⋅ 𝐮 𝐯 ∇ ⋅ 𝐯 𝐮 [\mathbf{v},\mathbf{u}]=\nabla\mathbf{u}\cdot\mathbf{v}-\nabla\mathbf{v}\cdot% \mathbf{u}, [ bold_v , bold_u ] = ∇ bold_u ⋅ bold_v - ∇ bold_v ⋅ bold_u ,

or in coordinates

[ 𝐯 , 𝐮 ] i = ∑ j = 1 n ( v j ⁢ ∂ ⁡ u i ∂ ⁡ x j - u j ⁢ ∂ ⁡ v i ∂ ⁡ x j ) . subscript 𝐯 𝐮 𝑖 superscript subscript 𝑗 1 𝑛 subscript 𝑣 𝑗 subscript 𝑢 𝑖 subscript 𝑥 𝑗 subscript 𝑢 𝑗 subscript 𝑣 𝑖 subscript 𝑥 𝑗 [\mathbf{v},\mathbf{u}]_{i}=\sum_{j=1}^{n}\left(v_{j}\frac{\partial u_{i}}{% \partial x_{j}}-u_{j}\frac{\partial v_{i}}{\partial x_{j}}\right). [ bold_v , bold_u ] start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG ∂ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG - italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) .
Definition 1.6 ( equivalent pentads, [ 9 , Definition 2.22] ) .

Let ( 𝔤 i , ρ i , V i , 𝒱 i , B 0 i ) superscript 𝔤 𝑖 superscript 𝜌 𝑖 superscript 𝑉 𝑖 superscript 𝒱 𝑖 superscript subscript 𝐵 0 𝑖 (\mathfrak{g}^{i},\rho^{i},V^{i},{\cal V}^{i},B_{0}^{i}) ( fraktur_g start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_V start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ( i = 1 , 2 ) 𝑖 1 2 (i=1,2) ( italic_i = 1 , 2 ) be standard pentads. We say that the pentads ( 𝔤 i , ρ i , V i , 𝒱 i , B 0 i ) superscript 𝔤 𝑖 superscript 𝜌 𝑖 superscript 𝑉 𝑖 superscript 𝒱 𝑖 superscript subscript 𝐵 0 𝑖 (\mathfrak{g}^{i},\rho^{i},V^{i},{\cal V}^{i},B_{0}^{i}) ( fraktur_g start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_V start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ( i = 1 , 2 ) 𝑖 1 2 (i=1,2) ( italic_i = 1 , 2 ) are equivalent if and only if there exist linear isomorphisms τ : 𝔤 1 → 𝔤 2 : 𝜏 → superscript 𝔤 1 superscript 𝔤 2 \tau:\mathfrak{g}^{1}\rightarrow\mathfrak{g}^{2} italic_τ : fraktur_g start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → fraktur_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , σ : V 1 → V 2 : 𝜎 → superscript 𝑉 1 superscript 𝑉 2 \sigma:V^{1}\rightarrow V^{2} italic_σ : italic_V start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → italic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ς : 𝒱 1 → 𝒱 2 : 𝜍 → superscript 𝒱 1 superscript 𝒱 2 \varsigma:{\cal V}^{1}\rightarrow{\cal V}^{2} italic_ς : caligraphic_V start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT → caligraphic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and a non-zero element c ∈ ℂ 𝑐 ℂ c\in{\mathbb{C}} italic_c ∈ blackboard_C such that

σ ⁢ ( ρ 1 ⁢ ( a 1 ⊗ v 1 ) ) = ρ 2 ⁢ ( τ ⁢ ( a 1 ) ⊗ σ ⁢ ( v 1 ) ) , 𝜎 superscript 𝜌 1 tensor-product superscript 𝑎 1 superscript 𝑣 1 superscript 𝜌 2 tensor-product 𝜏 superscript 𝑎 1 𝜎 superscript 𝑣 1 \displaystyle\sigma(\rho^{1}(a^{1}\otimes v^{1}))=\rho^{2}(\tau(a^{1})\otimes% \sigma(v^{1})), italic_σ ( italic_ρ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊗ italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ) = italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_τ ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ⊗ italic_σ ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ) , ς ⁢ ( ϱ 1 ⁢ ( a 1 ⊗ ϕ 1 ) ) = ϱ 2 ⁢ ( τ ⁢ ( a 1 ) ⊗ ς ⁢ ( ϕ 1 ) ) , 𝜍 superscript italic-ϱ 1 tensor-product superscript 𝑎 1 superscript italic-ϕ 1 superscript italic-ϱ 2 tensor-product 𝜏 superscript 𝑎 1 𝜍 superscript italic-ϕ 1 \displaystyle\varsigma(\varrho^{1}(a^{1}\otimes\phi^{1}))=\varrho^{2}(\tau(a^{% 1})\otimes\varsigma(\phi^{1})), italic_ς ( italic_ϱ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊗ italic_ϕ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ) = italic_ϱ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_τ ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ⊗ italic_ς ( italic_ϕ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ) ,
B 0 1 ⁢ ( a 1 , b 1 ) = c ⁢ B 0 2 ⁢ ( τ ⁢ ( a 1 ) , τ ⁢ ( b 1 ) ) , superscript subscript 𝐵 0 1 superscript 𝑎 1 superscript 𝑏 1 𝑐 superscript subscript 𝐵 0 2 𝜏 superscript 𝑎 1 𝜏 superscript 𝑏 1 \displaystyle B_{0}^{1}(a^{1},b^{1})=cB_{0}^{2}(\tau(a^{1}),\tau(b^{1})), italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) = italic_c italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_τ ( italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) , italic_τ ( italic_b start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ) , ⟨ v 1 , ϕ 1 ⟩ 1 = ⟨ σ ⁢ ( v 1 ) , ς ⁢ ( ϕ 1 ) ⟩ 2 , superscript superscript 𝑣 1 superscript italic-ϕ 1 1 superscript 𝜎 superscript 𝑣 1 𝜍 superscript italic-ϕ 1 2 \displaystyle\langle v^{1},\phi^{1}\rangle^{1}=\langle\sigma(v^{1}),\varsigma(% \phi^{1})\rangle^{2}, ⟨ italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_ϕ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⟩ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = ⟨ italic_σ ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) , italic_ς ( italic_ϕ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ⟩ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (1.5)

where ϱ italic-ϱ \varrho italic_ϱ is the representation of 𝔤 𝔤 \mathfrak{g} fraktur_g on 𝒱 𝒱 {\cal V} caligraphic_V (see Definition 1.1 ), for any a 1 , b 1 ∈ 𝔤 1 , v 1 ∈ V 1 , ϕ 1 ∈ 𝒱 1 formulae-sequence superscript 𝑎 1 superscript 𝑏 1 superscript 𝔤 1 formulae-sequence superscript 𝑣 1 superscript 𝑉 1 superscript italic-ϕ 1 superscript 𝒱 1 a^{1},b^{1}\in\mathfrak{g}^{1},v^{1}\in V^{1},\phi^{1}\in{\cal V}^{1} italic_a start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ fraktur_g start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_ϕ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ caligraphic_V start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT .

Definition 2.8 .

A lattice ( L , ∧ , ∨ ) 𝐿 (L,\wedge,\vee) ( italic_L , ∧ , ∨ ) is distributive if ∀ a , b , c ∈ L for-all 𝑎 𝑏 𝑐 𝐿 \forall a,b,c\in L ∀ italic_a , italic_b , italic_c ∈ italic_L :

a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) 𝑎 𝑏 𝑐 𝑎 𝑏 𝑎 𝑐 a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c) italic_a ∨ ( italic_b ∧ italic_c ) = ( italic_a ∨ italic_b ) ∧ ( italic_a ∨ italic_c )

(or equivalently, ∀ a , b , c ∈ L , a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) formulae-sequence for-all 𝑎 𝑏 𝑐 𝐿 𝑎 𝑏 𝑐 𝑎 𝑏 𝑎 𝑐 \forall a,b,c\in L,\ a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c) ∀ italic_a , italic_b , italic_c ∈ italic_L , italic_a ∧ ( italic_b ∨ italic_c ) = ( italic_a ∧ italic_b ) ∨ ( italic_a ∧ italic_c ) )

Definition 1.2 .

Let w = … ⁢ w - 2 ⁢ w - 1 ⋅ w 0 ⁢ w 1 ⁢ w 2 ⁢ … 𝑤 normal-⋅ normal-… subscript 𝑤 2 subscript 𝑤 1 subscript 𝑤 0 subscript 𝑤 1 subscript 𝑤 2 normal-… w=\ldots w_{-2}w_{-1}\cdot w_{0}w_{1}w_{2}\ldots italic_w = … italic_w start_POSTSUBSCRIPT - 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ⋅ italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … be a bi-infinite sequence in X φ subscript 𝑋 𝜑 X_{\varphi} italic_X start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT and let t ∈ [ 0 , 1 ) 𝑡 0 1 t\in[0,1) italic_t ∈ [ 0 , 1 ) , so that ( w , t ) 𝑤 𝑡 (w,t) ( italic_w , italic_t ) is an element of the tiling space Ω φ subscript normal-Ω 𝜑 \Omega_{\varphi} roman_Ω start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT . We define a map on the tiling spaces which we call φ : Ω φ → Ω φ normal-: 𝜑 normal-→ subscript normal-Ω 𝜑 subscript normal-Ω 𝜑 \varphi\colon\Omega_{\varphi}\to\Omega_{\varphi} italic_φ : roman_Ω start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT → roman_Ω start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT , given by

φ ⁢ ( w , t ) = ( σ ⌊ t ~ ⌋ ⁢ ( φ ⁢ ( w ) ) , t ~ - ⌊ t ~ ⌋ ) 𝜑 𝑤 𝑡 superscript 𝜎 ~ 𝑡 𝜑 𝑤 ~ 𝑡 ~ 𝑡 \varphi(w,t)=(\sigma^{\lfloor\tilde{t}\rfloor}(\varphi(w)),\tilde{t}-\lfloor% \tilde{t}\rfloor) italic_φ ( italic_w , italic_t ) = ( italic_σ start_POSTSUPERSCRIPT ⌊ ~ start_ARG italic_t end_ARG ⌋ end_POSTSUPERSCRIPT ( italic_φ ( italic_w ) ) , ~ start_ARG italic_t end_ARG - ⌊ ~ start_ARG italic_t end_ARG ⌋ )

where t ~ = | φ ⁢ ( w 0 ) | ⋅ t normal-~ 𝑡 normal-⋅ 𝜑 subscript 𝑤 0 𝑡 \tilde{t}=|\varphi(w_{0})|\cdot t ~ start_ARG italic_t end_ARG = | italic_φ ( italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | ⋅ italic_t and ⌊ - ⌋ \lfloor-\rfloor ⌊ - ⌋ is the floor function.