Definition 68 .

Consider the following terms.

s ⁢ u ⁢ c ⁢ ( x ) = 1 ; ( p ˘ ; x ; q ˘ ) 𝑠 𝑢 𝑐 𝑥 1 ˘ 𝑝 𝑥 ˘ 𝑞 suc(x)=1;(\breve{p};x;\breve{q}) italic_s italic_u italic_c ( italic_x ) = 1 ; ( ˘ start_ARG italic_p end_ARG ; italic_x ; ˘ start_ARG italic_q end_ARG )

and

p ⁢ r ⁢ e ⁢ d ⁢ ( x ) = p ˘ ; r ⁢ a ⁢ n ⁢ x ; q . 𝑝 𝑟 𝑒 𝑑 𝑥 ˘ 𝑝 𝑟 𝑎 𝑛 𝑥 𝑞 pred(x)=\breve{p};ranx;q. italic_p italic_r italic_e italic_d ( italic_x ) = ˘ start_ARG italic_p end_ARG ; italic_r italic_a italic_n italic_x ; italic_q .
Definition V.2 .

We say that functional ( 3 ) is invariant under an ε 𝜀 \varepsilon italic_ε -parameter group of infinitesimal transformations

q ¯ ⁢ ( t ) = q ⁢ ( t ) + ε ⁢ ξ ⁢ ( t , q ⁢ ( t ) ) + o ⁢ ( ε ) ¯ 𝑞 𝑡 𝑞 𝑡 𝜀 𝜉 𝑡 𝑞 𝑡 𝑜 𝜀 \bar{q}(t)=q(t)+\varepsilon\xi(t,q(t))+o(\varepsilon) ¯ start_ARG italic_q end_ARG ( italic_t ) = italic_q ( italic_t ) + italic_ε italic_ξ ( italic_t , italic_q ( italic_t ) ) + italic_o ( italic_ε ) (8)

if

∫ t a t b L ⁢ ( t , q ⁢ ( t ) , 𝐷 t α ⁢ ( t , τ ) a C ⁢ q ⁢ ( t ) , 𝐷 b β ⁢ ( t , τ ) t C ⁢ q ⁢ ( t ) ) ⁢ 𝑑 t = ∫ t a t b L ⁢ ( t , q ¯ ⁢ ( t ) , 𝐷 t α ⁢ ( t , τ ) a C ⁢ q ¯ ⁢ ( t ) , 𝐷 b β ⁢ ( t , τ ) t C ⁢ q ¯ ⁢ ( t ) ) ⁢ 𝑑 t superscript subscript subscript 𝑡 𝑎 subscript 𝑡 𝑏 𝐿 𝑡 𝑞 𝑡 superscript subscript subscript superscript 𝐷 𝛼 𝑡 𝜏 𝑡 𝑎 𝐶 𝑞 𝑡 superscript subscript subscript superscript 𝐷 𝛽 𝑡 𝜏 𝑏 𝑡 𝐶 𝑞 𝑡 differential-d 𝑡 superscript subscript subscript 𝑡 𝑎 subscript 𝑡 𝑏 𝐿 𝑡 ¯ 𝑞 𝑡 superscript subscript subscript superscript 𝐷 𝛼 𝑡 𝜏 𝑡 𝑎 𝐶 ¯ 𝑞 𝑡 superscript subscript subscript superscript 𝐷 𝛽 𝑡 𝜏 𝑏 𝑡 𝐶 ¯ 𝑞 𝑡 differential-d 𝑡 \int_{t_{a}}^{t_{b}}L\left(t,q(t),{{}^{C}_{a}}\textsl{D}^{\alpha(t,\tau)}_{t}q% (t),{{}^{C}_{t}}\textsl{D}^{\beta(t,\tau)}_{b}q(t)\right)dt\\ =\int_{t_{a}}^{t_{b}}L\left(t,\bar{q}(t),{{}^{C}_{a}}\textsl{D}^{\alpha(t,\tau% )}_{t}\bar{q}(t),{{}^{C}_{t}}\textsl{D}^{\beta(t,\tau)}_{b}\bar{q}(t)\right)dt ∫ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_L ( italic_t , italic_q ( italic_t ) , start_FLOATSUPERSCRIPT italic_C end_FLOATSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT D start_POSTSUPERSCRIPT italic_α ( italic_t , italic_τ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_q ( italic_t ) , start_FLOATSUPERSCRIPT italic_C end_FLOATSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT D start_POSTSUPERSCRIPT italic_β ( italic_t , italic_τ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_q ( italic_t ) ) italic_d italic_t = ∫ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_L ( italic_t , ¯ start_ARG italic_q end_ARG ( italic_t ) , start_FLOATSUPERSCRIPT italic_C end_FLOATSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT D start_POSTSUPERSCRIPT italic_α ( italic_t , italic_τ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ¯ start_ARG italic_q end_ARG ( italic_t ) , start_FLOATSUPERSCRIPT italic_C end_FLOATSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT D start_POSTSUPERSCRIPT italic_β ( italic_t , italic_τ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ¯ start_ARG italic_q end_ARG ( italic_t ) ) italic_d italic_t (9)

for any subinterval [ t a , t b ] ⊆ [ a , b ] subscript 𝑡 𝑎 subscript 𝑡 𝑏 𝑎 𝑏 [t_{a},t_{b}]\subseteq[a,b] [ italic_t start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ] ⊆ [ italic_a , italic_b ] .

Definition 4.1 .

An 𝔰 ⁢ 𝔩 2 ⁢ ( ℝ ) 𝔰 subscript 𝔩 2 ℝ \mathfrak{s}\mathfrak{l}_{2}(\mathbb{R}) fraktur_s fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_R ) triple ( a , n + , n - ) 𝑎 superscript 𝑛 superscript 𝑛 (a,n^{+},n^{-}) ( italic_a , italic_n start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_n start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) in a Lie algebra 𝔤 0 subscript 𝔤 0 \mathfrak{g}_{0} fraktur_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a triple satisfying the commutation relations

[ a , n ± ] = ± n ± , [ n + , n - ] = a . formulae-sequence 𝑎 superscript 𝑛 plus-or-minus plus-or-minus superscript 𝑛 plus-or-minus superscript 𝑛 superscript 𝑛 𝑎 [a,n^{\pm}]=\pm n^{\pm},\quad[n^{+},n^{-}]=a. [ italic_a , italic_n start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ] = ± italic_n start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT , [ italic_n start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_n start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ] = italic_a .
Definition 18.1

[ 2 , Line (2.2)] Assume that 𝔽 𝔽 \mathbb{F} blackboard_F has characteristic 0. For the Lie algebra 𝔰 ⁢ 𝔩 2 𝔰 subscript 𝔩 2 \mathfrak{sl}_{2} fraktur_s fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over 𝔽 𝔽 \mathbb{F} blackboard_F , the equitable basis x , y , z 𝑥 𝑦 𝑧 x,y,z italic_x , italic_y , italic_z satisfies

[ x , y ] = 2 ⁢ x + 2 ⁢ y , [ y , z ] = 2 ⁢ y + 2 ⁢ z , [ z , x ] = 2 ⁢ z + 2 ⁢ x . formulae-sequence 𝑥 𝑦 2 𝑥 2 𝑦 formulae-sequence 𝑦 𝑧 2 𝑦 2 𝑧 𝑧 𝑥 2 𝑧 2 𝑥 \displaystyle[x,y]=2x+2y,\qquad\qquad[y,z]=2y+2z,\qquad\qquad[z,x]=2z+2x. [ 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 .
Definition 5.4 .

Define a reflexive binary relation R 𝑅 R italic_R on { 0 , 1 } ℕ superscript 0 1 ℕ {\{0,1\}^{\mathbb{N}}} { 0 , 1 } start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT by setting x ⁢ R ⁢ y 𝑥 𝑅 𝑦 xRy italic_x italic_R italic_y if

∃ ∞ n , x ↾ [ n , 2 ⁢ n ) = y ↾ [ n , 2 ⁢ n ) . ↾ superscript 𝑛 𝑥 𝑛 2 𝑛 𝑦 ↾ 𝑛 2 𝑛 \exists^{\infty}n,\ x\upharpoonright[n,2n)=y\upharpoonright[n,2n). ∃ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_n , italic_x ↾ [ italic_n , 2 italic_n ) = italic_y ↾ [ italic_n , 2 italic_n ) .

For y ∈ { 0 , 1 } ℕ 𝑦 superscript 0 1 ℕ y\in{\{0,1\}^{\mathbb{N}}} italic_y ∈ { 0 , 1 } start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT , let

[ y ] R = { x ∈ { 0 , 1 } ℕ : x ⁢ R ⁢ y } . subscript delimited-[] 𝑦 𝑅 conditional-set 𝑥 superscript 0 1 ℕ 𝑥 𝑅 𝑦 [y]_{R}={\left\{x\in{\{0,1\}^{\mathbb{N}}}\;:\;xRy\right\}}. [ italic_y ] start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT = { italic_x ∈ { 0 , 1 } start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT : italic_x italic_R italic_y } .

For x → = ⟨ x n : n ∈ ℕ ⟩ ∈ ( { 0 , 1 } ℕ ) ℕ fragments → 𝑥 fragments ⟨ subscript 𝑥 𝑛 : n N ⟩ superscript fragments ( superscript fragments { 0 , 1 } ℕ ) ℕ \vec{x}={\langle x_{n}\;:\;n\in\mathbb{N}\rangle}\in({\{0,1\}^{\mathbb{N}}})^{% \mathbb{N}} → start_ARG italic_x end_ARG = ⟨ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n ∈ blackboard_N ⟩ ∈ ( { 0 , 1 } start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT and y ∈ { 0 , 1 } ℕ 𝑦 superscript 0 1 ℕ y\in{\{0,1\}^{\mathbb{N}}} italic_y ∈ { 0 , 1 } start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT , define

Match ⁡ ( x → , y ) = χ { n : x n ⁢ R ⁢ y } . Match → 𝑥 𝑦 subscript 𝜒 conditional-set 𝑛 subscript 𝑥 𝑛 𝑅 𝑦 \operatorname{Match}(\vec{x},y)=\chi_{{\left\{n\;:\;x_{n}Ry\right\}}}. roman_Match ( → start_ARG italic_x end_ARG , italic_y ) = italic_χ start_POSTSUBSCRIPT { italic_n : italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_R italic_y } end_POSTSUBSCRIPT .
Definition 4.4 .

Let X 𝑋 X italic_X be a smooth projective surface defined over an algebraically closed field and f : X ⇢ X : 𝑓 ⇢ 𝑋 𝑋 f:X\dashrightarrow X italic_f : italic_X ⇢ italic_X be a birational transformation. We say that ( X ′ , f ′ ) superscript 𝑋 ′ superscript 𝑓 ′ (X^{\prime},f^{\prime}) ( italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is a birational model of ( X , f ) 𝑋 𝑓 (X,f) ( italic_X , italic_f ) if there is a birational map π : X ′ ⇢ X : 𝜋 ⇢ superscript 𝑋 ′ 𝑋 \pi:X^{\prime}\dashrightarrow X italic_π : italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⇢ italic_X such that

f ′ = π - 1 ∘ f ∘ π . superscript 𝑓 ′ superscript 𝜋 1 𝑓 𝜋 f^{\prime}=\pi^{-1}\circ f\circ\pi. italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ italic_f ∘ italic_π .
Definition 2.1

Let 𝕂 𝕂 \mathbb{K} blackboard_K be a division ring and suppose that there exists a pair ( σ , ε ) 𝜎 𝜀 (\sigma,\varepsilon) ( italic_σ , italic_ε ) where σ 𝜎 \sigma italic_σ is an anti-automorphism of 𝕂 𝕂 \mathbb{K} blackboard_K and ε ∈ 𝕂 * 𝜀 superscript 𝕂 \varepsilon\in\mathbb{K}^{*} italic_ε ∈ blackboard_K start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT such that

  1. (i)

    ε σ = ε - 1 ; superscript 𝜀 𝜎 superscript 𝜀 1 \varepsilon^{\sigma}=\varepsilon^{-1}; italic_ε start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ;

  2. (ii)

    t σ 2 = ε ⁢ t ⁢ ε - 1 . superscript 𝑡 superscript 𝜎 2 𝜀 𝑡 superscript 𝜀 1 t^{{\sigma}^{2}}=\varepsilon t\varepsilon^{-1}. italic_t start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT = italic_ε italic_t italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT .

Let 𝕍 𝕍 \mathbb{V} blackboard_V be a right vector space over 𝕂 𝕂 \mathbb{K} blackboard_K . Then a function ϕ : 𝕍 × 𝕍 → 𝕂 normal-: italic-ϕ normal-→ 𝕍 𝕍 𝕂 \phi\colon\mathbb{V}\times\mathbb{V}\rightarrow\mathbb{K} italic_ϕ : blackboard_V × blackboard_V → blackboard_K is a reflexive ( σ , ε ) 𝜎 𝜀 (\sigma,\varepsilon) ( italic_σ , italic_ε ) -sesquilinear form (also ( σ , ε ) 𝜎 𝜀 (\sigma,\varepsilon) ( italic_σ , italic_ε ) -sesquilinear form for short) if

  1. (iii)

    ϕ ⁢ ( x , y ⁢ α + z ⁢ β ) = ϕ ⁢ ( x , y ) ⁢ α + ϕ ⁢ ( x , z ) ⁢ β ⁢ ∀ α , β ∈ 𝕂 , ∀ x , y ∈ 𝕍 formulae-sequence italic-ϕ 𝑥 𝑦 𝛼 𝑧 𝛽 italic-ϕ 𝑥 𝑦 𝛼 italic-ϕ 𝑥 𝑧 𝛽 for-all 𝛼 formulae-sequence 𝛽 𝕂 for-all 𝑥 𝑦 𝕍 \phi(x,y\alpha+z\beta)=\phi(x,y)\alpha+\phi(x,z)\beta\,\,\,\,\forall\alpha,% \beta\in\mathbb{K},\forall x,y\in\mathbb{V} italic_ϕ ( italic_x , italic_y italic_α + italic_z italic_β ) = italic_ϕ ( italic_x , italic_y ) italic_α + italic_ϕ ( italic_x , italic_z ) italic_β ∀ italic_α , italic_β ∈ blackboard_K , ∀ italic_x , italic_y ∈ blackboard_V ;

  2. (iv)

    ϕ ⁢ ( y , x ) = ϕ ⁢ ( x , y ) σ ⁢ ε , ∀ x , y ∈ 𝕍 . formulae-sequence italic-ϕ 𝑦 𝑥 italic-ϕ superscript 𝑥 𝑦 𝜎 𝜀 for-all 𝑥 𝑦 𝕍 \phi(y,x)=\phi(x,y)^{\sigma}\varepsilon,\,\,\,\,\forall x,y\in\mathbb{V}. italic_ϕ ( italic_y , italic_x ) = italic_ϕ ( italic_x , italic_y ) start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT italic_ε , ∀ italic_x , italic_y ∈ blackboard_V .

Definition 1 .

A triangle quadruple t = ( a , b , c , d ) 𝑡 𝑎 𝑏 𝑐 𝑑 t=(a,\,b,\,c,\,d) italic_t = ( italic_a , italic_b , italic_c , italic_d ) is a quadruple of nonnegative integers satisfying

3 ⁢ ( a 2 + b 2 + c 2 + d 2 ) = ( a + b + c + d ) 2 . 3 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 superscript 𝑑 2 superscript 𝑎 𝑏 𝑐 𝑑 2 3(a^{2}+b^{2}+c^{2}+d^{2})=(a+b+c+d)^{2}. 3 ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = ( italic_a + italic_b + italic_c + italic_d ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .
Definition 2.1 .

A vector space 𝒵 𝒵 \mathcal{Z} caligraphic_Z over a field K 𝐾 K italic_K with a bilinear operation “ ∘ \circ ∘ ” is called Zinbiel algebra if for any x , y , z ∈ 𝒵 𝑥 𝑦 𝑧 𝒵 x,y,z\in\mathcal{Z} italic_x , italic_y , italic_z ∈ caligraphic_Z the following identity

(2.1) ( x ∘ y ) ∘ z = x ∘ ( y ∘ z ) + x ∘ ( z ∘ y ) 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 𝑧 𝑦 (x\circ y)\circ z=x\circ(y\circ z)+x\circ(z\circ y) ( italic_x ∘ italic_y ) ∘ italic_z = italic_x ∘ ( italic_y ∘ italic_z ) + italic_x ∘ ( italic_z ∘ italic_y )

holds.

Definition 3.1 .

Let A 𝐴 A italic_A be an algebra with multiplication ∙ ∙ \bullet ∙ . A bilinear form σ 𝜎 \sigma italic_σ on A 𝐴 A italic_A is called invariant if it satisfies the condition

σ ⁢ ( u , v ∙ w ) = σ ⁢ ( u ∙ v , w ) 𝜎 𝑢 ∙ 𝑣 𝑤 𝜎 ∙ 𝑢 𝑣 𝑤 \sigma(u,v\bullet w)=\sigma(u\bullet v,w) italic_σ ( italic_u , italic_v ∙ italic_w ) = italic_σ ( italic_u ∙ italic_v , italic_w ) (5)

for all u , v , w ∈ A 𝑢 𝑣 𝑤 𝐴 u,v,w\in A italic_u , italic_v , italic_w ∈ italic_A .

Definition 4 .

(The Gyrogroup Cooperation (Coaddition)). Let ( G , ⊕ ) 𝐺 direct-sum (G,\mathbf{\oplus}) ( italic_G , ⊕ ) be a gyrogroup with gyrogroup operation (or, addition) ⊕ direct-sum \mathbf{\oplus} ⊕ . The gyrogroup cooperation (or, coaddition) ⊞ normal-⊞ \boxplus ⊞ is a second binary operation in G 𝐺 G italic_G given by the equation

(43) 𝐚 ⊞ 𝐛 = 𝐚 ⊕ gyr ⁢ [ 𝐚 , ⊖ 𝐛 ] ⁢ 𝐛 ⊞ 𝐚 𝐛 direct-sum 𝐚 gyr 𝐚 symmetric-difference 𝐛 𝐛 \mathbf{a}\boxplus\mathbf{b}=\mathbf{a}\mathbf{\oplus}{\rm gyr}[\mathbf{a},% \mathbf{\ominus}\mathbf{b}]\mathbf{b} bold_a ⊞ bold_b = bold_a ⊕ roman_gyr [ bold_a , ⊖ bold_b ] bold_b

for all 𝐚 , 𝐛 ∈ G 𝐚 𝐛 𝐺 \mathbf{a},\mathbf{b}\in G bold_a , bold_b ∈ italic_G .

Definition 6.1 .

A cross ratio on X 𝑋 X italic_X is a Hölder-continuous function 𝖻 : X ( 4 ) → ℝ : 𝖻 → superscript 𝑋 4 ℝ \mathsf{b}:X^{(4)}\to\mathbb{R} sansserif_b : italic_X start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT → blackboard_R that verifies the following relations for every x , y , z , t , w ∈ X : : 𝑥 𝑦 𝑧 𝑡 𝑤 𝑋 absent x,y,z,t,w\in X: italic_x , italic_y , italic_z , italic_t , italic_w ∈ italic_X :

Definition 19

For each fixed k 𝑘 k italic_k , let h ℎ h italic_h be the maximum common divisor of n 𝑛 n italic_n and k 𝑘 k italic_k . We define

n ¯ = n / h ⁢ and ⁢ k ¯ = k / h . ¯ 𝑛 𝑛 ℎ and ¯ 𝑘 𝑘 ℎ \bar{n}=n/h\text{ and }\bar{k}=k/h. ¯ start_ARG italic_n end_ARG = italic_n / italic_h and ¯ start_ARG italic_k end_ARG = italic_k / italic_h .
Definition 4.22 .

Let A ¯ ⊂ A ¯ 𝐴 𝐴 \bar{A}\subset A ¯ start_ARG italic_A end_ARG ⊂ italic_A be the kernel of the augmentation, define B ⁢ ( A ) = ( T ⁢ ( s ⁢ A ¯ ) , d ) 𝐵 𝐴 𝑇 𝑠 ¯ 𝐴 𝑑 B(A)=(T(s\bar{A}),d) italic_B ( italic_A ) = ( italic_T ( italic_s ¯ start_ARG italic_A end_ARG ) , italic_d ) ( s = 𝑠 absent s= italic_s = suspension) where d 𝑑 d italic_d is generated as a coderivation by

d ⁢ ( s ⁢ ( a ) ) = s ⁢ ( a ⊗ a ) - s ⁢ ( a ⊗ 1 ) - s ⁢ ( 1 ⊗ a ) . 𝑑 𝑠 𝑎 𝑠 tensor-product 𝑎 𝑎 𝑠 tensor-product 𝑎 1 𝑠 tensor-product 1 𝑎 d(s(a))=s(a\otimes a)-s(a\otimes 1)-s(1\otimes a). italic_d ( italic_s ( italic_a ) ) = italic_s ( italic_a ⊗ italic_a ) - italic_s ( italic_a ⊗ 1 ) - italic_s ( 1 ⊗ italic_a ) .

Dually, let C = C ¯ ⊕ ℚ 𝐶 direct-sum ¯ 𝐶 ℚ C=\bar{C}\oplus\mathbb{Q} italic_C = ¯ start_ARG italic_C end_ARG ⊕ blackboard_Q , and define Ω ⁢ C = ( T ⁢ ( s - 1 ⁢ C ¯ ) , d ) Ω 𝐶 𝑇 superscript 𝑠 1 ¯ 𝐶 𝑑 \Omega C=(T(s^{-1}\bar{C}),d) roman_Ω italic_C = ( italic_T ( italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ¯ start_ARG italic_C end_ARG ) , italic_d ) where d 𝑑 d italic_d is generated as a derivation by the equation

d ⁢ ( s - 1 ⁢ c ) = s - 1 ⁢ ( Δ ¯ ⁢ ( c ) ) = s - 1 ⁢ ( Δ ⁢ ( c ) - c ⊗ 1 - 1 ⊗ c ) . 𝑑 superscript 𝑠 1 𝑐 superscript 𝑠 1 ¯ Δ 𝑐 superscript 𝑠 1 Δ 𝑐 tensor-product 𝑐 1 tensor-product 1 𝑐 d(s^{-1}c)=s^{-1}(\bar{\Delta}(c))=s^{-1}(\Delta(c)-c\otimes 1-1\otimes c). italic_d ( italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_c ) = italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ¯ start_ARG roman_Δ end_ARG ( italic_c ) ) = italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( roman_Δ ( italic_c ) - italic_c ⊗ 1 - 1 ⊗ italic_c ) .