Definition 3.2 .

A string is called alternating if j i β‰  j i + 1 subscript 𝑗 𝑖 subscript 𝑗 𝑖 1 j_{i}\neq j_{i+1} italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT β‰  italic_j start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT for every i ∈ { 1 , … , β„“ - 1 } 𝑖 1 … β„“ 1 i\in\{1,\dots,\ell-1\} italic_i ∈ { 1 , … , roman_β„“ - 1 } . For a string s 𝑠 s italic_s , we define the alternating reduction red ⁑ ( s ) red 𝑠 \operatorname{red}(s) roman_red ( italic_s ) to be the alternating string obtained by replacing consecutive occurrences of the same letter by a single occurrence of that letter; for instance,

red ⁑ ( 112331 ) = 1231 , red ⁑ ( 1221311 ) = 12131 . formulae-sequence red 112331 1231 red 1221311 12131 \operatorname{red}(112331)=1231,\qquad\operatorname{red}(1221311)=12131. roman_red ( 112331 ) = 1231 , roman_red ( 1221311 ) = 12131 .
Definition 2.6 .

Let ( H , [ - - - ] ) fragments ( H , fragments [ ] ) (H,[---]) ( italic_H , [ - - - ] ) be a herd. A subset S βŠ† H 𝑆 𝐻 S\subseteq H italic_S βŠ† italic_H is a sub-herd , if it is closed under [ - - - ] fragments [ ] [---] [ - - - ] , i.e., for all x , y , z ∈ S π‘₯ 𝑦 𝑧 𝑆 x,y,z\in S italic_x , italic_y , italic_z ∈ italic_S , [ x , y , z ] ∈ S π‘₯ 𝑦 𝑧 𝑆 [x,y,z]\in S [ italic_x , italic_y , italic_z ] ∈ italic_S . A sub-herd S 𝑆 S italic_S of ( H , [ - - - ] ) fragments ( H , fragments [ ] ) (H,[---]) ( italic_H , [ - - - ] ) is said to be normal if there exists e ∈ S 𝑒 𝑆 e\in S italic_e ∈ italic_S such that, for all x ∈ H π‘₯ 𝐻 x\in H italic_x ∈ italic_H and s ∈ S 𝑠 𝑆 s\in S italic_s ∈ italic_S there exists t ∈ S 𝑑 𝑆 t\in S italic_t ∈ italic_S such that

[ x , e , s ] = [ t , e , x ] . π‘₯ 𝑒 𝑠 𝑑 𝑒 π‘₯ [x,e,s]=[t,e,x]. [ italic_x , italic_e , italic_s ] = [ italic_t , italic_e , italic_x ] . (2.16)
Definition 3.1 .

A truss is an algebraic system consisting of a set T 𝑇 T italic_T , a ternary operation [ - - - ] fragments [ ] [---] [ - - - ] making T 𝑇 T italic_T into an Abelian herd, and an associative binary operation β‹… β‹… \cdot β‹… (denoted by juxtaposition) which distributes over [ - - - ] fragments [ ] [---] [ - - - ] , that is, for all w , x , y , z ∈ T 𝑀 π‘₯ 𝑦 𝑧 𝑇 w,x,y,z\in T italic_w , italic_x , italic_y , italic_z ∈ italic_T ,

w ⁒ [ x , y , z ] = [ w ⁒ x , w ⁒ y , w ⁒ z ] , [ x , y , z ] ⁒ w = [ x ⁒ w , y ⁒ w , z ⁒ w ] . formulae-sequence 𝑀 π‘₯ 𝑦 𝑧 𝑀 π‘₯ 𝑀 𝑦 𝑀 𝑧 π‘₯ 𝑦 𝑧 𝑀 π‘₯ 𝑀 𝑦 𝑀 𝑧 𝑀 w[x,y,z]=[wx,wy,wz],\qquad[x,y,z]w=[xw,yw,zw]. italic_w [ italic_x , italic_y , italic_z ] = [ italic_w italic_x , italic_w italic_y , italic_w italic_z ] , [ italic_x , italic_y , italic_z ] italic_w = [ italic_x italic_w , italic_y italic_w , italic_z italic_w ] . (3.1)

A truss is said to be commutative if the binary operation β‹… β‹… \cdot β‹… is commutative.

Given trusses ( T , [ - - - ] , β‹… ) fragments ( T , fragments [ ] , β‹… ) (T,[---],\cdot) ( italic_T , [ - - - ] , β‹… ) and ( T ~ , [ - - - ] , β‹… ) fragments ( ~ 𝑇 , fragments [ ] , β‹… ) (\tilde{T},[---],\cdot) ( ~ start_ARG italic_T end_ARG , [ - - - ] , β‹… ) a function Ο† : T β†’ T ~ : πœ‘ β†’ 𝑇 ~ 𝑇 \varphi:T\to\tilde{T} italic_Ο† : italic_T β†’ ~ start_ARG italic_T end_ARG that is both a morphism of herds (with respect of [ - - - ] fragments [ ] [---] [ - - - ] ) and semigroups (with respect to β‹… β‹… \cdot β‹… ) is called a morphism of trusses or a truss homomorphism . The category of trusses is denoted by 𝐓𝐫𝐬 𝐓𝐫𝐬 \mathbf{Trs} bold_Trs .

By a sub-truss of ( T , [ - - - ] , β‹… ) fragments ( T , fragments [ ] , β‹… ) (T,[---],\cdot) ( italic_T , [ - - - ] , β‹… ) we mean a non-empty subset of T 𝑇 T italic_T closed under both operations.

Definition 3.8 .

A truss ( T , [ - - - ] , β‹… ) fragments ( T , fragments [ ] , β‹… ) (T,[---],\cdot) ( italic_T , [ - - - ] , β‹… ) is said to be unital , if ( T , β‹… ) 𝑇 β‹… (T,\cdot) ( italic_T , β‹… ) is a monoid (with the identity denoted by 1 1 1 1 ).

An element 0 0 of a truss ( T , [ - - - ] , β‹… ) fragments ( T , fragments [ ] , β‹… ) (T,[---],\cdot) ( italic_T , [ - - - ] , β‹… ) is called an absorber if, for all x ∈ T π‘₯ 𝑇 x\in T italic_x ∈ italic_T ,

x ⁒ 0 = 0 = 0 ⁒ x . π‘₯ 0 0 0 π‘₯ x0=0=0x. italic_x 0 = 0 = 0 italic_x . (3.8)
Definition 2.5 (Quasihomogeneous blow-up) .

Consider the vector field X : ℝ m + n + 1 β†’ ℝ m + n + 1 : 𝑋 β†’ superscript ℝ π‘š 𝑛 1 superscript ℝ π‘š 𝑛 1 X:\mathbb{R}^{m+n+1}\to\mathbb{R}^{m+n+1} italic_X : blackboard_R start_POSTSUPERSCRIPT italic_m + italic_n + 1 end_POSTSUPERSCRIPT β†’ blackboard_R start_POSTSUPERSCRIPT italic_m + italic_n + 1 end_POSTSUPERSCRIPT given by ( 2.6 ) and assume that X ⁒ ( 0 ) = 0 𝑋 0 0 X(0)=0 italic_X ( 0 ) = 0 . Let Ξ± = ( Ξ± 1 , … , Ξ± m ) ∈ β„• 0 m 𝛼 subscript 𝛼 1 … subscript 𝛼 π‘š subscript superscript β„• π‘š 0 \alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}_{0} italic_Ξ± = ( italic_Ξ± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Ξ± start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∈ blackboard_N start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , Ξ² = ( Ξ² 1 , … , Ξ² m ) ∈ β„• 0 n 𝛽 subscript 𝛽 1 … subscript 𝛽 π‘š subscript superscript β„• 𝑛 0 \beta=(\beta_{1},\ldots,\beta_{m})\in\mathbb{N}^{n}_{0} italic_Ξ² = ( italic_Ξ² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Ξ² start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∈ blackboard_N start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Ξ³ ∈ β„• 0 𝛾 subscript β„• 0 \gamma\in\mathbb{N}_{0} italic_Ξ³ ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . Let the generalized polar transformation Ο• : π•Š m + n Γ— I β†’ ℝ m + n + 1 : italic-Ο• β†’ superscript π•Š π‘š 𝑛 𝐼 superscript ℝ π‘š 𝑛 1 \phi:\mathbb{S}^{m+n}\times I\to\mathbb{R}^{m+n+1} italic_Ο• : blackboard_S start_POSTSUPERSCRIPT italic_m + italic_n end_POSTSUPERSCRIPT Γ— italic_I β†’ blackboard_R start_POSTSUPERSCRIPT italic_m + italic_n + 1 end_POSTSUPERSCRIPT be defined by

(2.7) Ο• ⁒ ( x , y , Ξ΅ ) = ( r Ξ± ⁒ x Β― , r Ξ² ⁒ y Β― , r Ξ³ ⁒ Ξ΅ Β― ) , italic-Ο• π‘₯ 𝑦 πœ€ superscript π‘Ÿ 𝛼 Β― π‘₯ superscript π‘Ÿ 𝛽 Β― 𝑦 superscript π‘Ÿ 𝛾 Β― πœ€ \phi(x,y,\varepsilon)=(r^{\alpha}\bar{x},r^{\beta}\bar{y},r^{\gamma}\bar{% \varepsilon}), italic_Ο• ( italic_x , italic_y , italic_Ξ΅ ) = ( italic_r start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT Β― start_ARG italic_x end_ARG , italic_r start_POSTSUPERSCRIPT italic_Ξ² end_POSTSUPERSCRIPT Β― start_ARG italic_y end_ARG , italic_r start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT Β― start_ARG italic_Ξ΅ end_ARG ) ,

where ( x Β― , y Β― , Ξ΅ Β― ) = ( x Β― 1 , … , x Β― m , y Β― 1 , … , y Β― n , Ξ΅ Β― ) ∈ π•Š m + n Β― π‘₯ Β― 𝑦 Β― πœ€ subscript Β― π‘₯ 1 … subscript Β― π‘₯ π‘š subscript Β― 𝑦 1 … subscript Β― 𝑦 𝑛 Β― πœ€ superscript π•Š π‘š 𝑛 (\bar{x},\bar{y},\bar{\varepsilon})=(\bar{x}_{1},\ldots,\bar{x}_{m},\bar{y}_{1% },\ldots,\bar{y}_{n},\bar{\varepsilon})\in\mathbb{S}^{m+n} ( Β― start_ARG italic_x end_ARG , Β― start_ARG italic_y end_ARG , Β― start_ARG italic_Ξ΅ end_ARG ) = ( Β― start_ARG italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , Β― start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , Β― start_ARG italic_y end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , Β― start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , Β― start_ARG italic_Ξ΅ end_ARG ) ∈ blackboard_S start_POSTSUPERSCRIPT italic_m + italic_n end_POSTSUPERSCRIPT , and I βŠ† ℝ 𝐼 ℝ I\subseteq\mathbb{R} italic_I βŠ† blackboard_R is an interval containing the origin. Here we use the multi-index notation r Ξ± ⁒ x Β― = ( r Ξ± 1 ⁒ x Β― 1 , … , r Ξ± m ⁒ x Β― m ) superscript π‘Ÿ 𝛼 Β― π‘₯ superscript π‘Ÿ subscript 𝛼 1 subscript Β― π‘₯ 1 … superscript π‘Ÿ subscript 𝛼 π‘š subscript Β― π‘₯ π‘š r^{\alpha}\bar{x}=(r^{\alpha_{1}}\bar{x}_{1},\ldots,r^{\alpha_{m}}\bar{x}_{m}) italic_r start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT Β― start_ARG italic_x end_ARG = ( italic_r start_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT Β― start_ARG italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_r start_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT Β― start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) , and similarly for r Ξ² ⁒ y Β― superscript π‘Ÿ 𝛽 Β― 𝑦 r^{\beta}\bar{y} italic_r start_POSTSUPERSCRIPT italic_Ξ² end_POSTSUPERSCRIPT Β― start_ARG italic_y end_ARG . The quasihomogeneous blow-up of the vector field X 𝑋 X italic_X , denoted as X Β― Β― 𝑋 \bar{X} Β― start_ARG italic_X end_ARG , is defined by

(2.8) X Β― = D ⁒ Ο• - 1 | ( x Β― , y Β― , Ξ΅ Β― , r ) ∘ X ∘ Ο• ⁒ ( x Β― , y Β― , Ξ΅ Β― , r ) . Β― 𝑋 evaluated-at D superscript italic-Ο• 1 Β― π‘₯ Β― 𝑦 Β― πœ€ π‘Ÿ 𝑋 italic-Ο• Β― π‘₯ Β― 𝑦 Β― πœ€ π‘Ÿ \bar{X}=\textnormal{D}\phi^{-1}|_{(\bar{x},\bar{y},\bar{\varepsilon},r)}\circ X% \circ\phi(\bar{x},\bar{y},\bar{\varepsilon},r). Β― start_ARG italic_X end_ARG = D italic_Ο• start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT ( Β― start_ARG italic_x end_ARG , Β― start_ARG italic_y end_ARG , Β― start_ARG italic_Ξ΅ end_ARG , italic_r ) end_POSTSUBSCRIPT ∘ italic_X ∘ italic_Ο• ( Β― start_ARG italic_x end_ARG , Β― start_ARG italic_y end_ARG , Β― start_ARG italic_Ξ΅ end_ARG , italic_r ) .
Definition 3.1 .

A composition algebra ( C , q ) 𝐢 π‘ž (C,q) ( italic_C , italic_q ) over R 𝑅 R italic_R is a faithfully projective R 𝑅 R italic_R -module C 𝐢 C italic_C endowed with a bilinear map C Γ— C β†’ C , ( x , y ) ↦ x β‹… y formulae-sequence β†’ 𝐢 𝐢 𝐢 maps-to π‘₯ 𝑦 β‹… π‘₯ 𝑦 C\times C\to C,(x,y)\mapsto x\cdot y italic_C Γ— italic_C β†’ italic_C , ( italic_x , italic_y ) ↦ italic_x β‹… italic_y (the multiplication) and a nonsingular quadratic form q = q C π‘ž subscript π‘ž 𝐢 q=q_{C} italic_q = italic_q start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT satisfying q ⁒ ( x β‹… y ) = q ⁒ ( x ) ⁒ q ⁒ ( y ) π‘ž β‹… π‘₯ 𝑦 π‘ž π‘₯ π‘ž 𝑦 q(x\cdot y)=q(x)q(y) italic_q ( italic_x β‹… italic_y ) = italic_q ( italic_x ) italic_q ( italic_y ) for all x , y ∈ C π‘₯ 𝑦 𝐢 x,y\in C italic_x , italic_y ∈ italic_C . The algebra is called symmetric if the symmetric bilinear form

( x , y ) ↦ ⟨ x , y ⟩ = q ⁒ ( x + y ) - q ⁒ ( x ) - q ⁒ ( y ) maps-to π‘₯ 𝑦 π‘₯ 𝑦 π‘ž π‘₯ 𝑦 π‘ž π‘₯ π‘ž 𝑦 (x,y)\mapsto\langle x,y\rangle=q(x+y)-q(x)-q(y) ( italic_x , italic_y ) ↦ ⟨ italic_x , italic_y ⟩ = italic_q ( italic_x + italic_y ) - italic_q ( italic_x ) - italic_q ( italic_y )

satisfies

⟨ x β‹… y , z ⟩ = ⟨ x , y β‹… z ⟩ β‹… π‘₯ 𝑦 𝑧 π‘₯ β‹… 𝑦 𝑧 \langle x\cdot y,z\rangle=\langle x,y\cdot z\rangle ⟨ italic_x β‹… italic_y , italic_z ⟩ = ⟨ italic_x , italic_y β‹… italic_z ⟩

for any x , y , z ∈ C π‘₯ 𝑦 𝑧 𝐢 x,y,z\in C italic_x , italic_y , italic_z ∈ italic_C .

Definition 3.10 (Elastica in ℍ 2 superscript ℍ 2 \mathbb{H}^{2} blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .

A curve Ξ³ ∈ C ∞ ⁒ ( ( 0 , L ) , ℍ 2 ) 𝛾 superscript 𝐢 0 𝐿 superscript ℍ 2 \gamma\in C^{\infty}((0,L),\mathbb{H}^{2}) italic_Ξ³ ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ( 0 , italic_L ) , blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) is called elastica or elastic curve in ℍ 2 superscript ℍ 2 \mathbb{H}^{2} blackboard_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT if it is parametrized with hyperbolic arclength and satisfies

2 ⁒ ΞΊ ⁒ [ Ξ³ ] β€²β€² + ΞΊ ⁒ [ Ξ³ ] 3 - ( Ξ» + 2 ) ⁒ ΞΊ ⁒ [ Ξ³ ] = 0 2 πœ… superscript delimited-[] 𝛾 β€²β€² πœ… superscript delimited-[] 𝛾 3 πœ† 2 πœ… delimited-[] 𝛾 0 2\kappa[\gamma]^{\prime\prime}+\kappa[\gamma]^{3}-(\lambda+2)\kappa[\gamma]=0 2 italic_ΞΊ [ italic_Ξ³ ] start_POSTSUPERSCRIPT β€² β€² end_POSTSUPERSCRIPT + italic_ΞΊ [ italic_Ξ³ ] start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - ( italic_Ξ» + 2 ) italic_ΞΊ [ italic_Ξ³ ] = 0

for some Ξ» ∈ ℝ πœ† ℝ \lambda\in\mathbb{R} italic_Ξ» ∈ blackboard_R . If Ξ» = 0 πœ† 0 \lambda=0 italic_Ξ» = 0 the curve is called free elastica , otherwise it is called Ξ» πœ† \lambda italic_Ξ» - constrained elastica .

Definition 2.1 .

[ References ] delimited-[] References {}^{[\ref{ref06}]} start_FLOATSUPERSCRIPT [ ] end_FLOATSUPERSCRIPT A Lie-Yamaguti algebra(LY-algebra for short) is a vector space T 𝑇 T italic_T over 𝕂 𝕂 \mathbb{K} blackboard_K with a bilinear composition [ β‹… , β‹… ] β‹… β‹… [\cdot,\cdot] [ β‹… , β‹… ] and a trilinear composition { β‹… , β‹… , β‹… } β‹… β‹… β‹… \{\cdot,\cdot,\cdot\} { β‹… , β‹… , β‹… } satisfying:

[ a , a ] = 0 , π‘Ž π‘Ž 0 [a,a]=0, [ italic_a , italic_a ] = 0 , ( L ⁒ Y ⁒ 1 ) 𝐿 π‘Œ 1 ( italic_L italic_Y 1 )
{ a , a , b } = 0 , π‘Ž π‘Ž 𝑏 0 \{a,a,b\}=0, { italic_a , italic_a , italic_b } = 0 , ( L ⁒ Y ⁒ 2 ) 𝐿 π‘Œ 2 ( italic_L italic_Y 2 )
{ a , b , c } + { b , c , a } + { c , a , b } + [ [ a , b ] , c ] + [ [ b , c ] , a ] + [ [ c , a ] , b ] = 0 , π‘Ž 𝑏 𝑐 𝑏 𝑐 π‘Ž 𝑐 π‘Ž 𝑏 π‘Ž 𝑏 𝑐 𝑏 𝑐 π‘Ž 𝑐 π‘Ž 𝑏 0 \{a,b,c\}+\{b,c,a\}+\{c,a,b\}+[[a,b],c]+[[b,c],a]+[[c,a],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 , ( L ⁒ Y ⁒ 3 ) 𝐿 π‘Œ 3 ( italic_L italic_Y 3 )
{ [ a , b ] , c , d } + { [ b , c ] , a , d } + { [ c , a ] , b , d } = 0 , π‘Ž 𝑏 𝑐 𝑑 𝑏 𝑐 π‘Ž 𝑑 𝑐 π‘Ž 𝑏 𝑑 0 \{[a,b],c,d\}+\{[b,c],a,d\}+\{[c,a],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 , ( L ⁒ Y ⁒ 4 ) 𝐿 π‘Œ 4 ( italic_L italic_Y 4 )
{ a , b , [ c , d ] } = [ { a , b , c } , d ] + [ c , { a , b , d } ] , π‘Ž 𝑏 𝑐 𝑑 π‘Ž 𝑏 𝑐 𝑑 𝑐 π‘Ž 𝑏 𝑑 \{a,b,[c,d]\}=[\{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 } ] , ( L ⁒ Y ⁒ 5 ) 𝐿 π‘Œ 5 ( italic_L italic_Y 5 )
{ a , b , { c , d , e } } = { { a , b , c } , d , e } + { c , { a , b , d } , e } + { c , d , { a , b , e } } , π‘Ž 𝑏 𝑐 𝑑 𝑒 π‘Ž 𝑏 𝑐 𝑑 𝑒 𝑐 π‘Ž 𝑏 𝑑 𝑒 𝑐 𝑑 π‘Ž 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 } } , ( L ⁒ Y ⁒ 6 ) 𝐿 π‘Œ 6 ( italic_L italic_Y 6 )

for any a , b , c , d , e ∈ T π‘Ž 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 3.6 .

Let L 𝐿 L italic_L be an algebra over 𝕂 𝕂 \mathbb{K} blackboard_K . If the multiplication satisfies the following identities:

x ⁒ y = y ⁒ x , ( x ⁒ y ) ⁒ ( x ⁒ x ) = x ⁒ ( y ⁒ ( x ⁒ x ) ) , formulae-sequence π‘₯ 𝑦 𝑦 π‘₯ π‘₯ 𝑦 π‘₯ π‘₯ π‘₯ 𝑦 π‘₯ π‘₯ xy=yx,\quad(xy)(xx)=x(y(xx)), italic_x italic_y = italic_y italic_x , ( italic_x italic_y ) ( italic_x italic_x ) = italic_x ( italic_y ( italic_x italic_x ) ) ,

for all x , y ∈ L π‘₯ 𝑦 𝐿 x,y\in L italic_x , italic_y ∈ italic_L , then we call L 𝐿 L italic_L a Jordan algebra.

Definition 3.11 .

A functional f 𝑓 f italic_f on π’ͺ 𝔾 ⁒ ( 𝕂 \ 𝔾 ) subscript π’ͺ 𝔾 \ 𝕂 𝔾 \mathscr{O}_{\mathbb{G}}(\mathbb{K}\backslash\mathbb{G}) script_O start_POSTSUBSCRIPT blackboard_G end_POSTSUBSCRIPT ( blackboard_K \ blackboard_G ) is called 𝕂 𝕂 \mathbb{K} blackboard_K -invariant if

( ΞΈ βŠ— f ) ∘ Ξ± = f . tensor-product πœƒ 𝑓 𝛼 𝑓 (\theta\otimes f)\circ\alpha=f. ( italic_ΞΈ βŠ— italic_f ) ∘ italic_Ξ± = italic_f .
Definition 2.8 .

Let ( A , B , Ξ± ) 𝐴 𝐡 𝛼 \left(A,B,\alpha\right) ( italic_A , italic_B , italic_Ξ± ) , ( M , P , ΞΌ ) 𝑀 𝑃 πœ‡ \left(M,P,\mu\right) ( italic_M , italic_P , italic_ΞΌ ) and ( C , D , Ξ³ ) 𝐢 𝐷 𝛾 \left(C,D,\gamma\right) ( italic_C , italic_D , italic_Ξ³ ) be three crossed modules and f = ⟨ f A , f B ⟩ : ( A , B , Ξ± ) β†’ ( M , P , ΞΌ ) : 𝑓 subscript 𝑓 𝐴 subscript 𝑓 𝐡 β†’ 𝐴 𝐡 𝛼 𝑀 𝑃 πœ‡ f=\left\langle{{f}_{A}},{{f}_{B}}\right\rangle:\left(A,B,\alpha\right)\to\left% (M,P,\mu\right) italic_f = ⟨ italic_f start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟩ : ( italic_A , italic_B , italic_Ξ± ) β†’ ( italic_M , italic_P , italic_ΞΌ ) and g = ⟨ g C , g D ⟩ : ( C , D , Ξ³ ) β†’ ( M , P , ΞΌ ) : 𝑔 subscript 𝑔 𝐢 subscript 𝑔 𝐷 β†’ 𝐢 𝐷 𝛾 𝑀 𝑃 πœ‡ g=\left\langle{{g}_{C}},{{g}_{D}}\right\rangle:\left(C,D,\gamma\right)\to\left% (M,P,\mu\right) italic_g = ⟨ italic_g start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ⟩ : ( italic_C , italic_D , italic_Ξ³ ) β†’ ( italic_M , italic_P , italic_ΞΌ ) be two crossed module morphisms. Then the pullback crossed module of f 𝑓 f italic_f and g 𝑔 g italic_g is ( A Γ— g C f A C , B Γ— g D f B D , Ξ± Γ— Ξ³ ) fragments ( A subscript subscript subscript 𝑔 𝐢 subscript 𝑓 𝐴 C , B subscript subscript subscript 𝑔 𝐷 subscript 𝑓 𝐡 D , Ξ± Ξ³ ) \left(A{}_{{{f}_{A}}}{{\times}_{{{g}_{C}}}}C,B{}_{{{f}_{B}}}{{\times}_{{{g}_{D% }}}}D,\alpha\times\gamma\right) ( italic_A start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_C , italic_B start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D , italic_Ξ± Γ— italic_Ξ³ ) where the action of B Γ— g D f B D fragments B subscript subscript subscript 𝑔 𝐷 subscript 𝑓 𝐡 D B{}_{{{f}_{B}}}{{\times}_{{{g}_{D}}}}D italic_B start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D on A Γ— g C f A C fragments A subscript subscript subscript 𝑔 𝐢 subscript 𝑓 𝐴 C A{}_{{{f}_{A}}}{{\times}_{{{g}_{C}}}}C italic_A start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_C is given by

( b , d ) β‹… ( a , c ) = ( b β‹… a , d β‹… c ) β‹… 𝑏 𝑑 π‘Ž 𝑐 β‹… 𝑏 π‘Ž β‹… 𝑑 𝑐 \left(b,d\right)\cdot\left(a,c\right)=\left(b\cdot a,d\cdot c\right) ( italic_b , italic_d ) β‹… ( italic_a , italic_c ) = ( italic_b β‹… italic_a , italic_d β‹… italic_c )

for all ( b , d ) ∈ B Γ— g D f B D fragments fragments ( b , d ) B subscript subscript subscript 𝑔 𝐷 subscript 𝑓 𝐡 D \left(b,d\right)\in B{}_{{{f}_{B}}}{{\times}_{{{g}_{D}}}}D ( italic_b , italic_d ) ∈ italic_B start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D and ( a , c ) ∈ A Γ— g C f A C fragments fragments ( a , c ) A subscript subscript subscript 𝑔 𝐢 subscript 𝑓 𝐴 C \left(a,c\right)\in A{}_{{{f}_{A}}}{{\times}_{{{g}_{C}}}}C ( italic_a , italic_c ) ∈ italic_A start_FLOATSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_FLOATSUBSCRIPT Γ— start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_C .

Definition 4 .

An ( n , k ) - limit-from 𝑛 π‘˜ (n,k)- ( italic_n , italic_k ) - linear code is called maximum distance separable (MDS) if it achieves the Singleton bound:

d = n - k + 1 , 𝑑 𝑛 π‘˜ 1 d=n-k+1, italic_d = italic_n - italic_k + 1 ,

with d 𝑑 d italic_d denoting the minimum distance 2 2 2 See [ 24 ] for more details. between the codewords of the code.

Definition 12 .

A barrier hinge loss is defined as

β„“ ⁒ ( z ) = max ⁒ ( - b ⁒ ( r + z ) + r , max ⁒ ( b ⁒ ( z - r ) , r - z ) ) ⁒ , β„“ 𝑧 max 𝑏 π‘Ÿ 𝑧 π‘Ÿ max 𝑏 𝑧 π‘Ÿ π‘Ÿ 𝑧 , \displaystyle\ell(z)=\mathrm{max}(-b(r+z)+r,\mathrm{max}(b(z-r),r-z))\text{,} roman_β„“ ( italic_z ) = roman_max ( - italic_b ( italic_r + italic_z ) + italic_r , roman_max ( italic_b ( italic_z - italic_r ) , italic_r - italic_z ) ) ,

where b > 1 𝑏 1 b>1 italic_b > 1 and r > 0 π‘Ÿ 0 r>0 italic_r > 0 .

Definition 1 (Level) .

Let G ⁒ ( V , E ) 𝐺 𝑉 𝐸 G(V,E) italic_G ( italic_V , italic_E ) be a network. For any vertex u 𝑒 u italic_u in G ⁒ ( V , E ) 𝐺 𝑉 𝐸 G(V,E) italic_G ( italic_V , italic_E ) , where s 𝑠 s italic_s is the source node, the l ⁒ e ⁒ v ⁒ e ⁒ l 𝑙 𝑒 𝑣 𝑒 𝑙 level italic_l italic_e italic_v italic_e italic_l of the node u 𝑒 u italic_u is

l ⁒ ( u ) = d ⁒ ( s , u ) 𝑙 𝑒 𝑑 𝑠 𝑒 l(u)=d(s,u) italic_l ( italic_u ) = italic_d ( italic_s , italic_u )

i.e., the level of u 𝑒 u italic_u is its geometric distance to s 𝑠 s italic_s . Let

S i = { u ∣ u ∈ V , l ⁒ ( u ) = i } subscript 𝑆 𝑖 conditional-set 𝑒 formulae-sequence 𝑒 𝑉 𝑙 𝑒 𝑖 S_{i}=\{u\mid u\in V,\ l(u)=i\} italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_u ∣ italic_u ∈ italic_V , italic_l ( italic_u ) = italic_i }

denote the set containing all the vertices at level i 𝑖 i italic_i .

Definition 1 .

A ternary term p ⁒ ( x , y , z ) 𝑝 π‘₯ 𝑦 𝑧 p(x,y,z) italic_p ( italic_x , italic_y , italic_z ) of an algebra 𝐀 𝐀 {\mathbf{A}} bold_A is a Maltsev term for 𝐀 𝐀 {\mathbf{A}} bold_A if it satisfies the equations

p ⁒ ( x , x , y ) β‰ˆ p ⁒ ( y , x , x ) β‰ˆ y 𝑝 π‘₯ π‘₯ 𝑦 𝑝 𝑦 π‘₯ π‘₯ 𝑦 p(x,x,y)\approx p(y,x,x)\approx y italic_p ( italic_x , italic_x , italic_y ) β‰ˆ italic_p ( italic_y , italic_x , italic_x ) β‰ˆ italic_y

and aΒ 6-ary term t ⁒ ( x 1 , … , x 6 ) 𝑑 subscript π‘₯ 1 … subscript π‘₯ 6 t(x_{1},\dots,x_{6}) italic_t ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) is a minority-majority term of 𝐀 𝐀 {\mathbf{A}} bold_A if it satisfies the equations

t ⁒ ( y , x , x , z , y , y ) 𝑑 𝑦 π‘₯ π‘₯ 𝑧 𝑦 𝑦 \displaystyle t(y,x,x,z,y,y) italic_t ( italic_y , italic_x , italic_x , italic_z , italic_y , italic_y ) β‰ˆ y absent 𝑦 \displaystyle\approx y β‰ˆ italic_y
t ⁒ ( x , y , x , y , z , y ) 𝑑 π‘₯ 𝑦 π‘₯ 𝑦 𝑧 𝑦 \displaystyle t(x,y,x,y,z,y) italic_t ( italic_x , italic_y , italic_x , italic_y , italic_z , italic_y ) β‰ˆ y absent 𝑦 \displaystyle\approx y β‰ˆ italic_y
t ⁒ ( x , x , y , y , y , z ) 𝑑 π‘₯ π‘₯ 𝑦 𝑦 𝑦 𝑧 \displaystyle t(x,x,y,y,y,z) italic_t ( italic_x , italic_x , italic_y , italic_y , italic_y , italic_z ) β‰ˆ y . absent 𝑦 \displaystyle\approx y. β‰ˆ italic_y .
Definition 2 (Rejection Calibration of a Rejector) .

We say that r : 𝒳 β†’ ℝ : π‘Ÿ β†’ 𝒳 ℝ r:\mathcal{X}\to\mathbb{R} italic_r : caligraphic_X β†’ blackboard_R is rejection calibrated if

sign ⁒ [ r ] = sign ⁒ [ r * ] sign delimited-[] π‘Ÿ sign delimited-[] superscript π‘Ÿ \displaystyle\mathrm{sign}[r]=\mathrm{sign}[r^{*}] roman_sign [ italic_r ] = roman_sign [ italic_r start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ]

holds almost everywhere on input space 𝒳 𝒳 \mathcal{X} caligraphic_X .

Definition 3 (Classification Calibration of a Classifier) .

We say that f : 𝒳 β†’ 𝒴 : 𝑓 β†’ 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y} italic_f : caligraphic_X β†’ caligraphic_Y is classification calibrated if

f = f * 𝑓 superscript 𝑓 \displaystyle f=f^{*} italic_f = italic_f start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT

holds almost everywhere on input space 𝒳 𝒳 \mathcal{X} caligraphic_X .

Definition 2.1 .

We say that a function u 𝑒 u italic_u is screw-symmetric if

u ⁒ ( r , ΞΈ + Ξ± , t + Ξ± ) = u ⁒ ( r , ΞΈ , t ) 𝑒 π‘Ÿ πœƒ 𝛼 𝑑 𝛼 𝑒 π‘Ÿ πœƒ 𝑑 u(r,\theta+\alpha,t+\alpha)=u(r,\theta,t) italic_u ( italic_r , italic_ΞΈ + italic_Ξ± , italic_t + italic_Ξ± ) = italic_u ( italic_r , italic_ΞΈ , italic_t )

for any Ξ± ∈ ℝ 𝛼 ℝ \alpha\in\mathbb{R} italic_Ξ± ∈ blackboard_R .

Definition 3.1 .

A commutative, maybe nonassociative, algebra with a positive definite associating form ⟨ , ⟩ fragments ⟨ , ⟩ \langle,\rangle ⟨ , ⟩ satisfying

(37) ⟨ x 2 , x 2 ⟩ = ⟨ x , x ⟩ 2 superscript π‘₯ 2 superscript π‘₯ 2 superscript π‘₯ π‘₯ 2 \langle x^{2},x^{2}\rangle=\langle x,x\rangle^{2} ⟨ italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = ⟨ italic_x , italic_x ⟩ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

is called an eiconal algebra.

Definition 1.1 .

An enhanced Leibniz algebra (ELA) is a triple
( t : π•Ž β†’ 𝕍 , [ β‹… , β‹… ] , ∘ ) : 𝑑 β†’ π•Ž 𝕍 β‹… β‹… (t\colon\mathbb{W}\to\mathbb{V},[\cdot,\cdot],\circ) ( italic_t : roman_π•Ž β†’ roman_𝕍 , [ β‹… , β‹… ] , ∘ ) consisting of a linear map

t : π•Ž β†’ 𝕍 : 𝑑 β†’ π•Ž 𝕍 t\colon{\mathbb{W}}\to{\mathbb{V}} italic_t : roman_π•Ž β†’ roman_𝕍

where ( 𝕍 , [ β‹… , β‹… ] ) 𝕍 β‹… β‹… (\mathbb{V},[\cdot,\cdot]) ( roman_𝕍 , [ β‹… , β‹… ] ) is a left Leibniz algebra, i.e. for all v 1 , v 2 , v 3 ∈ 𝕍 subscript 𝑣 1 subscript 𝑣 2 subscript 𝑣 3 𝕍 v_{1},v_{2},v_{3}\in{\mathbb{V}} italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ roman_𝕍 one has

[ v 1 , [ v 2 , v 3 ] ] = [ [ v 1 , v 2 ] , v 3 ] + [ v 2 , [ v 1 , v 3 ] ] , subscript 𝑣 1 subscript 𝑣 2 subscript 𝑣 3 subscript 𝑣 1 subscript 𝑣 2 subscript 𝑣 3 subscript 𝑣 2 subscript 𝑣 1 subscript 𝑣 3 [v_{1},[v_{2},v_{3}]]\,=\,[[v_{1},v_{2}],v_{3}]+[v_{2},[v_{1},v_{3}]], [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , [ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] ] = [ [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] , italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] + [ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] ] , (9)

and a (bilinear) product ∘ : 𝕍 Γ— 𝕍 β†’ π•Ž fragments : 𝕍 𝕍 β†’ π•Ž \circ\colon{\mathbb{V}}\times{\mathbb{V}}\to{\mathbb{W}} ∘ : roman_𝕍 Γ— roman_𝕍 β†’ roman_π•Ž such that for all v ∈ 𝕍 𝑣 𝕍 v\in{\mathbb{V}} italic_v ∈ roman_𝕍 and all w ∈ π•Ž 𝑀 π•Ž w\in{\mathbb{W}} italic_w ∈ roman_π•Ž

  1. (a)

    [ t ⁒ ( w ) , v ] = 0 𝑑 𝑀 𝑣 0 [t(w),v]\,=\,0 [ italic_t ( italic_w ) , italic_v ] = 0 ,

  2. (b)

    t ⁒ ( w ) ∘ t ⁒ ( w ) = 0 𝑑 𝑀 𝑑 𝑀 0 t(w)\circ t(w)\,=\,0 italic_t ( italic_w ) ∘ italic_t ( italic_w ) = 0 ,

  3. (c)

    u ∘ s [ v , v ] = v ∘ s [ u , v ] superscript 𝑠 𝑒 𝑣 𝑣 𝑣 superscript 𝑠 𝑒 𝑣 u\stackrel{{\scriptstyle s}}{{\circ}}[v,v]\,=\,v\stackrel{{\scriptstyle s}}{{% \circ}}[u,v] italic_u start_RELOP SUPERSCRIPTOP start_ARG ∘ end_ARG start_ARG italic_s end_ARG end_RELOP [ italic_v , italic_v ] = italic_v start_RELOP SUPERSCRIPTOP start_ARG ∘ end_ARG start_ARG italic_s end_ARG end_RELOP [ italic_u , italic_v ] ,

  4. (d)

    [ v , v ] = t ⁒ ( v ∘ v ) 𝑣 𝑣 𝑑 𝑣 𝑣 [v,v]\,=\,t(v\circ v) [ italic_v , italic_v ] = italic_t ( italic_v ∘ italic_v ) .

Definition 2.5 (Polygon Feynman amplitudes) .

Under the geometric setting from paragraph 2.0.1 , we set 𝐆 ∈ π’Ÿ β€² ⁒ ( M Γ— M , E ⊠ F * ) 𝐆 superscript π’Ÿ normal-β€² 𝑀 𝑀 normal-⊠ 𝐸 superscript 𝐹 \mathbf{G}\in\mathcal{D}^{\prime}\left(M\times M,E\boxtimes F^{*}\right) bold_G ∈ caligraphic_D start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ( italic_M Γ— italic_M , italic_E ⊠ italic_F start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) to be the Schwartz kernel of P - 1 superscript 𝑃 1 P^{-1} italic_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and π’œ = P + C ∞ ⁒ ( H ⁒ o ⁒ m ⁒ ( E , F ) ) π’œ 𝑃 superscript 𝐢 𝐻 π‘œ π‘š 𝐸 𝐹 \mathcal{A}=P+C^{\infty}(Hom(E,F)) caligraphic_A = italic_P + italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_H italic_o italic_m ( italic_E , italic_F ) ) , where P = Ξ” , k = 2 , E = F formulae-sequence 𝑃 normal-Ξ” formulae-sequence π‘˜ 2 𝐸 𝐹 P=\Delta,k=2,E=F italic_P = roman_Ξ” , italic_k = 2 , italic_E = italic_F in the bosonic case and P = D , k = 1 , E = E + , F = E - formulae-sequence 𝑃 𝐷 formulae-sequence π‘˜ 1 formulae-sequence 𝐸 subscript 𝐸 𝐹 subscript 𝐸 P=D,k=1,E=E_{+},F=E_{-} italic_P = italic_D , italic_k = 1 , italic_E = italic_E start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_F = italic_E start_POSTSUBSCRIPT - end_POSTSUBSCRIPT in the fermionic case. For every n β©Ύ 2 𝑛 2 n\geqslant 2 italic_n β©Ύ 2 , we formally set

t n = 𝐆 ⁒ ( x 1 , x 2 ) ⁒ … ⁒ 𝐆 ⁒ ( x n - 1 , x n ) ⁒ 𝐆 ⁒ ( x n , x 1 ) subscript 𝑑 𝑛 𝐆 subscript π‘₯ 1 subscript π‘₯ 2 … 𝐆 subscript π‘₯ 𝑛 1 subscript π‘₯ 𝑛 𝐆 subscript π‘₯ 𝑛 subscript π‘₯ 1 italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_G ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) … bold_G ( italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) bold_G ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) (2.4)

which is well–defined in C ∞ ⁒ ( M n βˆ– π·π‘–π‘Žπ‘”π‘œπ‘›π‘Žπ‘™π‘  , H ⁒ o ⁒ m ⁒ ( F , E ) ⊠ n ) superscript 𝐢 superscript 𝑀 𝑛 π·π‘–π‘Žπ‘”π‘œπ‘›π‘Žπ‘™π‘  𝐻 π‘œ π‘š superscript 𝐹 𝐸 normal-⊠ absent 𝑛 C^{\infty}\left(M^{n}\setminus\text{Diagonals},Hom(F,E)^{\boxtimes n}\right) italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT βˆ– Diagonals , italic_H italic_o italic_m ( italic_F , italic_E ) start_POSTSUPERSCRIPT ⊠ italic_n end_POSTSUPERSCRIPT ) .

There is a natural fiberwise pairing ⟨ t , Ο† ⟩ 𝑑 πœ‘ \left\langle t,\varphi\right\rangle ⟨ italic_t , italic_Ο† ⟩ between distributions t 𝑑 t italic_t in π’Ÿ β€² ⁒ ( M n , H ⁒ o ⁒ m ⁒ ( F , E ) ⊠ n ) superscript π’Ÿ normal-β€² superscript 𝑀 𝑛 𝐻 π‘œ π‘š superscript 𝐹 𝐸 normal-⊠ absent 𝑛 \mathcal{D}^{\prime}(M^{n},Hom(F,E)^{\boxtimes n}) caligraphic_D start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_H italic_o italic_m ( italic_F , italic_E ) start_POSTSUPERSCRIPT ⊠ italic_n end_POSTSUPERSCRIPT ) with elements Ο† πœ‘ \varphi italic_Ο† in C ∞ ⁒ ( M n , H ⁒ o ⁒ m ⁒ ( E , F ) ⊠ n ) superscript 𝐢 superscript 𝑀 𝑛 𝐻 π‘œ π‘š superscript 𝐸 𝐹 normal-⊠ absent 𝑛 C^{\infty}\left(M^{n},Hom(E,F)^{\boxtimes n}\right) italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_H italic_o italic_m ( italic_E , italic_F ) start_POSTSUPERSCRIPT ⊠ italic_n end_POSTSUPERSCRIPT ) to get an element ⟨ t , Ο† ⟩ ∈ π’Ÿ β€² ⁒ ( M n ) 𝑑 πœ‘ superscript π’Ÿ normal-β€² superscript 𝑀 𝑛 \left\langle t,\varphi\right\rangle\in\mathcal{D}^{\prime}(M^{n}) ⟨ italic_t , italic_Ο† ⟩ ∈ caligraphic_D start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and to get a number, we need to integrate this distribution against a density d ⁒ v ∈ | Ξ› t ⁒ o ⁒ p | ⁒ M n 𝑑 𝑣 superscript normal-Ξ› 𝑑 π‘œ 𝑝 superscript 𝑀 𝑛 dv\in|\Lambda^{top}|M^{n} italic_d italic_v ∈ | roman_Ξ› start_POSTSUPERSCRIPT italic_t italic_o italic_p end_POSTSUPERSCRIPT | italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT as ∫ M n ⟨ t , Ο† ⟩ ⁒ 𝑑 v subscript superscript 𝑀 𝑛 𝑑 πœ‘ differential-d 𝑣 \int_{M^{n}}\left\langle t,\varphi\right\rangle dv ∫ start_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ italic_t , italic_Ο† ⟩ italic_d italic_v .

Definition 4.5 (Decomposition) .

Under the assumptions of Lemma 4.4 . We set

I ⁒ ( s , V 1 , … , V k + 1 ) = ∫ [ 0 , ∞ ) k + 1 ( βˆ‘ e = 1 k + 1 u e ) s ⁒ T ⁒ r ⁒ ( e - u k + 1 ⁒ Ξ” ⁒ V 1 ⁒ … ⁒ e - u 1 ⁒ Ξ” ⁒ V k + 1 ) ⁒ ∏ e = 1 k + 1 d ⁒ u e 𝐼 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 subscript superscript 0 π‘˜ 1 superscript superscript subscript 𝑒 1 π‘˜ 1 subscript 𝑒 𝑒 𝑠 𝑇 π‘Ÿ superscript 𝑒 subscript 𝑒 π‘˜ 1 Ξ” subscript 𝑉 1 … superscript 𝑒 subscript 𝑒 1 Ξ” subscript 𝑉 π‘˜ 1 superscript subscript product 𝑒 1 π‘˜ 1 𝑑 subscript 𝑒 𝑒 italic_I ( italic_s , italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) = ∫ start_POSTSUBSCRIPT [ 0 , ∞ ) start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( βˆ‘ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_T italic_r ( italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_d italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT (4.6)

that we shall decompose in two pieces

I ⁒ ( s ; V 1 , … , V k + 1 ) = S ⁒ ( s ; V 1 , … , V k + 1 ) + R ⁒ ( s ; V 1 , … , V k + 1 ) 𝐼 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 𝑆 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 𝑅 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 \displaystyle I(s;V_{1},\dots,V_{k+1})=S(s;V_{1},\dots,V_{k+1})+R(s;V_{1},% \dots,V_{k+1}) italic_I ( italic_s ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) = italic_S ( italic_s ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) + italic_R ( italic_s ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) (4.7)

where

R ⁒ ( s ; V 1 , … , V k + 1 ) = ( k + 1 ) ! ⁒ ∫ { u k + 1 β©Ύ β‹― β©Ύ u 1 β©Ύ 0 , u k + 1 β©Ύ 1 } ( βˆ‘ e = 1 k + 1 u e ) s ⁒ T ⁒ r ⁒ ( e - u k + 1 ⁒ Ξ” ⁒ V 1 ⁒ … ⁒ e - u 1 ⁒ Ξ” ⁒ V k + 1 ) ⁒ ∏ e = 1 k + 1 d ⁒ u e , 𝑅 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 π‘˜ 1 subscript fragments { subscript 𝑒 π‘˜ 1 β‹― subscript 𝑒 1 0 , subscript 𝑒 π‘˜ 1 1 } superscript superscript subscript 𝑒 1 π‘˜ 1 subscript 𝑒 𝑒 𝑠 𝑇 π‘Ÿ superscript 𝑒 subscript 𝑒 π‘˜ 1 Ξ” subscript 𝑉 1 … superscript 𝑒 subscript 𝑒 1 Ξ” subscript 𝑉 π‘˜ 1 superscript subscript product 𝑒 1 π‘˜ 1 𝑑 subscript 𝑒 𝑒 R(s;V_{1},\dots,V_{k+1})=(k+1)!\int_{\{u_{k+1}\geqslant\dots\geqslant u_{1}% \geqslant 0,u_{k+1}\geqslant 1\}}(\sum_{e=1}^{k+1}u_{e})^{s}Tr\left(e^{-u_{k+1% }\Delta}V_{1}\dots e^{-u_{1}\Delta}V_{k+1}\right)\prod_{e=1}^{k+1}du_{e}, italic_R ( italic_s ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) = ( italic_k + 1 ) ! ∫ start_POSTSUBSCRIPT { italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT β©Ύ β‹― β©Ύ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β©Ύ 0 , italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT β©Ύ 1 } end_POSTSUBSCRIPT ( βˆ‘ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_T italic_r ( italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_d italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT , (4.8)

and

S ⁒ ( s ; V 1 , … , V k + 1 ) = ( k + 1 ) ! ⁒ ∫ { 1 β©Ύ u k + 1 β©Ύ β‹― β©Ύ u 1 β©Ύ 0 } ( βˆ‘ e = 1 k + 1 u e ) s ⁒ T ⁒ r ⁒ ( e - u k + 1 ⁒ Ξ” ⁒ V 1 ⁒ … ⁒ e - u 1 ⁒ Ξ” ⁒ V k + 1 ) ⁒ ∏ e = 1 k + 1 d ⁒ u e . 𝑆 𝑠 subscript 𝑉 1 … subscript 𝑉 π‘˜ 1 π‘˜ 1 subscript fragments { 1 subscript 𝑒 π‘˜ 1 β‹― subscript 𝑒 1 0 } superscript superscript subscript 𝑒 1 π‘˜ 1 subscript 𝑒 𝑒 𝑠 𝑇 π‘Ÿ superscript 𝑒 subscript 𝑒 π‘˜ 1 Ξ” subscript 𝑉 1 … superscript 𝑒 subscript 𝑒 1 Ξ” subscript 𝑉 π‘˜ 1 superscript subscript product 𝑒 1 π‘˜ 1 𝑑 subscript 𝑒 𝑒 S(s;V_{1},\dots,V_{k+1})=(k+1)!\int_{\{1\geqslant u_{k+1}\geqslant\dots% \geqslant u_{1}\geqslant 0\}}(\sum_{e=1}^{k+1}u_{e})^{s}Tr\left(e^{-u_{k+1}% \Delta}V_{1}\dots e^{-u_{1}\Delta}V_{k+1}\right)\prod_{e=1}^{k+1}du_{e}. italic_S ( italic_s ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) = ( italic_k + 1 ) ! ∫ start_POSTSUBSCRIPT { 1 β©Ύ italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT β©Ύ β‹― β©Ύ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β©Ύ 0 } end_POSTSUBSCRIPT ( βˆ‘ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_T italic_r ( italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_e start_POSTSUPERSCRIPT - italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_Ξ” end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_e = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_d italic_u start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT . (4.9)
Definition 2.1 .

Let n 1 subscript 𝑛 1 n_{1} italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT be a positive integer and q ∈ 𝐊 π‘ž 𝐊 q\in{\bf K} italic_q ∈ bold_K be an n 𝑛 n italic_n -th primitive root of unity. Suppose that H Ξ² subscript 𝐻 𝛽 H_{\beta} italic_H start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT is an algebra generated by a , b , c , x , y π‘Ž 𝑏 𝑐 π‘₯ 𝑦 a,b,c,x,y italic_a , italic_b , italic_c , italic_x , italic_y with the relations

a ⁒ b = b ⁒ a , a ⁒ c = c ⁒ a , b ⁒ c = c ⁒ b , x ⁒ a = q ⁒ a ⁒ x , y ⁒ a = q - 1 ⁒ a ⁒ y , b ⁒ x = x ⁒ b , c ⁒ x = x ⁒ c , b ⁒ y = y ⁒ b , c ⁒ y = y ⁒ c , formulae-sequence π‘Ž 𝑏 𝑏 π‘Ž formulae-sequence π‘Ž 𝑐 𝑐 π‘Ž formulae-sequence 𝑏 𝑐 𝑐 𝑏 formulae-sequence π‘₯ π‘Ž π‘ž π‘Ž π‘₯ formulae-sequence 𝑦 π‘Ž superscript π‘ž 1 π‘Ž 𝑦 formulae-sequence 𝑏 π‘₯ π‘₯ 𝑏 formulae-sequence 𝑐 π‘₯ π‘₯ 𝑐 formulae-sequence 𝑏 𝑦 𝑦 𝑏 𝑐 𝑦 𝑦 𝑐 ab=ba,ac=ca,bc=cb,xa=qax,ya=q^{-1}ay,bx=xb,cx=xc,by=yb,cy=yc, italic_a italic_b = italic_b italic_a , italic_a italic_c = italic_c italic_a , italic_b italic_c = italic_c italic_b , italic_x italic_a = italic_q italic_a italic_x , italic_y italic_a = italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a italic_y , italic_b italic_x = italic_x italic_b , italic_c italic_x = italic_x italic_c , italic_b italic_y = italic_y italic_b , italic_c italic_y = italic_y italic_c ,
y ⁒ x - q - n 1 ⁒ x ⁒ y = Ξ² 3 ⁒ ( a 2 ⁒ n 1 - b ⁒ c ) , x n = Ξ² 1 ⁒ ( a n ⁒ n 1 - b n ) , y n = Ξ² 2 ⁒ ( a n ⁒ n 1 - c n ) , formulae-sequence 𝑦 π‘₯ superscript π‘ž subscript 𝑛 1 π‘₯ 𝑦 subscript 𝛽 3 superscript π‘Ž 2 subscript 𝑛 1 𝑏 𝑐 formulae-sequence superscript π‘₯ 𝑛 subscript 𝛽 1 superscript π‘Ž 𝑛 subscript 𝑛 1 superscript 𝑏 𝑛 superscript 𝑦 𝑛 subscript 𝛽 2 superscript π‘Ž 𝑛 subscript 𝑛 1 superscript 𝑐 𝑛 yx-q^{-n_{1}}xy=\beta_{3}(a^{2n_{1}}-bc),\ x^{n}=\beta_{1}(a^{nn_{1}}-b^{n}),% \ y^{n}=\beta_{2}(a^{nn_{1}}-c^{n}), italic_y italic_x - italic_q start_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_x italic_y = italic_Ξ² start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT 2 italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_b italic_c ) , italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_Ξ² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_n italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_b start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , italic_y start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_Ξ² start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT italic_n italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_c start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ,

for Ξ² = ( Ξ² 1 , Ξ² 2 , Ξ² 3 ) ∈ 𝐊 3 𝛽 subscript 𝛽 1 subscript 𝛽 2 subscript 𝛽 3 superscript 𝐊 3 \beta=(\beta_{1},\beta_{2},\beta_{3})\in{\bf K}^{3} italic_Ξ² = ( italic_Ξ² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Ξ² start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_Ξ² start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ bold_K start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT . The coproduct Ξ” normal-Ξ” \Delta roman_Ξ” and counit of H Ξ² subscript 𝐻 𝛽 H_{\beta} italic_H start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT is determined by

Ξ” ⁒ ( a ) = a βŠ— a , Ξ” ⁒ ( b ) = b βŠ— b , Ξ” ⁒ ( c ) = c βŠ— c , Ξ” ⁒ ( x ) = x βŠ— a n 1 + b βŠ— x , Ξ” ⁒ ( y ) = y βŠ— a n 1 + c βŠ— y , formulae-sequence Ξ” π‘Ž tensor-product π‘Ž π‘Ž formulae-sequence Ξ” 𝑏 tensor-product 𝑏 𝑏 formulae-sequence Ξ” 𝑐 tensor-product 𝑐 𝑐 formulae-sequence Ξ” π‘₯ tensor-product π‘₯ superscript π‘Ž subscript 𝑛 1 tensor-product 𝑏 π‘₯ Ξ” 𝑦 tensor-product 𝑦 superscript π‘Ž subscript 𝑛 1 tensor-product 𝑐 𝑦 \Delta(a)=a\otimes a,\Delta(b)=b\otimes b,\Delta(c)=c\otimes c,\Delta(x)=x% \otimes a^{n_{1}}+b\otimes x,\Delta(y)=y\otimes a^{n_{1}}+c\otimes y, roman_Ξ” ( italic_a ) = italic_a βŠ— italic_a , roman_Ξ” ( italic_b ) = italic_b βŠ— italic_b , roman_Ξ” ( italic_c ) = italic_c βŠ— italic_c , roman_Ξ” ( italic_x ) = italic_x βŠ— italic_a start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT + italic_b βŠ— italic_x , roman_Ξ” ( italic_y ) = italic_y βŠ— italic_a start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT + italic_c βŠ— italic_y ,

and

Ξ΅ ⁒ ( a ) = Ξ΅ ⁒ ( b ) = Ξ΅ ⁒ ( c ) = 1 , Ξ΅ ⁒ ( x ) = Ξ΅ ⁒ ( y ) = 0 formulae-sequence πœ€ π‘Ž πœ€ 𝑏 πœ€ 𝑐 1 πœ€ π‘₯ πœ€ 𝑦 0 \varepsilon(a)=\varepsilon(b)=\varepsilon(c)=1,\varepsilon(x)=\varepsilon(y)=0 italic_Ξ΅ ( italic_a ) = italic_Ξ΅ ( italic_b ) = italic_Ξ΅ ( italic_c ) = 1 , italic_Ξ΅ ( italic_x ) = italic_Ξ΅ ( italic_y ) = 0

respectively. An anti-automorphism s 𝑠 s italic_s of H Ξ² subscript 𝐻 𝛽 H_{\beta} italic_H start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT is determined by

s ⁒ ( a ) = a - 1 , s ⁒ ( b ) = b - 1 , s ⁒ ( c ) = c - 1 , s ⁒ ( x ) = - q - n 1 ⁒ a - n 1 ⁒ b - 1 ⁒ x , s ⁒ ( y ) = - q n 1 ⁒ a - n 1 ⁒ c - 1 ⁒ y formulae-sequence 𝑠 π‘Ž superscript π‘Ž 1 formulae-sequence 𝑠 𝑏 superscript 𝑏 1 formulae-sequence 𝑠 𝑐 superscript 𝑐 1 formulae-sequence 𝑠 π‘₯ superscript π‘ž subscript 𝑛 1 superscript π‘Ž subscript 𝑛 1 superscript 𝑏 1 π‘₯ 𝑠 𝑦 superscript π‘ž subscript 𝑛 1 superscript π‘Ž subscript 𝑛 1 superscript 𝑐 1 𝑦 s(a)=a^{-1},s(b)=b^{-1},s(c)=c^{-1},s(x)=-q^{-n_{1}}a^{-n_{1}}b^{-1}x,s(y)=-q^% {n_{1}}a^{-n_{1}}c^{-1}y italic_s ( italic_a ) = italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_s ( italic_b ) = italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_s ( italic_c ) = italic_c start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_s ( italic_x ) = - italic_q start_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x , italic_s ( italic_y ) = - italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_y
Definition 3.1 .

A structure π’œ = ⟨ A Β― , βŠ“ , βŠ” , 𝟎 , 𝟏 ⟩ π’œ normal-Β― 𝐴 square-intersection square-union 0 1 \mathcal{A}=\langle\overline{A},\sqcap,\sqcup,{\bf 0},{\bf 1}\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” , bold_0 , bold_1 ⟩ , where A Β― normal-Β― 𝐴 \overline{A} Β― start_ARG italic_A end_ARG is a set and 𝟎 , 𝟏 ∈ A Β― 0 1 normal-Β― 𝐴 {\bf 0},{\bf 1}\in\overline{A} bold_0 , bold_1 ∈ Β― start_ARG italic_A end_ARG , is a distributive bounded lattice if for every a , b , c ∈ A Β― π‘Ž 𝑏 𝑐 normal-Β― 𝐴 a,b,c\in\overline{A} italic_a , italic_b , italic_c ∈ Β― start_ARG italic_A end_ARG :

For every lattice π’œ = ⟨ A Β― , βŠ“ , βŠ” ⟩ π’œ Β― 𝐴 square-intersection square-union \mathcal{A}=\langle\overline{A},\sqcap,\sqcup\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” ⟩ the order induced by π’œ π’œ \mathcal{A} caligraphic_A is the binary relation βŠ‘ βŠ† ( A Β― Γ— A Β― ) fragments square-image-of-or-equals fragments ( Β― 𝐴 Β― 𝐴 ) \sqsubseteq\;\subseteq(\overline{A}\times\overline{A}) βŠ‘ βŠ† ( Β― start_ARG italic_A end_ARG Γ— Β― start_ARG italic_A end_ARG ) such that

a βŠ‘ b ⁒ if and only if ⁒ a βŠ“ b = a ⁒ if and only if ⁒ a βŠ” b = b square-image-of-or-equals π‘Ž square-intersection 𝑏 if and only if π‘Ž 𝑏 square-union π‘Ž if and only if π‘Ž 𝑏 𝑏 a\sqsubseteq b\mbox{ if and only if }a\sqcap b=a\mbox{ if and only if }a\sqcup b=b italic_a βŠ‘ italic_b if and only if italic_a βŠ“ italic_b = italic_a if and only if italic_a βŠ” italic_b = italic_b

An involution on a lattice π’œ π’œ \mathcal{A} caligraphic_A is a unary operation - - - s.t. for every a , b ∈ A Β― π‘Ž 𝑏 Β― 𝐴 a,b\in\overline{A} italic_a , italic_b ∈ Β― start_ARG italic_A end_ARG :

A bounded, distributive, involutive lattice π’œ = ⟨ A Β― , βŠ“ , βŠ” , - , 𝟎 , 𝟏 ⟩ π’œ Β― 𝐴 square-intersection square-union 0 1 \mathcal{A}=\langle\overline{A},\sqcap,\sqcup,-,\mathbf{0},\mathbf{1}\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” , - , bold_0 , bold_1 ⟩ is a De Morgan algebra if for every a , b ∈ A Β― π‘Ž 𝑏 Β― 𝐴 a,b\in\overline{A} italic_a , italic_b ∈ Β― start_ARG italic_A end_ARG :

A De Morgan algebra π’œ = ⟨ A Β― , βŠ“ , βŠ” , - , 𝟎 , 𝟏 ⟩ π’œ Β― 𝐴 square-intersection square-union 0 1 \mathcal{A}=\langle\overline{A},\sqcap,\sqcup,-,\mathbf{0},\mathbf{1}\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” , - , bold_0 , bold_1 ⟩ is Kleene if, for every a , b ∈ A Β― π‘Ž 𝑏 Β― 𝐴 a,b\in\overline{A} italic_a , italic_b ∈ Β― start_ARG italic_A end_ARG :

A relative pseudocomplementation on a lattice π’œ π’œ \mathcal{A} caligraphic_A is a binary operation ↣ ↣ \rightarrowtail ↣ s.t. for every a , b ∈ A Β― π‘Ž 𝑏 Β― 𝐴 a,b\in\overline{A} italic_a , italic_b ∈ Β― start_ARG italic_A end_ARG :

A relatively pseudocompletemented Kleene algebra π’œ = ⟨ A Β― , βŠ“ , βŠ” , - , ↣ , 𝟎 , 𝟏 ⟩ π’œ Β― 𝐴 square-intersection square-union ↣ 0 1 \mathcal{A}=\langle\overline{A},\sqcap,\sqcup,-,\rightarrowtail,\mathbf{0},% \mathbf{1}\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” , - , ↣ , bold_0 , bold_1 ⟩ is an Ł ⁒ 3 italic-Ł 3 \L{}3 italic_Ł 3 algebra if for every a ∈ A Β― π‘Ž Β― 𝐴 a\in\overline{A} italic_a ∈ Β― start_ARG italic_A end_ARG :

An Ł ⁒ 3 italic-Ł 3 \L{}3 italic_Ł 3 algebra π’œ = ⟨ A Β― , βŠ“ , βŠ” , - , ↣ , β–Ά , 𝟎 , 𝟏 , 𝟏 / 𝟐 ⟩ π’œ Β― 𝐴 square-intersection square-union ↣ β–Ά 0 1 1 2 \mathcal{A}=\langle\overline{A},\sqcap,\sqcup,-,\rightarrowtail,% \blacktriangleright,\mathbf{0},\mathbf{1},\nicefrac{{{\bf 1}}}{{{\bf 2}}}\rangle caligraphic_A = ⟨ Β― start_ARG italic_A end_ARG , βŠ“ , βŠ” , - , ↣ , β–Ά , bold_0 , bold_1 , / start_ARG bold_1 end_ARG start_ARG bold_2 end_ARG ⟩ is De Finetti if:

Definition 6.5 .

By the V-monotone Gaussian operator associated with f ∈ β„‹ 1 𝑓 subscript β„‹ 1 f\in\mathcal{H}_{1} italic_f ∈ caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT we mean the operator of the form

Ο‰ ⁒ ( f ) = a ⁒ ( f ) + a * ⁒ ( f ) ⁒ . πœ” 𝑓 π‘Ž 𝑓 superscript π‘Ž 𝑓 . \omega(f)=a(f)+a^{*}(f)\text{.} italic_Ο‰ ( italic_f ) = italic_a ( italic_f ) + italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_f ) .

If f = πŸ™ [ 0 , 1 ] 𝑓 subscript 1 0 1 f=\mathbbm{1}_{[0,1]} italic_f = blackboard_1 start_POSTSUBSCRIPT [ 0 , 1 ] end_POSTSUBSCRIPT , this operator will be called standard and denoted by Ο‰ πœ” \omega italic_Ο‰ . By a π‘Ž a italic_a and a * superscript π‘Ž a^{*} italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT we denote the associated creation and annihilation operators, respectively.

Definition 3.9 (Composition of Homotopies) .

Suppose that f , f β€² : G β†’ H : 𝑓 superscript 𝑓 β€² β†’ 𝐺 𝐻 f,f^{\prime}:G\to H italic_f , italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT : italic_G β†’ italic_H and g , g β€² : H β†’ K : 𝑔 superscript 𝑔 β€² β†’ 𝐻 𝐾 g,g^{\prime}:H\to K italic_g , italic_g start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT : italic_H β†’ italic_K . Given Ξ± : f ≃ f β€² : 𝛼 similar-to-or-equals 𝑓 superscript 𝑓 β€² \alpha:f\simeq f^{\prime} italic_Ξ± : italic_f ≃ italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT and Ξ² : g ≃ g β€² : 𝛽 similar-to-or-equals 𝑔 superscript 𝑔 β€² \beta:g\simeq g^{\prime} italic_Ξ² : italic_g ≃ italic_g start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT , we define Ξ± ∘ Ξ² 𝛼 𝛽 \alpha\circ\beta italic_Ξ± ∘ italic_Ξ² from g ⁒ f 𝑔 𝑓 gf italic_g italic_f to g β€² ⁒ f β€² superscript 𝑔 β€² superscript 𝑓 β€² g^{\prime}f^{\prime} italic_g start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT as follows: let g ⁒ Ξ± = g * ⁒ Ξ± 𝑔 𝛼 subscript 𝑔 𝛼 g\alpha=g_{*}\alpha italic_g italic_Ξ± = italic_g start_POSTSUBSCRIPT * end_POSTSUBSCRIPT italic_Ξ± denote the homotopy from g ⁒ f 𝑔 𝑓 gf italic_g italic_f to g ⁒ f β€² 𝑔 superscript 𝑓 β€² gf^{\prime} italic_g italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT , and ( f β€² ) * ⁒ Ξ² = Ξ² ⁒ f β€² superscript superscript 𝑓 β€² 𝛽 𝛽 superscript 𝑓 β€² (f^{\prime})^{*}\beta=\beta f^{\prime} ( italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Ξ² = italic_Ξ² italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT denote the homotopy from g ⁒ f β€² 𝑔 superscript 𝑓 β€² gf^{\prime} italic_g italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT to g β€² ⁒ f β€² superscript 𝑔 β€² superscript 𝑓 β€² g^{\prime}f^{\prime} italic_g start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT , as defined in Lemma 3.5 . Then

Ξ± ∘ Ξ² = g ⁒ Ξ± * Ξ² ⁒ f β€² . 𝛼 𝛽 𝑔 𝛼 𝛽 superscript 𝑓 β€² \alpha\circ\beta=g\alpha*\beta f^{\prime}. italic_Ξ± ∘ italic_Ξ² = italic_g italic_Ξ± * italic_Ξ² italic_f start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT .
Definition 1.1 :

Given real Ξ΄ > 0 𝛿 0 \delta>0 italic_Ξ΄ > 0 , define π’Ÿ Ξ΄ = { | m | < Ξ΄ } fragments subscript π’Ÿ 𝛿 fragments normal-{ normal-| m normal-| Ξ΄ normal-} \mathcal{D}_{\delta}=\{|m|<\delta\} caligraphic_D start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT = { | italic_m | < italic_Ξ΄ } , and assume that Ο• italic-Ο• \phi italic_Ο• and Ο• + 1 italic-Ο• 1 \phi+1 italic_Ο• + 1 lie in π’Ÿ Ξ΄ subscript π’Ÿ 𝛿 \mathcal{D}_{\delta} caligraphic_D start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT . For such Ο• italic-Ο• \phi italic_Ο• , assume a π‘Ž a italic_a , p 𝑝 p italic_p are given initial values for a solution u 𝑒 u italic_u of q π‘ž q italic_q - P 𝐼 𝐼 {}_{\text{I}} start_FLOATSUBSCRIPT I end_FLOATSUBSCRIPT or q π‘ž q italic_q - P 𝐼𝐼𝐼 𝐼𝐼𝐼 {}_{\text{III}} start_FLOATSUBSCRIPT III end_FLOATSUBSCRIPT , such that

u ⁒ ( Ο• ) = a , u ⁒ ( Ο• + 1 ) = p , formulae-sequence 𝑒 italic-Ο• π‘Ž 𝑒 italic-Ο• 1 𝑝 u(\phi)=a,\;u(\phi+1)=p, italic_u ( italic_Ο• ) = italic_a , italic_u ( italic_Ο• + 1 ) = italic_p ,

where p β‰  a 𝑝 π‘Ž p\neq a italic_p β‰  italic_a and a β‰  0 π‘Ž 0 a\neq 0 italic_a β‰  0 for q π‘ž q italic_q - P 𝐼 𝐼 {}_{\text{I}} start_FLOATSUBSCRIPT I end_FLOATSUBSCRIPT and q π‘ž q italic_q - P 𝐼𝐼𝐼 𝐼𝐼𝐼 {}_{\text{III}} start_FLOATSUBSCRIPT III end_FLOATSUBSCRIPT , p β‰  0 𝑝 0 p\neq 0 italic_p β‰  0 for q π‘ž q italic_q - P 𝐼𝐼𝐼 𝐼𝐼𝐼 {}_{\text{III}} start_FLOATSUBSCRIPT III end_FLOATSUBSCRIPT , p β‰  1 ΞΎ 0 ⁒ q Ο• 𝑝 1 subscript πœ‰ 0 superscript π‘ž italic-Ο• p\neq\frac{1}{\xi_{0}q^{\phi}} italic_p β‰  divide start_ARG 1 end_ARG start_ARG italic_ΞΎ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_Ο• end_POSTSUPERSCRIPT end_ARG for q π‘ž q italic_q - P 𝐼 𝐼 {}_{\text{I}} start_FLOATSUBSCRIPT I end_FLOATSUBSCRIPT , and p β‰  Ξ± 𝑝 𝛼 p\neq\alpha italic_p β‰  italic_Ξ± , Ξ² 𝛽 \beta italic_Ξ² , Ξ³ 𝛾 \gamma italic_Ξ³ , or Ξ΄ 𝛿 \delta italic_Ξ΄ for q π‘ž q italic_q - P 𝐼𝐼𝐼 𝐼𝐼𝐼 {}_{\text{III}} start_FLOATSUBSCRIPT III end_FLOATSUBSCRIPT . We define the set of numbers Ο• italic-Ο• \phi italic_Ο• , a π‘Ž a italic_a , p 𝑝 p italic_p satisfying the above conditions to be admissible .

Definition 4.1 (wiring diagram) .

Let 𝐒 = ( i 1 , i 2 , … ⁒ i N ) ∈ 𝒲 ⁒ ( w 0 ) 𝐒 subscript 𝑖 1 subscript 𝑖 2 … subscript 𝑖 𝑁 𝒲 subscript 𝑀 0 \operatorname{\mathbf{i}}=(i_{1},i_{2},\ldots i_{N})\in\mathcal{W}(w_{0}) bold_i = ( italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) ∈ caligraphic_W ( italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) . The wiring diagram π’Ÿ 𝐒 subscript π’Ÿ 𝐒 \mathcal{D}_{\operatorname{\mathbf{i}}} caligraphic_D start_POSTSUBSCRIPT bold_i end_POSTSUBSCRIPT consists of a family of n 𝑛 n italic_n piecewise straight lines, called wires , which can be viewed as graphs of n 𝑛 n italic_n continuous piecewise linear functions defined on the same interval. The wires have labels in the set [ n ] delimited-[] 𝑛 [n] [ italic_n ] . Each vertex of π’Ÿ 𝐒 subscript π’Ÿ 𝐒 \mathcal{D}_{\operatorname{\mathbf{i}}} caligraphic_D start_POSTSUBSCRIPT bold_i end_POSTSUBSCRIPT (i.e. an intersection of two wires) represents a letter j 𝑗 j italic_j in 𝐒 𝐒 \operatorname{\mathbf{i}} bold_i . If the vertex corresponds to the letter j ∈ [ n - 1 ] 𝑗 delimited-[] 𝑛 1 j\in[n-1] italic_j ∈ [ italic_n - 1 ] , then j - 1 𝑗 1 j-1 italic_j - 1 is equal to the number of wires running below this intersection. We call j 𝑗 j italic_j the level of the vertex v 𝑣 v italic_v and write

level ⁒ ( v ) = j - 1 . level 𝑣 𝑗 1 \text{level}(v)=j-1. level ( italic_v ) = italic_j - 1 .

The word 𝐒 𝐒 \operatorname{\mathbf{i}} bold_i can be read off from π’Ÿ 𝐒 subscript π’Ÿ 𝐒 \mathcal{D}_{\operatorname{\mathbf{i}}} caligraphic_D start_POSTSUBSCRIPT bold_i end_POSTSUBSCRIPT by reading the levels of the vertices from left to right.

Definition 1 .

The k π‘˜ k italic_k -change graph of a language L βŠ‚ Ξ£ n 𝐿 superscript normal-Ξ£ 𝑛 L\subset\Sigma^{n} italic_L βŠ‚ roman_Ξ£ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT has nodes L 𝐿 L italic_L and edge ( u , v ) 𝑒 𝑣 (u,v) ( italic_u , italic_v ) whenever

u = w ⁒ u β€² ⁒ w β€² , v = w ⁒ v β€² ⁒ w β€² formulae-sequence 𝑒 𝑀 superscript 𝑒 β€² superscript 𝑀 β€² 𝑣 𝑀 superscript 𝑣 β€² superscript 𝑀 β€² u=wu^{\prime}w^{\prime},v=wv^{\prime}w^{\prime} italic_u = italic_w italic_u start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT , italic_v = italic_w italic_v start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT

where u , u β€² ∈ Ξ£ k β€² 𝑒 superscript 𝑒 normal-β€² superscript normal-Ξ£ superscript π‘˜ normal-β€² u,u^{\prime}\in\Sigma^{k^{\prime}} italic_u , italic_u start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∈ roman_Ξ£ start_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT for some k β€² ≀ k superscript π‘˜ normal-β€² π‘˜ k^{\prime}\leq k italic_k start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ≀ italic_k and w , w β€² ∈ Ξ£ * 𝑀 superscript 𝑀 normal-β€² superscript normal-Ξ£ w,w^{\prime}\in\Sigma^{*} italic_w , italic_w start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∈ roman_Ξ£ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT .

Definition 2.1 .

An A ∞ subscript 𝐴 A_{\infty} italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT -structure on a β„€ β„€ \mathbb{Z} blackboard_Z -graded vector space V 𝑉 V italic_V is given by an element m ∈ Hom 1 ⁒ ( T ⁒ ( V ) , V ) π‘š superscript Hom 1 𝑇 𝑉 𝑉 m\in\mathrm{Hom}^{1}(T(V),V) italic_m ∈ roman_Hom start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_T ( italic_V ) , italic_V ) obeying the Maurer–Cartan (MC) equation

(2.4) m ∘ m = 0 . π‘š π‘š 0 m\circ m=0\,. italic_m ∘ italic_m = 0 .

The pair ( V , m ) 𝑉 π‘š (V,m) ( italic_V , italic_m ) is called the A ∞ subscript 𝐴 A_{\infty} italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT -algebra.

Definition 2 (Pseudometric) .

A non-negative real-valued function d : X Γ— X β†’ ℝ + normal-: 𝑑 normal-β†’ 𝑋 𝑋 superscript ℝ d:X\times X\rightarrow\mathbb{R}^{+} italic_d : italic_X Γ— italic_X β†’ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT is a pseudometric if for any x , y , z ∈ X π‘₯ 𝑦 𝑧 𝑋 x,y,z\in X italic_x , italic_y , italic_z ∈ italic_X , the following condition holds:

d ⁒ ( x , x ) = 0 . 𝑑 π‘₯ π‘₯ 0 \displaystyle d(x,x)=0. italic_d ( italic_x , italic_x ) = 0 . (74)
d ⁒ ( x , y ) = d ⁒ ( y , x ) . 𝑑 π‘₯ 𝑦 𝑑 𝑦 π‘₯ \displaystyle d(x,y)=d(y,x). italic_d ( italic_x , italic_y ) = italic_d ( italic_y , italic_x ) . (75)
d ⁒ ( x , z ) ≀ d ⁒ ( x , y ) + d ⁒ ( y , z ) . 𝑑 π‘₯ 𝑧 𝑑 π‘₯ 𝑦 𝑑 𝑦 𝑧 \displaystyle d(x,z)\leq d(x,y)+d(y,z). italic_d ( italic_x , italic_z ) ≀ italic_d ( italic_x , italic_y ) + italic_d ( italic_y , italic_z ) . (76)

Unlike a metric, one may have d ⁒ ( x , y ) = 0 𝑑 π‘₯ 𝑦 0 d(x,y)=0 italic_d ( italic_x , italic_y ) = 0 for distinct values x β‰  y π‘₯ 𝑦 x\neq y italic_x β‰  italic_y .

Definition 3.1 .

Let V 𝑉 V italic_V and W π‘Š W italic_W be vector spaces over a field E 𝐸 E italic_E and Οƒ : E β†’ E normal-: 𝜎 normal-β†’ 𝐸 𝐸 \sigma:E\to E italic_Οƒ : italic_E β†’ italic_E a field automorphism. We say that a map Ο† : V β†’ W normal-: πœ‘ normal-β†’ 𝑉 π‘Š \varphi:V\to W italic_Ο† : italic_V β†’ italic_W is Οƒ 𝜎 \sigma italic_Οƒ -linear if

Ο† ⁒ ( Ξ» ⁒ v + ΞΌ ⁒ w ) = Οƒ ⁒ ( Ξ» ) ⁒ v + Οƒ ⁒ ( ΞΌ ) ⁒ w πœ‘ πœ† 𝑣 πœ‡ 𝑀 𝜎 πœ† 𝑣 𝜎 πœ‡ 𝑀 \varphi\left(\lambda v+\mu w\right)=\sigma(\lambda)v+\sigma(\mu)w italic_Ο† ( italic_Ξ» italic_v + italic_ΞΌ italic_w ) = italic_Οƒ ( italic_Ξ» ) italic_v + italic_Οƒ ( italic_ΞΌ ) italic_w

for all v , w ∈ V 𝑣 𝑀 𝑉 v,w\in V italic_v , italic_w ∈ italic_V and Ξ» , ΞΌ ∈ E πœ† πœ‡ 𝐸 \lambda,\mu\in E italic_Ξ» , italic_ΞΌ ∈ italic_E . If Ο† πœ‘ \varphi italic_Ο† is moreover bijective, we call it a Οƒ 𝜎 \sigma italic_Οƒ -isomorphism between the vector spaces V 𝑉 V italic_V and W π‘Š W italic_W .

In the special case where V = 𝔀 𝑉 𝔀 V=\mathfrak{g} italic_V = fraktur_g and W = π”₯ π‘Š π”₯ W=\mathfrak{h} italic_W = fraktur_h are Lie algebras, we say that Ο† πœ‘ \varphi italic_Ο† is a Οƒ 𝜎 \sigma italic_Οƒ -morphism if it is both Οƒ 𝜎 \sigma italic_Οƒ -linear and if it preserves the Lie bracket, i.e.

Ο† ⁒ ( [ X , Y ] ) = [ Ο† ⁒ ( X ) , Ο† ⁒ ( Y ) ] πœ‘ 𝑋 π‘Œ πœ‘ 𝑋 πœ‘ π‘Œ \varphi\big{(}\left[X,Y\right]\big{)}=\left[\varphi\left(X\right),\varphi\left% (Y\right)\right] italic_Ο† ( [ italic_X , italic_Y ] ) = [ italic_Ο† ( italic_X ) , italic_Ο† ( italic_Y ) ]

for all X , Y ∈ 𝔀 𝑋 π‘Œ 𝔀 X,Y\in\mathfrak{g} italic_X , italic_Y ∈ fraktur_g . If Ο† πœ‘ \varphi italic_Ο† is bijective we call it an Οƒ 𝜎 \sigma italic_Οƒ -isomorphism between the Lie algebras 𝔀 𝔀 \mathfrak{g} fraktur_g and π”₯ π”₯ \mathfrak{h} fraktur_h .

Definition 3.8 .

[ 7 ] Let Ξ· : π”ͺ β†’ 𝔀 : πœ‚ β†’ π”ͺ 𝔀 \eta\colon\mathfrak{m}\to\mathfrak{g} italic_Ξ· : fraktur_m β†’ fraktur_g and ΞΌ : 𝔫 β†’ 𝔀 : πœ‡ β†’ 𝔫 𝔀 \mu\colon\mathfrak{n}\to\mathfrak{g} italic_ΞΌ : fraktur_n β†’ fraktur_g be two Leibniz crossed modules. The non-abelian exterior product π”ͺ ⋏ 𝔫 π”ͺ 𝔫 \mathfrak{m}\curlywedge\mathfrak{n} fraktur_m ⋏ fraktur_n of π”ͺ π”ͺ \mathfrak{m} fraktur_m and 𝔫 𝔫 \mathfrak{n} fraktur_n is defined to be

π”ͺ ⋏ 𝔫 = π”ͺ ⋆ 𝔫 π”ͺ ⁒ β–‘ ⁒ 𝔫 . π”ͺ 𝔫 ⋆ π”ͺ 𝔫 π”ͺ β–‘ 𝔫 \mathfrak{m}\curlywedge\mathfrak{n}=\dfrac{\mathfrak{m}\star\mathfrak{n}}{% \mathfrak{m}\square\mathfrak{n}}. fraktur_m ⋏ fraktur_n = divide start_ARG fraktur_m ⋆ fraktur_n end_ARG start_ARG fraktur_m β–‘ fraktur_n end_ARG .

The cosets of m * 1 n subscript 1 π‘š 𝑛 m*_{1}n italic_m * start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_n and m * 2 n subscript 2 π‘š 𝑛 m*_{2}n italic_m * start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_n will be denoted by m ⋏ 1 n subscript 1 π‘š 𝑛 m\curlywedge_{1}n italic_m ⋏ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_n and m ⋏ 2 n subscript 2 π‘š 𝑛 m\curlywedge_{2}n italic_m ⋏ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_n , respectively.

Definition 1

Consider an autonomous system

x Λ™ ⁒ ( t ) = f ⁒ ( x ⁒ ( t ) ) , Λ™ π‘₯ 𝑑 𝑓 π‘₯ 𝑑 \dot{x}(t)=f(x(t)), Λ™ start_ARG italic_x end_ARG ( italic_t ) = italic_f ( italic_x ( italic_t ) ) , (18)

where f : 𝒳 β†’ ℝ n normal-: 𝑓 normal-β†’ 𝒳 superscript ℝ 𝑛 f:{\cal X}\rightarrow\mathbb{R}^{n} italic_f : caligraphic_X β†’ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is a locally Lipschitz map from a domain 𝒳 βŠ‚ ℝ n 𝒳 superscript ℝ 𝑛 {\cal X}\subset\mathbb{R}^{n} caligraphic_X βŠ‚ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT into ℝ n superscript ℝ 𝑛 \mathbb{R}^{n} blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . Let z 𝑧 z italic_z be an equilibrium of ( 18 ) and β„° βŠ‚ 𝒳 β„° 𝒳 {\cal E}\subset{\cal X} caligraphic_E βŠ‚ caligraphic_X be a domain containing z 𝑧 z italic_z . If the equilibrium z 𝑧 z italic_z is asymptotically stable such that for any x ⁒ ( 0 ) ∈ β„° π‘₯ 0 β„° x(0)\in{\cal E} italic_x ( 0 ) ∈ caligraphic_E we have lim t β†’ ∞ ⁑ x ⁒ ( t ) = z subscript normal-β†’ 𝑑 π‘₯ 𝑑 𝑧 \lim_{t\rightarrow\infty}x(t)=z roman_lim start_POSTSUBSCRIPT italic_t β†’ ∞ end_POSTSUBSCRIPT italic_x ( italic_t ) = italic_z , then β„° β„° {\cal E} caligraphic_E is said to be a domain of attraction for z 𝑧 z italic_z .