Definition 2 .

If M 𝑀 M italic_M and A 𝐴 A italic_A are semi-linear τ 𝜏 \tau italic_τ -modules, then a ( 𝚲 𝚲 \mathbf{\Lambda} bold_Λ -bilinear) A 𝐴 A italic_A -valued τ 𝜏 \tau italic_τ -Hermitian pairing π 𝜋 \pi italic_π is a 𝚲 𝚲 \mathbf{\Lambda} bold_Λ -module homomorphism π : M ⊗ 𝚲 M ( τ ) → A normal-: 𝜋 normal-→ subscript tensor-product 𝚲 𝑀 superscript 𝑀 𝜏 𝐴 \pi:M\otimes_{\mathbf{\Lambda}}M^{(\tau)}\to A italic_π : italic_M ⊗ start_POSTSUBSCRIPT bold_Λ end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ( italic_τ ) end_POSTSUPERSCRIPT → italic_A such that for every lift τ ~ normal-~ 𝜏 \tilde{\tau} ~ start_ARG italic_τ end_ARG of τ 𝜏 \tau italic_τ to Gal ⁢ ( 𝐊 ∞ / 𝐐 ) normal-Gal subscript 𝐊 𝐐 \mathrm{Gal}(\mathbf{K}_{\infty}/\mathbf{Q}) roman_Gal ( bold_K start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT / bold_Q )

π ⁢ ( m ⊗ n ) = π ⁢ ( n ⊗ m ) τ ~ = π ⁢ ( τ ~ ⁢ n ⊗ τ ~ ⁢ m ) . 𝜋 tensor-product 𝑚 𝑛 𝜋 superscript tensor-product 𝑛 𝑚 ~ 𝜏 𝜋 tensor-product ~ 𝜏 𝑛 ~ 𝜏 𝑚 \pi(m\otimes n)=\pi(n\otimes m)^{\tilde{\tau}}=\pi(\tilde{\tau}n\otimes\tilde{% \tau}m). italic_π ( italic_m ⊗ italic_n ) = italic_π ( italic_n ⊗ italic_m ) start_POSTSUPERSCRIPT ~ start_ARG italic_τ end_ARG end_POSTSUPERSCRIPT = italic_π ( ~ start_ARG italic_τ end_ARG italic_n ⊗ ~ start_ARG italic_τ end_ARG italic_m ) .
Definition 1.4

For the rest of this paper, we assume 𝕂 𝕂 {\mathbb{K}} blackboard_K is algebraically closed with characteristic 0. We fix a nonzero scalar q ∈ 𝕂 𝑞 𝕂 q\in{\mathbb{K}} italic_q ∈ blackboard_K which is not a root of unity. We let A , A * 𝐴 superscript 𝐴 A,A^{*} italic_A , italic_A start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT denote a tridiagonal pair on V 𝑉 V italic_V which satisfies ( 3 ), ( 4 ) where

β = q 2 + q - 2 , γ = γ * = ϱ = ϱ * = 0 . formulae-sequence 𝛽 superscript 𝑞 2 superscript 𝑞 2 𝛾 superscript 𝛾 italic-ϱ superscript italic-ϱ 0 \displaystyle\beta=q^{2}+q^{-2},\qquad\qquad\gamma=\gamma^{*}=\varrho=\varrho^% {*}=0. italic_β = italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_q start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT , italic_γ = italic_γ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = italic_ϱ = italic_ϱ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = 0 . (17)
Definition (Normalized Elementary Reversor) .

An elementary map of the form

e : ( x , y ) → ( p ⁢ ( y ) - δ ⁢ x , ϵ ⁢ y ) , p ⁢ ( ϵ ⁢ y ) = δ ⁢ p ⁢ ( y ) , : 𝑒 formulae-sequence → 𝑥 𝑦 𝑝 𝑦 𝛿 𝑥 italic-ϵ 𝑦 𝑝 italic-ϵ 𝑦 𝛿 𝑝 𝑦 e:(x,y)\rightarrow\left(p(y)-\delta\,x,\epsilon\,y\right),\qquad p(\epsilon\,y% )=\delta\,p(y), italic_e : ( italic_x , italic_y ) → ( italic_p ( italic_y ) - italic_δ italic_x , italic_ϵ italic_y ) , italic_p ( italic_ϵ italic_y ) = italic_δ italic_p ( italic_y ) , (35)

with p ⁢ ( y ) 𝑝 𝑦 p(y) italic_p ( italic_y ) a normal polynomial and ϵ 2 , δ 2 superscript italic-ϵ 2 superscript 𝛿 2 \epsilon^{2},\delta^{2} italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT some primitive n t ⁢ h superscript 𝑛 𝑡 ℎ n^{th} italic_n start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT -roots of unity, will be called a normalized elementary reversor of order 2 ⁢ n 2 𝑛 2n 2 italic_n . Note that e 2 = ( δ 2 ⁢ x , ϵ 2 ⁢ y ) superscript 𝑒 2 superscript 𝛿 2 𝑥 superscript italic-ϵ 2 𝑦 e^{2}=(\delta^{2}x,\epsilon^{2}y) italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x , italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y ) .

Definition 2 .

A cocycle is a continuous function d 𝑑 d italic_d from G n subscript 𝐺 𝑛 G_{n} italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT to ℝ ℝ \mathbb{R} blackboard_R which satisfies the identity

d ⁢ ( x , k , y ) + d ⁢ ( y , l , z ) = d ⁢ ( x , k + l , z ) 𝑑 𝑥 𝑘 𝑦 𝑑 𝑦 𝑙 𝑧 𝑑 𝑥 𝑘 𝑙 𝑧 d(x,k,y)+d(y,l,z)=d(x,k+l,z) italic_d ( italic_x , italic_k , italic_y ) + italic_d ( italic_y , italic_l , italic_z ) = italic_d ( italic_x , italic_k + italic_l , italic_z )

for ( x , k , y ) 𝑥 𝑘 𝑦 (x,k,y) ( italic_x , italic_k , italic_y ) and ( y , l , z ) 𝑦 𝑙 𝑧 (y,l,z) ( italic_y , italic_l , italic_z ) in G n subscript 𝐺 𝑛 G_{n} italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .

If P = { ( x , k , y ) ∣ d ⁢ ( x , k , y ) ≥ 0 } 𝑃 conditional-set 𝑥 𝑘 𝑦 𝑑 𝑥 𝑘 𝑦 0 P=\{(x,k,y)\mid d(x,k,y)\geq 0\} italic_P = { ( italic_x , italic_k , italic_y ) ∣ italic_d ( italic_x , italic_k , italic_y ) ≥ 0 } is open, then we call A ⁢ ( P ) 𝐴 𝑃 A(P) italic_A ( italic_P ) the analytic subalgebra associated with d 𝑑 d italic_d .

Definition 3.4 .

A Lie crossed module ( G , H , t , α ) 𝐺 𝐻 𝑡 𝛼 (G,H,t,\alpha) ( italic_G , italic_H , italic_t , italic_α ) consists of two Lie groups G 𝐺 G italic_G and H 𝐻 H italic_H with Lie group homomorphisms t : H → G : 𝑡 → 𝐻 𝐺 t\colon H\to G italic_t : italic_H → italic_G and α : G → Aut ( H ) : 𝛼 → 𝐺 Aut 𝐻 \alpha\colon G\to\mathop{\rm Aut}\nolimits(H) italic_α : italic_G → roman_Aut ( italic_H ) ( i.e. α 𝛼 \alpha italic_α is an action of G 𝐺 G italic_G on H 𝐻 H italic_H that is compatible with the group structure of H 𝐻 H italic_H ; we write it as α ⁢ ( g ) ⁢ [ h ] 𝛼 𝑔 delimited-[] ℎ \alpha(g)[h] italic_α ( italic_g ) [ italic_h ] for g ∈ G 𝑔 𝐺 g\in G italic_g ∈ italic_G , h ∈ H ℎ 𝐻 h\in H italic_h ∈ italic_H ), such that

t ⁢ ( α ⁢ ( g ) ⁢ [ h ] ) = g ⁢ t ⁢ ( h ) ⁢ g - 1 , 𝑡 𝛼 𝑔 delimited-[] ℎ 𝑔 𝑡 ℎ superscript 𝑔 1 \displaystyle t(\alpha(g)[h])=gt(h)g^{-1}, italic_t ( italic_α ( italic_g ) [ italic_h ] ) = italic_g italic_t ( italic_h ) italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , (3.14)
α ⁢ ( t ⁢ ( h ) ) ⁢ [ h ′ ] = h ⁢ h ′ ⁢ h - 1 , 𝛼 𝑡 ℎ delimited-[] superscript ℎ ′ ℎ superscript ℎ ′ superscript ℎ 1 \displaystyle\alpha(t(h))[h^{\prime}]=hh^{\prime}h^{-1}, italic_α ( italic_t ( italic_h ) ) [ italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] = italic_h italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , (3.15)

for all g ∈ G 𝑔 𝐺 g\in G italic_g ∈ italic_G and h , h ′ ∈ H ℎ superscript ℎ ′ 𝐻 h,h^{\prime}\in H italic_h , italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_H .

Definition 2

A ring R 𝑅 R italic_R is 1 1 1 1 -dimensional if it is hereditary and noetherian, or equivalently if every submodule of a f.g. projective R 𝑅 R italic_R -module is f.g. projective.

∎ absent italic_∎
Definition 2

The FIN (Fontes-Isopi-Newman) singular diffusion Z ⁢ ( s ) 𝑍 𝑠 Z(s) italic_Z ( italic_s ) .

Let W ⁢ ( t ) 𝑊 𝑡 W(t) italic_W ( italic_t ) be a standard one-dimensional Brownian Motion, and l( t 𝑡 t italic_t , y 𝑦 y italic_y ) its local time at y 𝑦 y italic_y . Define the random time-change:

ϕ ρ ⁢ ( t ) = ∫ l ⁢ ( t , y ) ⁢ ρ ⁢ ( d ⁢ y ) superscript italic-ϕ 𝜌 𝑡 𝑙 𝑡 𝑦 𝜌 𝑑 𝑦 \phi^{\rho}(t)=\int l(t,y)\rho(dy) italic_ϕ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ( italic_t ) = ∫ italic_l ( italic_t , italic_y ) italic_ρ ( italic_d italic_y )

and its inverse

ψ ρ ( t ) = i n f ( s , ϕ ρ ( s ) = t ) . fragments superscript 𝜓 𝜌 fragments ( t ) i n f fragments ( s , superscript italic-ϕ 𝜌 fragments ( s ) t ) . \psi^{\rho}(t)=inf(s,\phi^{\rho}(s)=t). italic_ψ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ( italic_t ) = italic_i italic_n italic_f ( italic_s , italic_ϕ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ( italic_s ) = italic_t ) .

Then the FIN singular diffusion is Z ⁢ ( s ) = W ⁢ ( ψ ρ ⁢ ( t ) ) . 𝑍 𝑠 𝑊 superscript 𝜓 𝜌 𝑡 Z(s)=W(\psi^{\rho}(t)). italic_Z ( italic_s ) = italic_W ( italic_ψ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ( italic_t ) ) .

Definition 2.7

We say that a symmetric multicategory Q 𝑄 Q italic_Q is freely symmetric if and only if for every arrow α ∈ Q 𝛼 𝑄 \alpha\in Q italic_α ∈ italic_Q and permutation σ 𝜎 \sigma italic_σ

α ⁢ σ = α ⇒ σ = ι . 𝛼 𝜎 𝛼 ⇒ 𝜎 𝜄 \alpha\sigma=\alpha\Rightarrow\sigma=\iota. italic_α italic_σ = italic_α ⇒ italic_σ = italic_ι .