A Schrödinger representation of the Weyl algebra is a representation on a space of functions ψ : ℝ → ℂ normal-: 𝜓 normal-→ ℝ ℂ \psi:\mathbbm{R}\rightarrow\mathbbm{C} italic_ψ : blackboard_R → blackboard_C , with the position operator q 𝑞 q italic_q and momentum operator p 𝑝 p italic_p represented as
Let f ∈ L t 2 C k ( Σ d ) 𝑓 subscript superscript 𝐿 2 𝑡 superscript 𝐶 𝑘 subscript Σ 𝑑 f\in L^{2}_{t}C^{k}(\Sigma_{d}) italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) , σ ∈ Σ s t ( k ) 𝜎 superscript subscript Σ 𝑠 𝑡 𝑘 \sigma\in\Sigma_{st}^{(k)} italic_σ ∈ roman_Σ start_POSTSUBSCRIPT italic_s italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT .
(1) \qua If there exists ⟨ α ⟩ ∈ Σ d ( k ) delimited-⟨⟩ 𝛼 superscript subscript Σ 𝑑 𝑘 \langle\alpha\rangle\in\Sigma_{d}^{(k)} ⟨ italic_α ⟩ ∈ roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT such that σ ⊆ ⟨ α ⟩ 𝜎 delimited-⟨⟩ 𝛼 \sigma\subseteq\langle\alpha\rangle italic_σ ⊆ ⟨ italic_α ⟩ (there is at most one such ⟨ α ⟩ delimited-⟨⟩ 𝛼 \langle\alpha\rangle ⟨ italic_α ⟩ ), then
(2) \qua If there is no ⟨ α ⟩ delimited-⟨⟩ 𝛼 \langle\alpha\rangle ⟨ italic_α ⟩ as in (1), we put θ f ( σ ) = 0 𝜃 𝑓 𝜎 0 \theta\!f(\sigma)=0 italic_θ italic_f ( italic_σ ) = 0 .
Given U × U 𝑈 𝑈 U\times U italic_U × italic_U -variety 𝐗 𝐗 {\bf X} bold_X , subgroups U ′ , U ′′ ⊂ U superscript 𝑈 ′ superscript 𝑈 ′′ 𝑈 U^{\prime},U^{\prime\prime}\subset U italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⊂ italic_U , and a character χ : U → 𝔸 1 : 𝜒 → 𝑈 superscript 𝔸 1 \chi:U\to\mathbb{A}^{1} italic_χ : italic_U → blackboard_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , we say that a regular function f : X → 𝔸 1 : 𝑓 → 𝑋 superscript 𝔸 1 f:X\to\mathbb{A}^{1} italic_f : italic_X → blackboard_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT is ( U ′ × U ′′ , χ ) superscript 𝑈 ′ superscript 𝑈 ′′ 𝜒 (U^{\prime}\times U^{\prime\prime},\chi) ( italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT × italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT , italic_χ ) - linear if
for all u ′ ∈ U ′ superscript 𝑢 ′ superscript 𝑈 ′ u^{\prime}\in U^{\prime} italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , u ′′ ∈ U ′′ superscript 𝑢 ′′ superscript 𝑈 ′′ u^{\prime\prime}\in U^{\prime\prime} italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT , x ∈ X 𝑥 𝑋 x\in X italic_x ∈ italic_X . In particular, we refer to each ( e × U , χ ) 𝑒 𝑈 𝜒 (e\times U,\chi) ( italic_e × italic_U , italic_χ ) -linear (resp. ( U × e , χ ) 𝑈 𝑒 𝜒 (U\times e,\chi) ( italic_U × italic_e , italic_χ ) -linear) function f 𝑓 f italic_f as a right ( U , χ ) 𝑈 𝜒 (U,\chi) ( italic_U , italic_χ ) -linear (resp. left ( U , χ ) 𝑈 𝜒 (U,\chi) ( italic_U , italic_χ ) -linear ) function.
Given a commutative ℂ ℂ \mathbb{C} blackboard_C -algebra 𝒜 𝒜 {\mathcal{A}} caligraphic_A without zero divisors and a totally ordered free abelian group Γ Γ \Gamma roman_Γ , a map ν : 𝒜 ∖ { 0 } → Γ : 𝜈 → 𝒜 0 Γ \nu:{\mathcal{A}}\setminus\{0\}\to\Gamma italic_ν : caligraphic_A ∖ { 0 } → roman_Γ is said to be a valuation if
for all x , y ∈ 𝒜 ∖ { 0 } 𝑥 𝑦 𝒜 0 x,y\in{\mathcal{A}}\setminus\{0\} italic_x , italic_y ∈ caligraphic_A ∖ { 0 } ,
for any x , y ∈ 𝒜 ∖ { 0 } 𝑥 𝑦 𝒜 0 x,y\in{\mathcal{A}}\setminus\{0\} italic_x , italic_y ∈ caligraphic_A ∖ { 0 } such that ν ( x ) ≠ ν ( y ) 𝜈 𝑥 𝜈 𝑦 \nu(x)\neq\nu(y) italic_ν ( italic_x ) ≠ italic_ν ( italic_y ) (we use the convention that ν ( 0 ) = + ∞ 𝜈 0 \nu(0)=+\infty italic_ν ( 0 ) = + ∞ , where + ∞ +\infty + ∞ is greater than any element of Γ Γ \Gamma roman_Γ ).
We say that ν 𝜈 \nu italic_ν is saturated if for each λ ∈ ν ( 𝒜 ∖ { 0 } ) , 𝜆 𝜈 𝒜 0 \lambda\in\nu({\mathcal{A}}\setminus\{0\}), italic_λ ∈ italic_ν ( caligraphic_A ∖ { 0 } ) , the entire “half-line” ℚ ≥ 0 ⋅ λ ∩ Γ ⋅ subscript ℚ absent 0 𝜆 Γ \mathbb{Q}_{\geq 0}\cdot\lambda\cap\Gamma blackboard_Q start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT ⋅ italic_λ ∩ roman_Γ also belongs to the semi-group ν ( 𝒜 ∖ { 0 } ) 𝜈 𝒜 0 \nu({\mathcal{A}}\setminus\{0\}) italic_ν ( caligraphic_A ∖ { 0 } ) .)
A triangular r 𝑟 r italic_r -matrix of a Lie algebra 𝔤 𝔤 {\mathfrak{g}} fraktur_g is an element r ∈ ∧ 2 𝔤 𝑟 superscript 2 𝔤 r\in\wedge^{2}{\mathfrak{g}} italic_r ∈ ∧ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT fraktur_g such that
We define U λ ( 𝐡𝐞𝐢𝐬 ) subscript 𝑈 𝜆 𝐡𝐞𝐢𝐬 U_{\lambda}({\bf heis}) italic_U start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( bold_heis ) to be the Hopf subalgebra of U CM λ superscript subscript 𝑈 normal-CM 𝜆 U_{\mathrm{CM}}^{\lambda} italic_U start_POSTSUBSCRIPT roman_CM end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_λ end_POSTSUPERSCRIPT generated by z := z 2 assign 𝑧 subscript 𝑧 2 z:={z_{2}} italic_z := italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , α 𝛼 \alpha italic_α , β 𝛽 \beta italic_β . The presentation of U λ ( 𝐡𝐞𝐢𝐬 ) subscript 𝑈 𝜆 𝐡𝐞𝐢𝐬 U_{\lambda}({\bf heis}) italic_U start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( bold_heis ) is then: