Definition 6.1 .

Let 𝐜 = ( c 1 , … , c d - 1 ) ∈ β„‚ d - 1 𝐜 subscript 𝑐 1 normal-… subscript 𝑐 𝑑 1 superscript β„‚ 𝑑 1 \mathbf{c}=(c_{1},\ldots,c_{d-1})\in\mathbb{C}^{d-1} bold_c = ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT ) ∈ blackboard_C start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT . The rational Cherednik algebra 𝖧 t , 𝐜 ⁒ ( Ξ“ ) subscript 𝖧 𝑑 𝐜 normal-Ξ“ \mathsf{H}_{t,\mathbf{c}}(\Gamma) sansserif_H start_POSTSUBSCRIPT italic_t , bold_c end_POSTSUBSCRIPT ( roman_Ξ“ ) of rank one is the algebra generated by elements u , v , Ξ³ 𝑒 𝑣 𝛾 u,v,\gamma italic_u , italic_v , italic_Ξ³ with Ξ³ ∈ Ξ“ = β„€ / d ⁒ β„€ 𝛾 normal-Ξ“ β„€ 𝑑 β„€ \gamma\in\Gamma=\mathbb{Z}/d\mathbb{Z} italic_Ξ³ ∈ roman_Ξ“ = blackboard_Z / italic_d blackboard_Z and the following relations:

Ξ³ ⁒ u ⁒ Ξ³ - 1 = ΞΆ ⁒ u , Ξ³ ⁒ v ⁒ Ξ³ - 1 = ΞΆ - 1 ⁒ v formulae-sequence 𝛾 𝑒 superscript 𝛾 1 𝜁 𝑒 𝛾 𝑣 superscript 𝛾 1 superscript 𝜁 1 𝑣 \gamma u\gamma^{-1}=\zeta u,\;\;\gamma v\gamma^{-1}=\zeta^{-1}v italic_Ξ³ italic_u italic_Ξ³ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_ΞΆ italic_u , italic_Ξ³ italic_v italic_Ξ³ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_ΞΆ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_v
(6) v ⁒ u - u ⁒ v = t + βˆ‘ i = 1 d - 1 c i ⁒ ΞΎ i ⁒ where ⁒ ΞΎ ⁒ is a generator of ⁒ Ξ“ . 𝑣 𝑒 𝑒 𝑣 𝑑 superscript subscript 𝑖 1 𝑑 1 subscript 𝑐 𝑖 superscript πœ‰ 𝑖 where πœ‰ is a generator of Ξ“ vu-uv=t+\sum_{i=1}^{d-1}c_{i}\xi^{i}\mbox{ where }\xi\mbox{ is a generator of % }\Gamma. italic_v italic_u - italic_u italic_v = italic_t + βˆ‘ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ΞΎ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT where italic_ΞΎ is a generator of roman_Ξ“ .