Definition 1.4 .

(The dual Poisson–Lie group) Let G = 𝔮 × 𝔪 = 𝔤 𝐺 𝔮 𝔪 𝔤 G=\mathfrak{q}\times\mathfrak{m}=\mathfrak{g} italic_G = fraktur_q × fraktur_m = fraktur_g as a vector space. Define the multiplication law on it by

( p , q , r ) ⁢ ( p ′ , q ′ , r ′ ) = ( e λ ⁢ r ′ ⁢ p + p ′ , e λ ⁢ r ′ ⁢ q + q ′ , r + r ′ ) . 𝑝 𝑞 𝑟 superscript 𝑝 ′ superscript 𝑞 ′ superscript 𝑟 ′ superscript 𝑒 𝜆 superscript 𝑟 ′ 𝑝 superscript 𝑝 ′ superscript 𝑒 𝜆 superscript 𝑟 ′ 𝑞 superscript 𝑞 ′ 𝑟 superscript 𝑟 ′ (p,q,r)(p^{\prime},q^{\prime},r^{\prime})=(e^{\lambda r^{\prime}}p+p^{\prime},% e^{\lambda r^{\prime}}q+q^{\prime},r+r^{\prime}). ( italic_p , italic_q , italic_r ) ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( italic_e start_POSTSUPERSCRIPT italic_λ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_p + italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_e start_POSTSUPERSCRIPT italic_λ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_q + italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_r + italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .