LAUSR

AbstractWe discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms: 1)the first one is between the characteristic Lie algebra χ(sinhu)$\chi (\sinh {u})$ of the sinh-Gordon equation uxy=sinhu$u_{xy}=\sinh {u}$ and the non-negative part ℒ(????????(2,ℂ))≥0${\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}$ of the loop algebra of ????????(2,ℂ)${\mathfrak {sl}}(2,{\mathbb {C}})$ that corresponds to the Kac-Moody algebra A1(1)$A_{1}^{(1)}$χ(sinhu)≅ℒ(????????(2,ℂ))≥0=????????(2,ℂ)⊗ℂ[t].$$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$2)the second isomorphism is for the Tzitzeica equation uxy = eu + e− 2uχ(eu+e−2u)≅ℒ(????????(3,ℂ),μ)≥0=⊕j=0+∞????j(mod2)⊗tj,$$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$ where ℒ(????????(3,ℂ),μ)=⊕j∈ℤ????j(mod2)⊗tj${\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}$ is the twisted loop algebra of the simple Lie algebra ????????(3,ℂ)${\mathfrak {sl}}(3,{\mathbb {C}})$ that corresponds to the Kac-Moody algebra A2(2)$A_{2}^{(2)}$.Hence the Lie algebras χ(sinhu)$\chi (\sinh {u})$ and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates 32$\frac {3}{2}$ and 43$\frac {4}{3}$ respectively.