Bialgebra structures on a real semisimple Lie algebra
The Manin algebra \({\mathcal L}\) of the Lie bialgebra structure of a Lie algebra \(L\) with a 1-cocycle \(\psi: L\to \Lambda^2 L\) is defined in the following way: \({\mathcal L}\) has a nondegenerate invariant bilinear symmetric form \(\langle \langle\;,\;\rangle \rangle\); \({\mathcal L}\) is a direct sum (as vector spaces) of subalgebras \(L_1\) and \(L_2\); \(L_1\) and \(L_2\) are isotropic for \(\langle \langle\;,\;\rangle \rangle\). Theorem. Let \(L\) be a semisimple real Lie algebra; \(L= \bigoplus_{1\leq k\leq p} L_k\), where \(L_k\) are simple ideals of \(L\). Then up to isomorphism the Manin algebra \({\mathcal L}\) associated to \(L\) is of the form \({\mathcal L}= \bigoplus_{1\leq k\leq p} {\mathcal L}_k\), where \({\mathcal L}_k\) is one of the following: \({\mathcal L}_k= \text{Lie} (T^* (I_k))\), where \(I_k\) is the Lie group associated to the Lie algebra \(L_k\); \({\mathcal L}_k= (L_k )^C\); \({\mathcal L}_k= L_k \oplus L_k\). The bialgebra structures on a real semisimple Lie algebra for the nonzero standard modified Yang-Baxter equation are determined.
scientific article; zbMATH DE number 817246