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$$(H,R)$$-Lie coalgebras and $$(H,R)$$-Lie bialgebras. (Russian, English) Zbl 1135.17302
Sib. Mat. Zh. 47, No. 4, 932-945 (2006); translation in Sib. Math. J. 47, No. 4, 767-778 (2006).
Summary: Given an $$(H,R)$$-Lie coalgebra $$\Gamma$$, we construct $$(H,R_T)$$-Lie coalgebra $$\Gamma^T$$ through a right cocycle $$T$$, where $$(H,R)$$ is a triangular Hopf algebra, and prove that there exists a bijection between the set of $$(H,R)$$-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if $$(L,[\,,\,],\Delta_T,R)$$ is an $$(H,R)$$-Lie bialgebra of an ordinary Lie algebra then $$(L^T,[\,,\,],\Delta_T,R_T)$$ is an $$(H,R_T)$$-Lie bialgebra of an ordinary Lie algebra.
##### MSC:
 17B62 Lie bialgebras; Lie coalgebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000)
##### Keywords:
Lie coalgebra; triangular Hopf algebra; right cocycle
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