TY - JOUR

T1 - Braided Lie bialgebras associated to Kac-Moody algebras

AU - Grabowski, Jan

PY - 2008

Y1 - 2008

N2 - Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra sl3.

AB - Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra sl3.

KW - Kac-Moody algebra, braided Lie bialgebra.

M3 - Journal article

VL - 18

SP - 125

EP - 140

JO - Journal of Lie Theory

JF - Journal of Lie Theory

IS - 1

ER -