Central extensions of current algebras

The second cohomology group of Lie algebras of kind <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L circled-times upper U"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo> ⊗ </mml:mo> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L \otimes U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with trivial coefficients is investigated, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a decomposition with one-dimensional root spaces and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an arbitrary associative commutative algebra with unit. This paper gives a unification of some recent results of C. Kassel and A. Haddi and provides a determination of central extensions of certain modular semisimple Lie algebras.

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