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On the semisimplicity of the outer derivations of monomial algebras
We show that the Hochschild cohomology of a monomial algebra over a field of
characteristic zero vanishes from degree two if the first Hochschild cohomology
is semisimple as a Lie algebra. We also prove that first Hochschild cohomology
of a radical square zero algebra is reductive as a Lie algebra. In the case of
the multiple loops quiver, we obtain the Lie algebra of square matrices of size
equal to the number of loops
The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero
We study the Lie module structure given by the Gerstenhaber bracket on the
Hochschild cohomology groups of a monomial algebra with radical square zero.
The description of such Lie module structure will be given in terms of the
combinatorics of the quiver. The Lie module structure will be related to the
classification of finite dimensional modules over simple Lie algebras when the
quiver is given by the two loops and the ground field is the complex numbers
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