The Lie Module Structure on the Hochschild Cohomology Groups of Monomial Algebras with Radical Square Zero
نویسنده
چکیده
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. Introduction Let A be an associative unital k-algebra where k is a field. The n Hochschild cohomology group of A , denoted by HH(A), refers to HH(A) := HH(A,A) = ExtnAe (A,A) where A is the enveloping algebra A⊗kA of A. Thus, for example, HH (A) is the center of A and the first Hochschild cohomology group HH(A) is the vector space of the outer derivations. Note that the first Hochschild cohomology group has a Lie algebra structure given by the commutator bracket. In [Ger63], Gerstenhaber introduced two operations on the Hochschild cohomology groups: the cup product and the bracket [ − , − ] : HH(A) ×HH(A) −→ HH(A). He proved that the Hochschild cohomology of A ,
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