Explore a seminar on algebraic geometry focusing on representations of the diagonal for moduli spaces of sheaves. Delve into the cohomology and Chow rings of moduli spaces, examining Lie algebra actions in terms of Chern classes of universal sheaves. Investigate the Lefschetz sl(2) action associated with ample divisor classes on projective varieties, and learn about Grothendieck's standard conjectures for moduli spaces of sheaves over curves and surfaces. Discover the importance of explicit algebraic constructions of Lefschetz operators and their applications to holomorphic symplectic geometry. Cover topics including Mukai's theorem, Child decomposition, defining representations, double push forward, and multiplication by an ample divisor.