Talk: The semi-differential reduction algebra of sp(4)

Speaker: Dwight Anderson Williams II (Pomona College)

Joint Work: Jonas T. Hartwig

TItle: The semi-differential reduction algebra of $s p (4)$

Abstract: A reduction algebra $Z$ is an associative algebra that can be realized as a certain quotient of an associative algebra $A$ containing the universal enveloping algebra $U (g)$ of a Lie algebra $g$ . We require a triangular decomposition of $g = g_{-} \oplus h \oplus g_{+},$ with nilpotent parts $g_{\pm}$ and Cartan subalgebra $h$ . Then the representation theory of $Z$ sheds light on the representation theory of $g$ , including calculation/recovery of Clebsch-Gordan coefficients, branching rules, and intertwining operators. The algebra $Z$ is also realized as a space of double cosets carrying an associative product determined by the so-called extremal projector $P$ of $g$ . There is substantial literature on extremal projectors and their applications to reduction algebras when $g$ is $g l (n)$ or of rank one.
In this talk, we demonstrate the extremal projector of the rank two symplectic Lie algebra $s p (4)$ ; furthermore, using $A_{2}$ to express the second Weyl algebra, we study $Z = D (s p (4))$ constructed from a localized version of the tensor algebra $A = A_{2} \otimes U (s p (4)) .$ We provide generators and relations of $D (s p (4))$ as an example of a semi-differential reduction algebra.