# ROOM FOUR: Some Coupled Supersymmetries and Their Associated Bargmann Transforms

In this talk, I will define coupled supersymmetries, discuss the Lie algebras associated to them, and establish eigenvalues of the associated Hamiltonian-like operators from the $\mathfrak{su}(1,1)$ Lie algebra structure. Due to the Lie-algebraic structure coupled supersymmetries enjoy, Bargmann transforms can be established for some coupled supersymmetries on $\mathbb{R}$. I will develop these for the special classes of supersymmetries given by $\left\{\frac{1}{\sqrt{2}}\left(\frac{1}{x^{n-1}}\frac{d}{dx}+x^n\right), \frac{1}{\sqrt{2}}\left(\frac{d}{dx}\frac{1}{x^{n-1}}+x^n\right), -1, 2n-1\right\}$, where $n\in\mathbb{N}$. These Bargmann transforms are associated to holomorphic function spaces and generalize the standard Bargmann transform associated to the (harmonic) oscillator algebra on $\mathbb{R}$.

This is joint work with Bernhard G. Bodmann and Donald J. Kouri. It was supported in part by R. A. Welch Grant E-0608 and in part by NSF grant DMS-1412524.