Representations of Heisenberg Lie super algebras (Q917676)

The Heisenberg Lie superalgebra \(H=H^{(m,n)}\) is given by \(H^{(m,n)}=H_{\bar 0}+H_{\bar 1}\), \(H_{\bar 0}=<x_ 1,...,x_{2m},z>\), \(H_{\bar 1}=<y_ 1,...,y_ n>\) and the commutation rules \([x_{2k-1},x_{2k}]=z\), \([y_ i,y_ j]=2\cdot \delta_{ij}\cdot z\) (other products are zero). The author regards faithful finite-dimensional representations of \(H^{(m,n)}\) for \(m>0\) (the case \(m=0\) is considered in another paper). Since every irreducible representation T is a linear functional on H with \(T(H_{\bar 1})=T(z)=0\), the representations regarded here are reducible in general. The essential tools are inductions \(T=ind(T_ 0,H_{\bar 0}\uparrow H)\), where \(T_ 0\) is a representation of \(H_ 0\), and the existence of flags with one-dimensional factors for every finite- dimensional super representation of H. Some informations concerning the size of matrix representations are given.