I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson.
The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $n = 1$ the definition makes perfect sense.
My question is, $n = 1$ case is not considered because this Lie superalgebra is actually belonged to earlier defined types and so in ordered to avoid overcounting? or $P(1)$ is not simple? (In this case, is it easy to find the Ideal?)
Thanks for your time.
$P(1)$ is not simple:
To see why, consider the strange, type I, classical, simple, complex, LS $P(n)$, $n\geq 2$ realized as the set of complex, $(2n+2)\times(2n+2)$ matrices $\mathbf{M}$, with grading partitioning: $$ \mathbf{M}=\begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & -\mathbf{A} \end{bmatrix} $$ where $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ are complex $(n+1)\times(n+1)$ matrices, with $tr\mathbf{A}=0$, $\mathbf{B}$ symmetric and $\mathbf{C}$ antisymmetric. The dimensions of the odd and the even part are $n(n+2)$ and $(n+1)^2$ respectively.
Thus, the even subalgebra is $A_n=sl_{n+1}$ i.e. the set of complex, traceless matrices $\mathbf{A}$.
Now for $n=1$, the above definition makes sense however, the resulting LS $P(1)$ is not simple, because the basis elements of the even part together with $\mathbf{e}_{13}$ (that is the matrix of suitable dimension with a sole non-zero entry: $\big(\mathbf{e}_{13}\big)_{ij}=\delta_{1i}\delta_{3j}$), form the basis of a proper, graded ideal.
