I'm reading a book on Mathemathical Physics and speaking of the Quantum harmonic oscillator says (this is a translation in english, hope is right):
The commutation relations between operators are: $[N, a] = a$, $[N,a^*] = a^*$, $[a, a^*] = 1$ So this is an algebra generated by the operators $a$, $a^*$, $N$ e $1$. It's a solvable Lie Algebra which is a unidimensional right extension of the Heisenberg Algebra. Consequently N is a positive operator in H.
I'd like to understand better the phrase "unidimensional right extension of the Heisenberg Algebra" and why does it implies that N is a positive operator... Can you help me out?