Noncommutative pfaffians and classification of states of five-dimensional quasi-spin

Noncommutative pfaffians associated with an orthogonal algebra ${\mathfrak{o}}_N$ are some special elements of the universal enveloping algebra $U({\mathfrak{o}}_N)$. Using pfaffians we construct the fourth quantum number which together with the naturally defined three quantum numbers allow to classify the states of a five-dimensional quasi-spin. The pfaffians are treated as creation operators for the new quantum number.