Quantum Superalgebra

An algebra of rank r that is characterized by a Cartan matrix (a _ij ) and a subset τ ⊂ I ≡ {1,...,r} that identifies the odd generators. The Cartan matrix can be normalized so that a _ii = 2 if i ∉ τ and a _ii = 1 or 0 if i ∈ τ. Let q ∈ C \ {0} be the deformation parameter and q _i = q ^d _i , where the rational numbers d _i , i = 1,...,r, are such that: d _i a _ij = d _j a _ji . The quantum superalgebra is generated by 3r elements e _i , f _i and h _i , i ∈ I, which satisfy [1,2]

with the following assignements of parity

It is convenient to introduce the quantities k _i = in terms of which the above defining relations become

The quantum superalgebra is endowed with a Hopf algebra structure. The action of the coproduct , antipode and counit

on the generators is as follows [1]

One can define the q-analog ad _q of the adjoint operation by [3]

with id the identity operator and μ _L , μ _R the left and right (graded) multiplications: μ _L (x) y = xy, μ _R (x) y = (−)^{p(x) p(y)} yx. The quantum...