The q-Deformed Creation and Annihilation Operators as a Realization of the Quantum Superalgebra Bq(0∣l)
The Quantum Groups are mathematical structures, lately developed as deformation of Lie algebras. They retain many, but not all, the properties of the original structures; for instance the commutators are no more linear (or antisymmetric). Anyway because we ask that an underlying Hopf algebra is saved (i. e. an application Δ from the algebra A in A ⊗ A called coproduct, a counit ∈: A → C and an antipode γ: A → A with suitable properties ) no substantial changes are found at the level of the enveloping algebra, however because the coproducts in general are not any more primitive (commutative) the group structure by the exponential mapping is lost and the quantum Hopf algebra is only connected to pseudo-groups of matrices of not commuting representative functions .
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