Quantum Commutative Algebra

An associative algebra A equipped with an action ▹: H⊗ A→ A of a quasitriangular Hopf algebra H, and the R–matrix R = Σ R ⁽¹⁾⊗ R ⁽²⁾∈ H⊗ H satisfying the quantum Yang–Baxter equation

and the following condition

For a superalgebra A=A ₀⊕ A ₁ this condition takes the form

where |a| is the parity of a and provide a simple example of quantum commutative algebra corresponding to the Hopf algebra H:=spn{1, g} generating by 1 and g and relation g ²=1. The coproduct Δ, the counit ɛ and the antipode S are given by the following formulae

respectively. In this case [2]

The dual version corresponding for an associative algebra A equipped with an coaction ρ: A→ A⊗ H of an co quasitriangular Hopf algebra H with a bilinear form 〈 −,− 〉 : H⊗ H→ C, is given by the following relation

where the standard notation ρ (a) = Σ a ₍₀₎ ⊗ a ₍₁₎∈ A ⊗ H, and ρ (b) = Σ b ₍₀₎⊗ b ₍₁₎∈ A⊗ H for every a, b∈ A, has been used [1]. This means that Σ indicate a summation of terms which are (i)-th factors in the given...