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Over the -dimensional real superspace , we classify -invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second -relative cohomology space of with coefficients in the module of -densities on , where is the Lie superalgebra of contact vector fields on and is the affine Lie superalgebra. This result allows us to compute the second -relative cohomology space of with coefficients in the Poisson superalgebra . We explicitly give 2-cocycles spanning these cohomology spaces. [ABSTRACT FROM AUTHOR]

We construct a two-parameter bicovariant differential calculus on with the help of the covariance point of view using the Hopf algebra structure of . To achieve this, we first use the consistency of calculus and the approach of -matrix which satisfies both ungraded and graded Yang-Baxter equations. In particular, based on this differential calculus, we investigate Cartan-Maurer forms for this -superspace. Finally, we obtain the quantum Lie superalgebra corresponding the Cartan-Maurer forms. [ABSTRACT FROM AUTHOR]

Over the (l,n)-dimensional real superspace, we classify aff(n/1)-invariant linear differential operators acting on the superspaces of weighted densities, where κ(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of κ(n) with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra aff(n/1). We explicitly give 1-cocycles spanning these cohomology spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the operator forms of the classical Yang–Baxter equation (CYBE) in Lie superalgebras. We introduce the notion of super $\mathcal{O}$-operators and give their relationship with the (standard) tensor form. There are close relationships between the CYBE in Lie superalgebras and left-symmetric superalgebras which can be interpreted through the super $\mathcal{O}$-operators. In particular, there are natural and explicit solutions of the CYBE in Lie superalgebras constructed from left-symmetric superalgebras. [ABSTRACT FROM AUTHOR]