Let \(\Bbbk\) be an algebraically closed field of characteristic zero. A twisted generalized Weyl (TGW) algebra is an associative \(\Bbbk\)-algebra \(\mathcal{A}_\mu(R, \sigma, t)\) depending on the following parameters:

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\(\mu\) – a \(I\times I\) matrix without diagonal with entries in \(\Bbbk\setminus \{0\}\);

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\(R\) – an associative \(\Bbbk\)-algebra;

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\(\sigma = (\sigma_i)_{i\in I}\) – a sequence of commuting \(\Bbbk\)-algebra automorphisms of \(R\);

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\(t = (t_i)_{i\in I}\) – a sequence of central elements of \(R\).

They were introduced in [V. Mazorchuk and L. Turowska, Commun. Algebra 27, No. 6, 2613–2625 (1999; Zbl 0949.17003)] as a generalization of the so called generalized Weyl algebras.
In the paper under review, the authors prove that some supersymmetric analogs of some classical algebras are also examples of TGW Algebras.
In section 3, they prove that the supersymmetric analogs of the Weyl-Clifford superalgebras, \(A_{p|q}^\pm\) can be realized as TGW algebras.
Later, in section 4, they introduce and study the main object of the paper: a family of TGW algebras \(\mathcal{A}(\gamma)^{\pm}\), depending on a matrix \(\gamma\) with integer coefficients. This is a supersymmetric generalization of the construction given in [J. T. Hartwig and V. Serganova, Algebr. Represent. Theory 19, No. 2, 277–313 (2016; Zbl 1352.16027)].
In section 5, the authors apply their results to prove that for appropriate \(\gamma\), the TGW algebras \(\mathcal{A}(\gamma)^{\pm}\), fit into commutative diagrams involving the spinor representation of \(U(\mathfrak{g})\) for \(g = \mathfrak{gl}(m|n)\) and \(\mathfrak{osp}(m|2n)\) which allow them to conclude that some quotients of \(U(\mathfrak{g})\) are examples of TGW algebras for such \(\mathfrak{g}\) and for for classical Lie algebras as well.
It is worth mentioning that the structure of rings (and algebras) with involution and the structure of graded algebras play an important role in the paper.