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| By a general principle in noncommutative algebraic geometry (as formulated by M. Kontsevich and A. Rosenberg), a property of an associative algebra $A$ is `geometric' if it induces standard geometric properties on its representation spaces $\text{Rep}(A,V)$. Here, $\text{Rep}(A,V)$ is the space of all representations of $A$ in a finite-dimensional vector space $V$, which has a well-known affine-scheme structure. According to this principle, the family of affine schemes $\text{Rep}(A,V)$ should be regarded as a substitute for a hypothetical affine noncommutative scheme `$\text{Spec}(A)$'.
This principle has been applied successfully to symplectic structures and Poisson structures on quiver algebras by W. Crawley-Boevey, P. Etingof, V. Ginzburg, and M. Van den Bergh, among others. In this talk we will consider noncommutative analogues of other standard geometric structures. More precisely, we will introduce bi-symplectic $\mathbb{N}Q$-algebras. These are noncommutative counterparts of symplectic $\mathbb{N}Q$-manifolds, which are basic ingredients encoding higher Lie algebroid structures in the Batalin-Vilkovisky formulation of Topological Quantum Field Theories. The natural context to study these structures is graded noncommutative algebraic geometry. In particular, we will explain how bi-symplectic $\mathbb{N}Q$-algebras of weight 1 are closely related to Van den Bergh's double Poisson algebras. To establish this result, we will explain an analogue of the odd Darboux Theorem in this setting. Joint work with Luis \'Alvarez-C\'onsul |
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