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A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$-quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮,\beta )$ together with a left $𝔤$-module structure which acts on $𝔮$ via derivations and for which $\beta$ is $𝔤$-invariant. Geometrically, $𝔤$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic...

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$-symplectic Lie groupoid; the “$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $𝒢\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid...