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Lie supergroups obtained from 3-dimensional Lie superalgebras associated to the adjoint representation and having a 2-dimensional derived ideal. (English) Zbl 1155.17009
In the paper, the authors use an earlier result by R. Peniche and O. A. Sánchez-Valenzuela [ J. Geom. Phys. 56, No. 6, 999–1028 (2006; Zbl 1151.17012)] to give the explicit multiplication law of the Lie supergroups for which the base manifold is a \(3\)-dimensional Lie group and whose underlying Lie superalgebra \(\mathfrak g = \mathfrak g_0 \bigoplus \mathfrak g_1\), which satisfies \(\mathfrak g_1 = \mathfrak g_0\), \(\mathfrak g_0\) acts on \(\mathfrak g_1\) via the adjoint representation and \(\mathfrak g_0\) has a \(2\)-dimensional derived ideal. For the authors, it represents one step forward in the understanding of the real and complex \((3,3)-\)dimensional Lie supergroups. They describe the multiplication law before mentioned in terms of tetrads \((\varsigma, \upsilon, \sigma, \theta)\), where \((\varsigma, \upsilon) \in \mathbb{F} \times \mathbb{F}^2\) \((\mathbb{F}\) denotes either the real or the complex number field) are the local coordinates in the \(3\)-dimensional Lie group \(G_0(A)\), and \((\sigma, \theta)\) are the odd coordinates on the supergroup \((G_0(A), \bigwedge(E))\), where \(\bigwedge(E)\) stands for the sheaf of sections of the exterior algebra bundle associated to the rank-\(3\) vector bundle \(E \mapsto G_0(A)\), whose typical filter \(\mathfrak g_0\) can be decomposed as \(\mathcal{S} \bigoplus \mathfrak g'_0\), with \(\mathcal{S} = \mathfrak g_0 \, / \, \mathfrak g'_0\). Thus, \(\sigma\) is a local section of \(G_0(A) \times \mathcal{S} \rightarrow G_0(A)\), and \(\theta\) is a local section of \(G_0(A) \times \mathfrak g'_0 \rightarrow G_0(A)\). So, they can obtain an expression for the product \((\varsigma', \upsilon', \sigma', \theta') \ast (\varsigma, \upsilon, \sigma, \theta)\). Once they have obtained the multiplication law for the different Lie supergroups, they compute the left-invariant supervector fields associated to the built Lie supergroups.
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
58A50 Supermanifolds and graded manifolds
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