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- W2024261551 abstract "The Killing form is a powerful tool in the study of semi-simple Lie algebras. Unfortunately, it is non-degenerate only in the semi-simple case, therefore one has to look more generally for a form which has all the useful properties of the Killing form: bilinearity, symmetry, invariance, nondegeneracy. Such a form will be a scalar multiple of the Killing form in the central-simple case but in the general case it may not even exist: just take the solvable Lie algebra of dimension 2. In this work, we study what we call regular quadratic Lie algebra (g, cp), i.e., Lie algebra g equipped with a symmetric, invariant, non-degenerate bilinear form cp. If d is a derivation of g belonging to the orthogonal Lie algebra o(cp) then, following the ideas of V. G. Kac, we define a regular quadratic Lie algebra of dimension 2 + dim g, denoted (g,, (Pi), which we call the extension of (g, cp) by d (Proposition 2.3). The interesting point of this extension is the fact that any indecomposable regular quadratic Lie algebra with nontrivial center is of the form (g,, (Pi) (Proposition 2.9), which allows an inductive classification. Moreover, a regular quadratic solvable Lie algebra has a non-zero center (Lemma 2.8). Therefore, the solvable case is completely covered. Proposition 2.11 classified all extensions of a given (g, cp)" @default.
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- W2024261551 date "1987-02-01" @default.
- W2024261551 modified "2023-10-14" @default.
- W2024261551 title "Symmetric, invariant, non-degenerate bilinear form on a Lie algebra" @default.
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- W2024261551 doi "https://doi.org/10.1016/0021-8693(87)90209-2" @default.
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