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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.01650 |
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| _version_ | 1866929574704578560 |
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| author | Boucetta, Mohamed Essoufi, Hasna |
| author_facet | Boucetta, Mohamed Essoufi, Hasna |
| contents | Let $(\mathfrak{g}, \bullet)$ be a real left symmetric algebra, and $(\mathfrak{g}^-, [\;,\;])$ the corresponding Lie algebra. We denote by $L$ the left multiplication operator associated with the product $\bullet$. The symmetric bilinear form $\mathrm{B}(X, Y) = \mathrm{tr}(L_{X \bullet Y})$, referred to as the Koszul form of $(\mathfrak{g}, \bullet)$, is introduced. We provide a complete characterization, along with a broad class of examples, of real left symmetric algebras that possess a positive definite Koszul form. In particular, we show that for a left symmetric algebra with positive definite Koszul form being commutative or associative or Novikov implies that this algebra is isomorphic to $\mathbb{R}^n$ endowed with its canonical product.
Beyond their algebraic interest, we show that any real left symmetric algebra $(\mathfrak{g}, \bullet)$ with a positive definite Koszul form induces a Kähler-Einstein structure with negative scalar curvature on the tangent bundle $TG$ of any connected Lie group $G$ associated to $(\mathfrak{g}^-, [\;,\;])$. Furthermore, the characterization of left symmetric algebras with a positive definite Koszul form leads to a new class of non-associative algebras, which are of independent interest and generalize Hessian Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Real Left-Symmetric Algebras with Positive Definite Koszul Form and Kähler-Einstein Structures Boucetta, Mohamed Essoufi, Hasna Differential Geometry Let $(\mathfrak{g}, \bullet)$ be a real left symmetric algebra, and $(\mathfrak{g}^-, [\;,\;])$ the corresponding Lie algebra. We denote by $L$ the left multiplication operator associated with the product $\bullet$. The symmetric bilinear form $\mathrm{B}(X, Y) = \mathrm{tr}(L_{X \bullet Y})$, referred to as the Koszul form of $(\mathfrak{g}, \bullet)$, is introduced. We provide a complete characterization, along with a broad class of examples, of real left symmetric algebras that possess a positive definite Koszul form. In particular, we show that for a left symmetric algebra with positive definite Koszul form being commutative or associative or Novikov implies that this algebra is isomorphic to $\mathbb{R}^n$ endowed with its canonical product. Beyond their algebraic interest, we show that any real left symmetric algebra $(\mathfrak{g}, \bullet)$ with a positive definite Koszul form induces a Kähler-Einstein structure with negative scalar curvature on the tangent bundle $TG$ of any connected Lie group $G$ associated to $(\mathfrak{g}^-, [\;,\;])$. Furthermore, the characterization of left symmetric algebras with a positive definite Koszul form leads to a new class of non-associative algebras, which are of independent interest and generalize Hessian Lie algebras. |
| title | Real Left-Symmetric Algebras with Positive Definite Koszul Form and Kähler-Einstein Structures |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2411.01650 |