Real Left-Symmetric Algebras with Positive Definite Koszul Form and K\"ahler-Einstein Structures

TL;DR

This paper characterizes real left symmetric algebras with positive definite Koszul forms and shows they induce Kähler-Einstein structures on tangent bundles, revealing new algebraic classes and geometric insights.

Contribution

It provides a complete characterization of such algebras, links them to Kähler-Einstein geometry, and introduces new non-associative algebra classes.

Findings

Algebras with positive definite Koszul form are either trivial or isomorphic to alculus.

These algebras induce Ke4hler-Einstein structures with negative scalar curvature.

New classes of non-associative algebras generalize Hessian Lie algebras.

Abstract

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 K\"ahler-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.