论文标题
当前谎言代数的不变指标
Invariant metrics on current Lie algebras
论文作者
论文摘要
在这项工作中，我们指出了当前谎言代数$ \ g \ otimes \ mathcal {s} $的条件，以接纳一个不变的度量，其中$ \ g $是二次谎言代数，$ \ nathcal {s} $是与单位的联想和通勤的代数。我们还考虑倒数：如果$ \ g \ otimes \ Mathcal {s} $允许不变的度量，那么我们声明$ \ g $的必要条件，以便承认不变的度量。特别是，我们表明，如果$ \ g $是一种难以解否的二次谎言代数，则$ \ g \ otimes \ nathcal {s} $仅在$ \ mathcal {s} $时也接受不变的度量，同时也承认一种不变的，对称性，对称性和非级别的Bilinerate Bilinerate Bilinerate Bilinerate。此外，我们证明了一个类似于$ \ g \ otimes \ Mathcal {s} $的双重扩展的定理，其中$ \ g $是一种不可分解的，nilpotent和quadratic lie elgebra。
In this work we state conditions for a current Lie algebra $\g \otimes \mathcal{S}$ to admit an invariant metric, where $\g$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. We also consider the reciprocal: if $\g \otimes \mathcal{S}$ admits an invariant metric, we state necessary and sufficient conditions for $\g$ to admit an invariant metric. In particular, we show that if $\g$ is an indecomposable quadratic Lie algebra, then $\g \otimes \mathcal{S}$ admits an invariant metric if and only if $\mathcal{S}$ also admits an invariant, symmetric and non-degenerate bilinear form. In addition, we prove a theorem similar to the double extension for $\g \otimes \mathcal{S}$, where $\g$ is an indecomposable, nilpotent and quadratic Lie algebra.
