论文标题
朝向约旦的张力分解
Toward Jordan Decompositions of Tensors
论文作者
论文摘要
我们扩展了Vinberg的想法，以占用张量的空间和作用于其的自然谎言代数，并将其直接汇总到辅助代数中。被视为该代数的内态性，我们将伴随操作员与张量相关联。我们表明，张量空间和伴随操作员上的组动作是一致的，这意味着张量的伴随算子的不变式（例如约旦分解）是张量的不变性。我们表明，存在一种基本独特的代数结构，可以保留张量结构并具有有意义的约旦分解。我们利用这些伴随操作员的各个方面在与张量分解和量子信息有关的示例中研究轨道分离和分类。
We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.
