A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.

中文翻译：

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通过单个形式计算的运算符身份的形式证明

从已知算子证明新算子身份的形式计算原则上受到所涉及的线性算子的域和余域的限制，因为不能添加或组合任何两个算子。在代数上，恒等式可以用非交换多项式建模，这样的形式计算证明了对应于新恒等式的多项式位于由对应于已知恒等式的多项式生成的理想中。然而，为了证明一个算子身份，仅仅证明理想中多项式的成员资格是不够的，因为非交换多项式环忽略了域和共域。我们表明，另外验证该多项式和理想生成器与标记的颤动的兼容性就足够了，颤抖对哪些多项式可以实现为线性算子进行编码。然后，对于线性范畴中这种箭袋的每一个一致表示，在范畴中都存在一个证明对应的恒等式实例的计算。此外，通过为箭袋的几个边分配相同的标签，所开发的代数框架允许通过非交换多项式中的相同不确定性对不同版本的算子进行建模。