Free loci of matrix pencils and domains of noncommutative rational functions
Free loci of matrix pencils and domains of noncommutative rational functions
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矩阵铅笔的自由轨迹和非交换有理函数的域
DOI：
10.4171/cmh/408
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发表时间：
2015
影响因子：
0.9
通讯作者：
Jurij Volčič
中科院分区：
文献类型：
--
作者：
I. Klep;Jurij Volčič
考虑一元线性铅笔$L(X)=i-A_1x_1-\cdots-A_gx_g$，其系数$A_j$是$d\×d$矩阵。用Kronecker张量积很自然地求出它的自由轨迹$Z(L)={X：\Det L(X)=0$，它自然地被求为$g$-矩阵的元组$X$。本文证明了由两支自由轨迹相等的线性铅笔L和L的系数分别生成的代数$A‘和$A’‘分别同构于根。此外，$Z(L)\子集Z(L‘)$当且仅当将$L’$的系数发送到$L$的系数的自然映射诱导同态$A‘/A’到A/Rmrad}A$.由于线性铅笔是通过实现理论研究非交换有理函数的关键因素，上述结果导致了具有给定区域的所有非交换有理函数的刻画。最后，给出了线性铅笔上Kippenhahn猜想的量子形式，并证明了：如果埃尔米特矩阵$A_1，\点，A_g$生成$M_d(\mathbb{C})$作为代数，则存在埃尔米特矩阵$X_1，\点，X_g$使得$\sum_i A_i\o乘X_i$有一个简单本征值.
Consider a monic linear pencil $L(x) = I - A_1x_1 - \cdots - A_gx_g$ whose coefficients $A_j$ are $d \times d$ matrices. It is naturally evaluated at $g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise to its free locus $Z(L) = \{ X: \det L(X) = 0 \}$. In this article it is shown that the algebras $A$ and $A'$ generated by the coefficients of two linear pencils $L$ and $L'$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L) \subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ induces a homomorphism $A'/{\rm rad} A' \to A/{\rm rad} A$. Since linear pencils are a key ingredient in studying noncommutative rational functions via realization theory, the above results lead to a characterization of all noncommutative rational functions with a given domain. Finally, a quantum version of Kippenhahn's conjecture on linear pencils is formulated and proved: if hermitian matrices $A_1, \dots, A_g$ generate $M_d(\mathbb{C})$ as an algebra, then there exist hermitian matrices $X_1, \dots, X_g$ such that $\sum_i A_i \otimes X_i$ has a simple eigenvalue.