Pedro S. Fagundes
The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.
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We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.
We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3\times 3$ upper triangular matrix algebra over an infinite field.
In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear pol…
In this paper we prove that the image of multilinear polynomials evaluated on the algebra $UT_n(K)$ of $n\times n$ upper triangular matrices over an infinite field $K$ equals $J^r$, a power of its Jacobson ideal $J=J(UT_n(K))$. In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for $UT_n(K)$ is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree $r$ in terms of its coefficients. It turns out that the image of a multilinear…
A note on the image of graded multilinear polynomials on upper triangular matrices
研究上三角矩阵代数上分次变量多元多项式的像，证明在恒等分量交换时像总是子空间，并推广到逆极限和直极限。
Values of multilinear graded $*$-polynomials on upper triangular matrices of small dimension
本文研究多重线性分次*-多项式在小维数上三角矩阵上的像，证明其在 UT_2 及非平凡分次 UT_3 上为齐次向量空间，并说明该结论最优，从而表明 L'vov-Kaplansky 猜想在带分次对合的代数中不成立。
Images of linear polynomials on upper triangular matrix algebras
本文描述了在基础域温和条件下上三角矩阵代数上零常数项线性多项式的像，并以此改进了Fagundes-Mello猜想及多重线性分次多项式像的相关结果。
Upper triangular matrices with superinvolution: identities and images of multilinear polynomials
本文完整刻画了UT_4(F)上由序列(0,1,0,1)诱导分次及超辛超级对合的恒等式，并证明UT_n(F)上多重线性多项式像为向量空间当且仅当n≤3，推进了L'vov-Kaplansky猜想的类比研究。
$σ$-matching and interchangeable structures on the strictly upper triangular matrix algebra
本文研究严格上三角矩阵代数UT_n(K)上的σ-匹配、可互换及完全兼容结构，并对所有n≥3的情形给出了完整刻画。
In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of …
We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic $\ne 2$. It is proved that the algebras of upper triangular $3\times 3$ or $4\times 4$ matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding $6$ contains a non-commutative local centrally essential subalgebra. The paper will appear in Linear and Multilinear Algebra. The work of O.V. Lyubimtsev is done under the state assignment No~0729-2020-0055. A.A. Tuganbaev is supported by Russian Scientif…
Power commuting and centralizing maps on the ring of strictly upper triangular matrices
本文研究了严格上三角矩阵环上满足特定交换和中心化条件的映射性质，揭示了2幂交换映射与中心化映射的结构特征。
An element $a$ in a Banach algebra $\mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $ab=ba, b=bab$ and $a-a^2b \in \mathcal{A}^{qnil}$. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix $\left( \begin{array}{cc} E&I F&0 \end{array} \right)$. We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear $\&$ Multilinear Algebra, 14(1983), 195--198].
Bilinear maps on the ring of strictly upper triangular matrices
该研究在严格上三角矩阵环的背景下，推广了满足交换条件的线性映射的已有结论，完整刻画了满足对所有矩阵$X$都有$[f(X,X),X]=0$的双线性映射的结构。
Let ${\mathcal N}$ be the Lie algebra of all $n\times n$ strictly block upper triangular matrices over a field ${\mathbb F}$ relative to a given partition. In this paper, we give an explicit description of all derivations of ${\mathcal N}$.
We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.
The $2\times 2$ upper triangular matrix algebra and its generalized polynomial identities
本文研究特征零域上 $2\ imes 2$ 上三角矩阵代数 $UT_2$ 的广义多项式恒等式，确定其生成元、计算广义余维数，并用杨图描述多重线性恒等式空间，同时证明其广义簇无几乎多项式增长但给出两个例外。
We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of $\mathbb ZS_n$-submodules in the $\mathbb ZS_n$-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding $\mathbb ZS_n$-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper tr…
Makar-Limanov's problem on values of polynomials on matrices
该研究通过构造特定矩阵，证明了非交换多项式在矩阵上的值可构造上三角矩阵，并探讨了其特征值的代数无关性。
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