On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) Boolean independence and OVI monotone independence. Then we show that OVI Boolean (respectively, monotone) independence is equivalent to the operator-valued (OV) Boolean (respectively, monotone) independence over an algebra of [Formula: see text] upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (respectively, monotone) additive convolution by reducing it to the OV case. We also define OVI Boolean and monotone cumulants and study their basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence.

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

Jordan triple homomorphisms on $${\mathcal {T}}_\infty (F)$$

Let $${\mathcal {T}}_\infty (F)$$ T ∞ ( F ) be the algebra of all $${\mathbb {N}}\times {\mathbb {N}}$$ N × N upper triangular matrices defined over a field F of characteristic different from 2. We consider the Jordan triple homomorphisms of $${\mathcal {T}}_\infty (F)$$ T ∞ ( F ) , i.e. the additive maps that satisfy the condition $$\phi (xyx)=\phi (x)\phi (y)\phi (x)$$ ϕ ( x y x ) = ϕ ( x ) ϕ ( y ) ϕ ( x ) for all $$x,y\in {\mathcal {T}}_\infty (F)$$ x , y ∈ T ∞ ( F ) . For the case when F is a prime field we find the form of all such maps $$\phi $$ ϕ . For the general case we present the form of the surjective maps $$\phi $$ ϕ .

Functions Preserving Orthogonality of Hermitian Matrices

Inspired by the results that functions preserve orthogonality of full matrices, upper triangular matrices, and symmetric matrices. We finish the work by finding special orthogonal matrices which satisfy the conditions of preserving orthogonality functions. We give a characterization of functions preserving orthogonality of Hermitian matrices.

The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices

We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.

Freeness Problem for Matrix Semigroups of Parikh Matrices

Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

Regimes and mechanisms of transient amplification in abstract and biological networks

We use upper triangular matrices as abstract representations of neuronal networks and directly manipulate their eigenspectra and non-normality to explore different regimes of transient amplification. Counter-intuitively, manipulating the imaginary distribution can lead to highly amplifying regimes. This is noteworthy, because biological networks are constrained by Dale's law and the non-existence of neuronal self-loops, limiting the range of manipulations in the real dimension. Within these constraints we can further manipulate transient amplification by controlling global inhibition.