Related papers: The Convolution Algebra
In this paper, we introduce the comaximal graph $\Gamma(L)$ of a finite-dimensional Lie algebra $L$, whose vertices are the nontrivial proper Lie subalgebras of $L$ over a field $\mathbb{F}$, and two vertices $A$ and $B$ are adjacent if and…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
Loday's dendriform algebras and its siblings pre-Lie and zinbiel have received attention over the past two decades. In recent literature, there has been interest in a generalization of these types of algebra in which each individual…
We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…
From a Lie algebra $\mathfrak{g}$ satisfying $\mathcal{Z}(\mathfrak{g})=0$ and $\Lambda^2(\mathfrak{g})^\mathfrak{g}=0$ (in particular, for $\g$ semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form…
Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case…
Neighborhood regression has been a successful approach in graphical and structural equation modeling, with applications to learning undirected and directed graphical models. We extend these ideas by defining and studying an algebraic…
We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for…
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand…
We construct a three parameter deformation of the Hopf algebra $\mathbf{LDIAG}$. This new algebra is a true Hopf deformation which reduces to $\mathbf{LDIAG}$ on one hand and to $\mathbf{MQSym}$ on the other, relating $\mathbf{LDIAG}$ to…
In this paper, we provide a unified approach to study the cohomology theories and deformation theories of various types of operators in the category of Lie algebras, including modified $r$-matrices, crossed homomorphisms, derivations,…
In this paper, we extend the concept of Lie superalgebras to a more generalized framework called Super-Lie superalgebras. In addition, they seem to be exploring various supergeneralizations of other algebraic structures, such as…
Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped…
Let F be a non-Archimedean local field and let G be a connected reductive affine algebraic F-group. Let I be an Iwahori subgroup of G(F) and denote by H(G; I) the Iwahori-Hecke algebra, i.e. the convolution algebra of complex-valued…
Let $T_X$ be the full transformation monoid over a finite set $X$, and fix some $a\in T_X$ of rank $r$. The variant $T_X^a$ has underlying set $T_X$, and operation $f\star g=fag$. We study the congruences of the subsemigroup $P=Reg(T_X^a)$…
In this series of two papers we will generalise the concept of extending a Lie algebroid by a Lie algebra bundle, leading to a notion of extending a Lie algebroid by another Lie algebroid whose orbits lie in the orbits of the former…
The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of $n$-Lie algebras and superalgebras, commutative Leibniz $n$-ary algebras and superalgebras, among others. We focus on the class of…
We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(\alpha)$ at the point $\alpha$ corresponds to the characteristic space associated to the left invariant…