Related papers: Left symmetric algebras from DNA insertion
Neural sequence models exhibit limited compositional generalization ability in semantic parsing tasks. Compositional generalization requires algebraic recombination, i.e., dynamically recombining structured expressions in a recursive…
Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are…
An involution $#$ on an associative ring $R$ is \textit{formally real} if a sum of nonzero elements of the form $r^# r$ where $r \in R$ is nonzero. Suppose that $R$ is a central simple algebra (i.e. $R=M_n(D)$ for some integer $n$ and…
We call an objective function or algorithm symmetric with respect to an input if after swapping two parts of the input in any algorithm, the solution of the algorithm and the output remain the same. More formally, for a permutation $\pi$ of…
The diagonal of a multivariate power series F is the univariate power series Diag(F) generated by the diagonal terms of F. Diagonals form an important class of power series; they occur frequently in number theory, theoretical physics and…
A new method to construct Hamiltonian functions in involution is presented. We show that on left-symmetric algebras a Nijenhuis-tensor is given in a natural manner by the usual right-multiplication. Furthermore we prove that symplectic…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
Zero mode waveguide technology of next generation sequencing demonstrated sequence-dependence of the enzymatic reaction, incorporating a base into the genomic DNA. We show that these experimental results indicate existence of a previously…
The derived algebra of a symmetrizible Kac-Moody algebra $\lie g$ is generated (as a Lie algebra) by its root spaces corresponding to real roots. In this paper, we address the natural reverse question: given any subset of real root vectors,…
Polylogrithmic functions, such as the logarithm or dilogarithm, satisfy a number of algebraic identities. For the logarithm, all the identities follow from the product rule. For the dilogarithm and higher-weight classical polylogarithms,…
A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered…
Cohen and Taylor, following an idea of Plesken, introduced a Lie algebra to the complex group algebra of a finite group and determined its structure, based on the character theory of the group. We show how the definition of this Plesken Lie…
Let $G$ be a group with involution * and $\sigma\colon G\to\{\pm1\}$ a group homomorphism. The map $\sharp$ that sends $\alpha=\sum\alpha_gg$ in a group ring $RG$ to $\alpha^{\sharp}=\sum\sigma(g)\alpha_gg^*$ is an involution of $RG$ called…
We consider the subalgebra $\Delta$ in the group algebra of the symmetric group $G=S_{n_1+\dots+n_\nu}$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\times \dots…
Symmetry is a fundamental tool in the exploration of a broad range of complex systems. In machine learning symmetry has been explored in both models and data. In this paper we seek to connect the symmetries arising from the architecture of…
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and…
We start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring $FR$ and here we consider the right bounded algebra filtration $FU(\mathfrak{g})$…
This work classifies three-dimensional simple evolution algebras over arbitrary fields. For this purpose, we use tools such as the associated directed graph, the moduli set, inductive limit group, Zariski topology and the dimension of the…
We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and…
Non-negative matrix factorization (NMF) has become a popular machine learning approach to many problems in text mining, speech and image processing, bio-informatics and seismic data analysis to name a few. In NMF, a matrix of non-negative…