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We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of…
Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families…
We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra…
A Novikov conformal algebra is a conformal algebra such that its coefficient algebra is right-symmetric and left commutative (i.e., it is an ``ordinary'' Novikov algebra). We prove that every Novikov conformal algebra with a uniformly…
We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module…
Let $K$ be an algebraically closed field with characteristic zero, and $\mathfrak{g}$ a Lie algebra. Let $Y(\mathfrak{g})$ be the subalgebra of the symmetric algebra $S(\mathfrak{g})=K[\mathfrak{g}^*]$ made of the polynomials which are…
The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are…
We introduce a symmetric operad whose algebras are the Operator Product Expansion (OPE) Algebras of quantum fields. There is a natural classical limit for the algebras over this operad and they are commutative associative algebras with…
Family algebraic structures indexed by a semigroup arise naturally in renormalizations of quantum field theory. In this paper, we first define the notion of $\Omega$-associative $H$-pseudoalgebra, where the operations are indexed by pairs…
An antithetical concept, adaptive symmetry, to conservative symmetry in physics is proposed to understand the deep neural networks (DNNs). It characterizes the invariance of variance, where a biotic system explores different pathways of…
The linear span P_n of the sums of all permutations in the symmetric group S_n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D_n; and the…
There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ to the Weyl-Clifford superalgebra $W(2n|n)$ with $2n$ even Weyl algebra generators and $n$ odd…
Many successful deep learning architectures are equivariant to certain transformations in order to conserve parameters and improve generalization: most famously, convolution layers are equivariant to shifts of the input. This approach only…
In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry…
Symmetry in integer programming causes redundant search and is often handled with symmetry breaking constraints that remove as many equivalent solutions as possible. We propose an algebraic method which allows to generate a random family of…
We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra…
A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e. even or odd) and associative bilinear form on A. In this paper, we show that any associative superalgebra with non…
Deep learning was recently successfully used in deriving symmetry transformations that preserve important physics quantities. Being completely agnostic, these techniques postpone the identification of the discovered symmetries to a later…
Many numerical algorithms in scientific computing -- particularly in areas like numerical linear algebra, PDE simulation, and inverse problems -- produce outputs that can be represented by semialgebraic functions; that is, the graph of the…
The notion of an F-manifold algebra is the underlying algebraic structure of an $F$-manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F-manifold algebras are the corresponding…