Related papers: Monomial algebras defined by Lyndon words

We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the…

We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…

Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of…

We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the…

It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…

It is shown that the ${\cal N}=1$ supersymmetric quantum mechanics (SQM) can be extended to a $\mathbb{Z}_2^n$-graded superalgebra. This is done by presenting quantum mechanical models which realize, with the aid of Clifford gamma matrices,…

Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some…

Let $S_\alpha = k\langle x_1,\dots,x_n\rangle /(x_i x_j - \alpha_{ij} x_j x_i)$ be a standard graded skew polynomial algebra over an algebraically closed field $k$ of characteristic not equal to $2$. We show the following results. When $n$…

Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to usual Clifford algebra. In turn multicomplex…

It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the…

We study criteria for deciding when the normal subgroup generated by a single polynomial automorphism of $\mathbb{A}^n$ is as large as possible, namely equal to the normal closure of the special linear group in the special automorphism…

There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety $\mathrm{Perm}_n$ as computed by Klyachko. Toward understanding this better, we…

We classify all non-abelian groups G such that there exists a pair (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional under two assumptions: the square…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

A foundational result in the theory of Lyndon words (words that are strictly earlier in lexicographic order than their cyclic permutations) is the Chen-Fox-Lyndon theorem which states that every word has a unique non-increasing…

We define and compute explicitly the classical limit of the realizations of $W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The classical limit is obtained by taking the commutative limit of the ring of…

We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group…

By a generalized Delsarte polynomial we mean a Laurent polynomial whose exponent vectors are linearly independent. We consider certain monomial deformations of generalized Delsarte polynomials and study their associated differential…

We find a generalization of the restricted PBW basis for pointed Hopf algebras over abelian groups constructed by Kharchenko. We obtain a factorization of the Hilbert series for a wide class of graded Hopf algebras. These factors are…