Universal Lyndon Words

A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon words}, which are words over an $n$-letter alphabet that have length $n!$ and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every $n$ and exhibit combinatorial and structural properties of these words. We then define particular prefix codes, which we call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in bijection with the set of the shortest unrepeated prefixes of the conjugates of a universal Lyndon word. This allows us to give an algorithm for constructing all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201

The Rough Veronese variety

We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Amendola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics

Algebras defined by Lyndon words and Artin-Schelter regularity

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially we do not impose restrictions on $W$ and investigate the case when all algebras in $\mathfrak{C} (X, W)$ have polynomial growth and finite global dimension $d$. Next we consider classes $\mathfrak{C} (X, W)$ of algebras whose sets of obstructions $W$ are antichains of Lyndon words. The central question is "when a class $\mathfrak{C} (X, W)$ contains Artin-Schelter regular algebras?" Each class $\mathfrak{C} (X, W)$ defines a Lyndon pair $(N,W)$ which determines uniquely the global dimension, $gl\dim A$, and the Gelfand-Kirillov dimension, $GK\dim A$, for every $A \in \mathfrak{C}(X, W)$. We find a combinatorial condition in terms of $(N,W)$, so that the class $\mathfrak{C}(X, W)$ contains the enveloping algebra $U\mathfrak{g}$ of a Lie algebra $\mathfrak{g}$. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Groebner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimensions $6$ and $7$ occurring as enveloping $U = U\mathfrak{g}$ of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs $(N, W)$, each of which determines also the explicit relations of $U$.Comment: 49 page

The homotopy type of the loops on (n−1)(n-1)-connected (2n+1)(2n+1)-manifolds

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math., Birkhauser/Springer, 2018. arXiv admin note: text overlap with arXiv:1510.0519