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Register a new userBases of Quantum Group Algebras in Terms of Lyndon Words
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Authors
Eremey Valetov
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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 construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the root of unity is non-restricted. As an additional, applied part of this work, we have implemented a Wolfram Mathematica package with functions for quantum shuffle multiplication and constructions of bases in terms of Lyndon words.
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We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations.
In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a relaxation of the condition used in the case of finite words. In the annex, we prove that the factorization of a rational word has a special form and that it can be computed from a rational expression describing the word.
We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra $mathcal E_n$ and Nichols algebra $mathfrak{B} ({mathcal O}_{{(1, 2)}} , epsilon otimes {rm sgn})$ of transposition over symmetry group by means of quiver Hopf algebras. We generalize {rm FK } algebra. The characteristic of finiteness of Nichols algebras in thirteen ways and of {rm FK } algebras ${mathcal E}_n$ in nine ways is given. All irreducible representations of finite dimensional Nichols algebras %({rm FK } algebras ${mathcal E}_n$) and a complete set of hard super- letters of Nichols algebras of finite Cartan types are found. The sufficient and necessary condition for Nichols algebra $mathfrak B(M)$ of reducible {rm YD} module $M$ over $A rtimes mathbb{S}_n$ with ${rm supp } (M) subseteq A$ to be finite dimensional is given. % Some conditions for a braided vector space to become a {rm YD} module over finite commutative group are obtained. It is shown that hard braided Lie Lyndon word, standard Lyndon word, Lyndon basis path, hard Lie Lyndon word and standard Lie Lyndon word are the same with respect to $ mathfrak B(V)$, Cartan matrix $A_c$ and $U(L^+)$, respectively, where $V$ and $L$ correspond to the same finite Cartan matrix $A_c$.
A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincreasing factorization of generalized Lyndon words. We give new characterizations of the first and the last factor in this factorization as well as new characterization of generalized Lyndon words. We also give more specific results on two special cases: the classical one and the one arising from the alternating lexicographical order.
In this paper we compare two finite words $u$ and $v$ by the lexicographical order of the infinite words $u^omega$ and $v^omega$. Informally, we say that we compare $u$ and $v$ by the infinite order. We show several properties of Lyndon words expressed using this infinite order. The innovative aspect of this approach is that it allows to take into account also non trivial conditions on the prefixes of a word, instead that only on the suffixes. In particular, we derive a result of Ufnarovskij [V. Ufnarovskij, Combinatorial and asymptotic methods in algebra, 1995] that characterizes a Lyndon word as a word which is greater, with respect to the infinite order, than all its prefixes. Motivated by this result, we introduce the prefix standard permutation of a Lyndon word and the corresponding (left) Cartesian tree. We prove that the left Cartesian tree is equal to the left Lyndon tree, defined by the left standard factorization of Viennot [G. Viennot, Alg`ebres de Lie libres et monoides libres, 1978]. This result is dual with respect to a theorem of Hohlweg and Reutenauer [C. Hohlweg and C. Reutenauer, Lyndon words, permutations and trees, 2003].
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