arXiv subjects

Yunnan Li

Publications and source records attributed to Yunnan Li.

31 records · Page 2Linked to original sources

arXiv2020

Coherent unit actions on braided operads and Hopf algebras

The notion of a coherent unit action on algebraic operads was first introduced by Loday for binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the free objects of the operads carry a Hopf algebra structure. There was also a classification of such operads in the binary quadratic nonsymmetric case. We generalize the notion of coherent unit action to braided operads and show that free objects of braided operads with such an action carries a braided Hopf algebra structure. Under the conditions of binary, quadratic and nonsymmetric, we give a characterization and classification of the braided operads that allow a coherent unit action and thus carry a braided Hopf algebra structure on their free objects.

math.QA↗

arXiv2020

Cohomologies, extensions and deformations of differential algebras with any weights

As an algebraic study of differential equations, differential algebras have been studied for a century and and become an important area of mathematics. In recent years the area has been expended to the noncommutative associative and Lie algebra contexts and to the case when the operator identity has a weight in order to include difference operators and difference algebras. This paper provides a cohomology theory for differential algebras of any weights. This gives a uniform approach to both the zero weight case which is similar to the earlier study of differential Lie algebras, and the non-zero weight case which poses new challenges. As applications, abelian extensions of a differential algebra are classified by the second cohomology group. Furthermore, formal deformations of differential algebras are obtained and the rigidity of a differential algebra is characterized by the vanishing of the second cohomology group.

math.RA↗

arXiv2019

Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees

This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra of Loday and Ronco on planar rooted trees.

math.QA↗

arXiv2019

Braided Rota-Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras

Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota-Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota-Baxter algebras and braided dendriform algebras, under the commutativity condition. We further generalize the notion of dendriform Hopf algebras to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative Rota-Baxter algebras of braided commutative dendriform algebras are obtained.

math.QA↗

arXiv2017

On weak peak quasisymmetric functions

In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting Stembridge's peak functions. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient algebra of WCQSym. On the other hand, we prove that WPQSym embeds as a Rota-Baxter subalgebra of WCQSym, thus of the free commutative Rota-Baxter algebra of weight 1 on one generator. Moreover, WPQSym can also be a Rota-Baxter quotient algebra of WCQSym.

math.CO↗

arXiv2016

The distance spectra of the derangement graphs

In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all their extremal distance eigenvalues.

math.CO↗

arXiv2016

Towards a polynomial basis of the algebra of peak quasisymmetric functions

Hazewinkel proved the Ditters conjecture that the algebra of quasisymmetric functions over the integers is free commutative by constructing a nice polynomial basis. In this paper we prove a structure theorem for the algebra of peak quasisymmetric functions (PQSym) over the integers. It provides a polynomial basis of PQSym over the rational field, different from Hsiao's basis, and implies the freeness of PQSym over its subring of symmetric functions spanned by Schur's Q-functions.

math.CO↗

arXiv2015

Representation theory of 0-Hecke-Clifford algebras

The representation theory of 0-Hecke-Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron et al. have proved that the Grothendieck ring of the category of finitely generated supermodules of 0-Hecke-Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by Stembridge. In this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups. In particular, two kinds of restriction rules for induced projective supermodules are obtained. After that, we consider the corresponding Heisenberg double and its Fock representation to prove that the ring of peak quasisymmetric functions is free over the subring of symmetric functions spanned by Schur's Q-functions.

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arXiv2015

A lift of Schur's Q-functions to the peak algebra

We construct a lift of Schur's Q-functions to the peak algebra of the symmetric group, called the noncommutative Schur Q-functions, and extract from them a new natural basis with several nice properties such as the positive right-Pieri rule, combinatorial expansion, etc. Dually, we get a basis of the Stembridge algebra of peak functions refining Schur's P-functions in a simple way.

math.CO↗

arXiv2014

The shifted Poirier-Reutenauer algebra

Based on the shifted Schensted correspondence and the shifted Knuth equivalence, a shifted analog of the Poirier-Reutenauer algebra as a higher lift of Schur's P-functions and a right coideal subalgebra of the Poirier-Reutenauer algebra is constructed. Its close relations with the peak subalgebra and the Stembridge algebra of peak functions are also uncovered.

math.CO↗

arXiv2014

The Green rings of the 2-rank Taft algebra and its two relatives twisted

In the paper, the representation rings (or the Green rings) for a family of Hopf algebras of tame type, the 2-rank Taft algebra (at $q=-1$) and its two relatives twisted by 2-cocycles are explicitly described via a representation theoretic analysis. It turns out that the Green rings can serve to detect effectively the twist-equivalent Hopf algebras here.

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arXiv2013

Multi-parameter Quantum Groups via Quantum Quasi-Symmetric Algebras

It is proved that the entire multi-parameter (small-)quantum groups of symmetrizable Kac-Moody algebras can be realized as certain subquotients of the cotensor Hopf algebras. This is an axiomatic construction. Hopf 2-cocycle deformations variation for the construction machinery is described, moreover, the integrable irreducible modules can be constructed using this setting, even available for those in the fundamental alcove at root of unity case.

math.QA↗

arXiv2013

q-symmetric functions and q-quasisymmetric functions

In this paper, we construct the q-analogue of Poirier-Reutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined in \cite{EK}, and naturally obtain the odd Littlewood-Richardson rule concerned in \cite{Ell}. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration in \cite{HLMW1}. All the q-Hopf algebras we discuss here provide the corresponding q-dual graded graphs in the sense of \cite{BLL}.

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