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Chengming Bai

Publications and source records attributed to Chengming Bai.

83 records · Page 5Linked to original sources
arXiv2009

${\mathcal O}$-operators of Loday algebras and analogues of the classical Yang-Baxter equation

We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and necessary condition on the existence of the $2^{n+1}$ operations on an algebra with the $2^{n}$ operations in an associative cluster. The analogues of the classical Yang-Baxter equation in these algebras can be understood as the $\mathcal O$-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.

math-ph↗
arXiv2009

Pre-alternative algebras and pre-alternative bialgebras

We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform dialgebra or a pre-Lie algebra. The left and right multiplication operators of a pre-alternative algebra give a bimodule structure of the associated alternative algebra. There exists a (coboundary) bialgebra theory for pre-alternative algebras, namely, pre-alternative bialgebras, which exhibits all the familiar properties of the famous Lie bialgebra theory. In particular, a pre-alternative bialgebra is equivalent to a phase space of an alternative algebra and our study leads to what we called $PA$-equations in a pre-alternative algebra, which are analogues of the classical Yang-Baxter equation.

math-ph↗
arXiv2008

Principal Realization of the Yangian Y(gl(n))

Motivated to simplify the structure of tensor representations we give a new set of generators for the Yangian $Y(sl(n))$ using the principal realization in simple Lie algebras. The isomorphism between our new basis and the standard Cartan-Weyl basis is also given. We show by example that the principal basis simplifies the Yangian action significantly in the tensor product of the fundamental representation and its dual.

math.QA↗
arXiv2008

Some Non-Abelian Phase Spaces in Low Dimensions

A non-abelian phase space, or a phase space of a Lie algebra is a generalization of the usual (abelian) phase space of a vector space. It corresponds to a parakähler structure in geometry. Its structure can be interpreted in terms of left-symmetric algebras. In particular, a solution of an algebraic equation in a left-symmetric algebra which is an analogue of classical Yang-Baxter equation in a Lie algebra can induce a phase space. In this paper, we find that such phase spaces have a symplectically isomorphic property. We also give all such phase spaces in dimension 4 and some examples in dimension 6. These examples can be a guide for a further development.

math-ph↗
arXiv2007

From Rota-Baxter Algebras to Pre-Lie Algebras

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension $\leq 3$ and their corresponding pre-Lie algebras.

math-ph↗
arXiv2007

Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras

Left-symmetric algebras have close relations with many important fields in mathematics and mathematical physics. Their classification is very complicated due to the nonassociativity. In this paper, we re-study the correspondence between left-symmetric algebras and the bijective 1-cocycles. Then a procedure is provided to classify left-symmetric algebras in terms of classification of equivalent classes of bijective 1-cocycles. As an example, the 3-dimensional complex left-symmetric algebras are classified.

math.QA↗
arXiv2007

Classification of Graded Left-symmetric Algebra Structures on Witt and Virasoro Algebras

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they include the examples given by Chapoton and Kupershmidt. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebra structures on the Virasoro algebra. They coincide with the examples given by Kupershmidt.

math.QA↗
arXiv2007

Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "$S$-equation", which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the $S$-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "${\cal O}$-operator", whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${\cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakähler Lie algebras) from ${\cal O}$-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson-Lie groups in this sense.

math.QA↗
arXiv2007

A Unified Algebraic Approach to Classical Yang-Baxter Equation

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parakähler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a classical r-matrix.

math.QA↗
arXiv2004

Left-symmetric Algebras From Linear Functions

In this paper, some left-symmetric algebras are constructed from linear functions. They include a kind of simple left-symmetric algebras and some examples appearing in mathematical physics. Their complete classification is also given, which shows that they can be regarded as generalization of certain 2-dimensional left-symmetric algebras.

math.QA↗