Li Guo
Central in the Hopf algebra approach to the renormalization of perturbative quantum field theory of Connes and Kreimer is their Algebraic Birkhoff Decomposition. In this tutorial article, we introduce their decomposition and prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give some applications of this decomposition in the study of divergent integrals and multiple zeta values.
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In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is views as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values.
In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.
We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while…
For an arbitrary group, the subgroups form a lattice with order determined by set inclusion. Not every lattice is isomorphic to the subgroup lattice for a group. However, Birkhoff and Frink proved that any compactly generated lattice is isomorphic to a subalgebra lattice for some algebraic structure. An algebraic structure is a set A with operations from A^n to A where n$is a non-negative integer. Although the proof by Birkhoff and Frink is constructive, many of the operations described are not needed for an algebraic structure to represent a given lattice. In this paper we utilize concepts in…
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
We consider Lie algebroids over algebraic spaces (in short we call it as $a$-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over an $a$-space $(X, \mathcal{O}_X)$. This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special $a$-spaces) resembles a sheaf of bialgebras. In the sequel we present a…
Triangular decompositions: Reedy algebras and quasi-hereditary algebras
本文证明有限维Reedy代数等价于具有三角分解的拟遗传代数，并给出Reedy代数的幂等理想刻画。
This survey article is devoted to the notions of purity, algebraic and $Σ$-algebraic compactness, direct sum decompositions, and representation type in the category of modules over a ring. It begins with basic definitions, a brief history, and a discussion of global decomposition problems going back to work of Köthe and Cohen-Kaplansky; these are strongly tied to algebraic compactness. Characterisations of ($Σ$-)algebraically compact modules are presented, as well as the functorial underpinnings on which they are based. In particular, product-compatible functors, matrix functors and $p$-functo…
Lie Algebra Decomposition Classes for Reductive Algebraic Groups in Arbitrary Characteristic
研究任意特征下约化代数群李代数分解类，扩展相关结果，引入列维型分解类，建立非幂零轨道诱导性质，确定良好特征下分解类闭包序覆盖关系。
Let $R$ denote a 2-fir. The notions of F-independence and algebraic subsets of R are defined. The decomposition of an algebraic subset into similarity classes gives a simple way of translating the F-independence in terms of dimension of some vector spaces. In particular to each element $a \in R$ is attached a certain algebraic set of atoms and the above decomposition gives a lower bound of the length of the atomic decompositions of $a$ in terms of dimensions of certain vector spaces. A notion of rank is introduced and fully reducible elements are studied in details.
Dynamics Decomposition of Boolean Networks: An algebraic foundation
研究布尔网络动力学分解，通过赋予可能动力学空间半环结构，实现对布尔网络动力学的系统分解，为网络模块化建立代数基础，引入新数学工具，推动代数方法在网络相关问题中的应用。
We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to ob…
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