经典三值代数与量子三值代数

三元霍普夫代数的基本概念

Basic concepts of ternary Hopf algebras

三元半群、群和代数的理论用抽象箭头语言重新表述。然后利用反向箭头定义了三元余乘、双代数和霍普夫代数，并研究了它们的性质。的主要属性“二进制派生”被认为是详细的。给出了Post定理的共类比形式。结果表明，存在3种类型的三元余结合，3种类型的三元计数器和2种类型的三元对极。文中还列举了一些实例。

Ternary Z2 x Z3 graded algebras and ternary Dirac equation

The wave equation generalizing the Dirac operator to the Z3-graded case is introduced, whose diagonalization leads to a sixth-order equation. It intertwines not only quark and anti-quark state as well as the "u" and "d" quarks, but also the three colors, and is therefore invariant under the product group Z2 x Z2 x Z3. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry and of the SU(2) x U(1) that arise …

Clifford代数的$Z_3$ -分次类似及超对称性的推广

$Z_3$-graded analogues of Clifford algebras and generalization of supersymmetry

定义并研究了Clifford代数的三元类似代数。证明了具有$N$生成元的三值Clifford代数与具有$N+1$生成元的三值Clifford代数的零级元素的子代数同构.在$N=3$情形下，导出了由零级元素的三元交换子导出的立方矩阵的三元交换子。利用单生成元的三元Clifford代数构造了最简单超对称代数的$Z_3$ -分次推广.

Grassmann代数的三值推广

Ternary generalizations of Grassmann algebra

提出了经典反交换性的三元推广，并研究了生成元是三元反交换的代数。定义了具有任意生成元数N的代数上的积分，并证明了变量变换公式。与费米子积分类似，我们用高斯型积分定义了立方矩阵的Pfizer积分，并计算了N=3时的显式形式。

Algebras with ternary law of composition combining Z_2 and Z_3 gradings

In the present article we investigate the possibility of combining the usual Grassmann algebras with their ternary Z_3-graded counterpart, thus creating a more general algebra with coexisting quadratic and cubic constitutive relations. We recall the classification of ternary and cubic algebras according to the symmetry properties of ternary products under the action of the S_3 permutation group. Instead of only two kinds of binary algebras, symmetric or antisymmetric, here we get four different generalizations of each of these two cases. Then we study a particular case of algebras generated by…

三值代数的必要条件

Necessary conditions for Ternary Algebras

三值代数，由三值算子构成，或者我们称之为三值算子，定义为三个算子的乘积的交替和，已经证明满足三次恒等式作为它们存在的必要条件。在这里，我们研究的情况下，我们允许的身份，而不仅仅是从ternutator或嵌套的ternutator构建，我们发现，在一般情况下，这些施加额外的限制，例如，反的算子或算子必须遵守一些线性关系。

Ternary algebras associated with irreducible tensor representations of SO(3) and quark model

We show that each irreducible tensor representation of weight 2 of the rotation group of three-dimensional space in the space of rank 3 covariant tensors gives rise to an associative algebra with unity. We find the algebraic relations that the generators of these algebras must satisfy. Part of these relations has a form of binary relations and another part has a form of ternary relations. The structure of ternary relations is based on the cyclic group Z_3 and the primitive cubic root of unity q=\exp(2πi/3). The subspace of each algebra spanned by the triple products of generators is 5-dimensio…

Chern-Simons规范理论的新规范不变量表述：经典和量子分析

New Gauge Invariant Formulation of the Chern-Simons Gauge Theory: Classical and Quantal Analysis

最近提出的新的规范不变陈-西蒙斯规范理论的提法被详细考虑。该公式与规格固定公式一致。此外，我们还发现正则（Noether）Poincaré生成元在约束曲面上也不是规范不变的，与通常情况不同，它们不满足Poincaré代数。它是由对称能量动量张量构造的改进生成元，它（明显）是规范不变的，并且服从量子和经典庞加莱代数。在薛定谔图像中发现，规范场的规范不变纵模对于构造（纯）Chern-Simons理论的物理波泛函是至关重要的。在与规范固定公式的匹配中，我们考虑了三种典型的规范，库仑规范，轴向规范和Weyl规范作为显式例子。澄清了最近关于狄拉克缀饰函数和规范固定效应的几个混淆。根据旧的规范无关公式a' la狄拉克的分析总结在附录中。

三体可积系统的隐代数

Hidden Algebra of Three-Body Integrable Systems

结果表明，所有的三体量子可积系统中出现的哈密顿约化方法具有相同的隐藏代数结构。它们都是由微分算子的无限维李代数的生成元中的二次多项式给出的。它导致新的家庭的正交多项式在两个变量。