与SO（3）和夸克模型不可约张量表示相关的三元代数
Ternary algebras associated with irreducible tensor representations of SO(3) and quark model
摘要：
证明了三维空间的旋转群在秩为3的协变张量空间中的每个权为2的不可约张量表示都产生一个具有单位元的结合代数。我们发现这些代数的生成元必须满足的代数关系。这些关系的一部分具有二元关系的形式，另一部分具有三元关系的形式。三元关系的结构基于循环群Z_3和单位的原立方根q= \exp（2πi/3）。生成元的三重积所张成的每个代数的子空间是5维的，并且它是旋转群SO（3）的权为2的不可约张量表示的空间。我们在这个5维子空间中定义了一个厄米特标量积，并构造了它的一个正交基，然后求出了一个无穷小旋转的表示矩阵。我们表明，构造代数的二元和三元关系可以有应用的夸克模型和大统一理论。
英文摘要：
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-dimensional and it is the space of an irreducible tensor representation of weight 2 of the rotation group SO(3). We define a Hermitian scalar product in this 5-dimensional subspace and construct an orthonormal basis for it. Then we find the representation matrix of an infinitesimal rotation. We show that constructed algebras with binary and ternary relations can have applications in the quark model and Grand Unification Theories.
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