标题： 复数李，实物理：代数复化的作用 显示英文标题

标题： Complex Lies, Real Physics: The Role of Algebra Complexification

摘要： 在物理学中，李群表示描述给定系统对称变换的代数结构。 然后，这些群的下降李代数是必要的实数。 在大多数情况下，为了推导李代数的不可约表示，进而推导对称群的不可约表示，需要对这些李代数进行复化。 在本文中，我们给出了该概念的精确定义，并逐步证明了一个重要的结果$$\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}. $$。这个结果用于确定适当洛伦兹群的不可约表示，从而确定当存在这种对称性时允许的物理对象。 结果显示，适当洛伦兹群的有限表示由一对半整数$(j_1,j_2)$表征，这些半整数明确地决定了与给定表示相关的物理对象。 例如，维度为$1$的表示$(0,0)$称为标量表示，它对应于希格斯场，而维度为$4$的表示$(\frac{1}{2},0) \oplus (0,\frac{1}{2})$称为狄拉克旋量表示，它对应于称为费米子的物质粒子。 这意味着根据这种代数结构，数学群结构决定了宇宙的物质内容。 显示英文摘要

摘要： In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessary real. In most cases, the complexification of those Lie algebra is necessary in order to derive irreductible representations of the Lie algebra and subsequently of the symmetry group. In this paper, we give a precise definition of the concept and prove step by step an important result $$\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}. $$ This result is used to determine the irreductible representations of the proper Lorentz group and thus the physical objects admissible when this symmetry is present. It is shown that finite representations of the proper Lorentz group are characterized by pairs of half-integers $(j_1,j_2)$, which determine unambiguously the physical object associated to the given representation. For example, the representation $(0,0)$ of dimension $1$ is called the scalar representation, it corresponds to the Higgs field, and $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of dimension $4$ is called the Dirac spinor representation, it corresponds to matter particle called fermions. This means that the mathematical group structure determines the material content of the universe following this algebraic structure.