关于任意双线性型的Clifford代数的分解
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
摘要：
Clifford代数自然地与二次型联系在一起。这些代数通过构造是Z_2-分次的。然而，只有通过基的选择，或者更好地，通过Chevalley向量空间同构Cl（V）<->\bigwedge V和排序，才能保证多向量分解为标量、向量、张量等，这在物理学中是强制性的。我们证明了Chevalley同构定理不能推广到代数，如果Z_n-分次或其他结构被添加，例如，线性形式。我们研究由Clifford代数和线性形式或Z_n-分次组成的对，我们现在称之为“多向量Clifford代数”或“量子Clifford代数”。结果表明，在这个意义下，所有具有相同二次型但不同双线性型的多向量Clifford代数是非同构的。这种代数在量子场论和超导性中的有用性在其他地方得到了证明。然而，允许任意的双线性型破坏了它们的可对角化性，这对由周期性定理（包括Atiyah-Bott-Shapiro mod 8周期性）所支配的Clifford代数的张量分解有相当大的影响。我们考虑实代数Cl_{p,q}，它可以在对称的情况下分解成张量积Cl_{p-1,q-1} \otimes Cl_{1,1}。在量子场论中使用的一般情况缺少这个特征。然而，在相互作用理论中分析多粒子态时，需要具有非对称双线性型的理论。通过非平凡的真空态在量子理论中的q-变形结构的连接概述。
英文摘要：
Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) <-> \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford algebras'. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1} \otimes Cl_{1,1}. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q-deformed structures through nontrivial vacuum states in quantum theories is outlined.
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