Graphical tensor product reduction scheme for the Lie algebras , , and

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• 作者：N.D. Vlasii^a ; ^b ; ^{vlasii@itp.unibe.ch" class="auth_mail" title="E-mail the corresponding author} ; F. von Rü ; tte^a ; U.-J. Wiese^a
• 关键词：Simple Lie algebras ; Tensor product reduction ; Graphical method ; Girdle method
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：371
• 期：Complete
• 页码：199-227
• 全文大小：1813 K

文摘

We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5)=sp(2)$so\left(5\right)=sp\left(2\right)$, su(3)$su\left(3\right)$, and g(2)$g\left(2\right)$. This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a “landscape” of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic “girdle” method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.