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Extra resources for A treatise on quantum Clifford algebras
Hence we identify with the branching product is extended to ✑ ❛ coefficients. It would be misleading to recognize ‘cliffordization’ as closely tied to ‘Clifford’ algebras. Rota and Stein showed in Refs. g. the Littlewood-Richardson rule emerges as a special case. Cliffordization provides a direct and computational very efficient approach to various product formulas of deformed structures. The language of cliffordization is that of Hopf gebras. The above example using Young tableaux is not far away from our topic.
T. an underlying Graßmann algebra. The question if such representations are equivalent is known as isomorphy problem in the theory of group presentations . g. using CLIFFORD , non grade preserving transformations of generators. This is well known from the group theory. g. t. t. ✾ . This observation is crucial for any attempt to identify algebraic expressions with geometric objects. The same will hold in QFT when identifying operator products. 4 Clifford algebras by factorization Clifford algebras can be approached in a basis free manner which for obvious reasons avoids the problems discussed in the previous section.
The Clifford map ➧ ❅ introduced by Chevalley is a mapping ➢ ✸❦✆✓✒ ❆❅ ➧ ➢ ✆❊✼ ✆ (2-49) and thus quite asymmetric in the structure of its factors. Stressing an analogy, we will call the process induced by the Clifford map as Pieri formula of Clifford algebra. g. ➮ ✄✝➮ ➮ ✡ ✤ ❨ ✲✴✲✴✲❈❨ . Denote a partition of the natural number ❻ into ➙ parts as ✩ ➣④❨ ❲ , ➮ ✩ ❻ . Young operators can be constructed which are projection operators allowing a with ❮ ❡ decomposition of the representation space. The formulas which allow to add one box (possibly in each row) to a Young tableau is a Pieri formula ⑥⑤ ⑤ ➊ ✤ ♣ ④æ ♣ ➱✚Ø➼Ø➼Ø ✃ ➋ ✩ ✿ ❞ ✾ ✼ ✼ ♣ ⑤❀⑦ (2-50) where runs over all partitions of the standard Young tableaux obtained by adding the box.