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Internal names (indices) have no particular meaning at all. g. ✄ ✡ that of groups, simply double the entry element Õ ✾ ✩ ✾r➛➹✾ . e. co-associativity, is derived from the associativity if we replace ➏➑➐ and ❿ ➏ Õ and it reads: Õ ✩ Õ (3-31) Õ Õ 40 A Treatise on Quantum Clifford Algebras If a product has an unit, this is pictographically represented as ✄ ❿ ❿ ✩ ✄ and dualizing yields the definition of a counit ✩ (3-32) ❶ ❶ ✩ Õ ✩ Õ (3-33) ❶ Ð A further prominent structure element is the antipode , an endomorphism, which, if it exists, fulfils the following defining relations Ð ❿ Õ ❶ ✄ ✩ ✩ ❿ Õ (3-34) Ð ✂ Ð ❇ ✆ ✶ ➅➠ .

Two extensors × ✄ ❣ ✤ æ ✡ ✩ ❣ ❛ ➜❃❣ ➒ . As an example we compute ✖ ✩ ✄ ✡ ✙ ❣ ✤ æ ❣ ❛ ❩ æ ❣ ➒ ❧⑥❣ ➒ æ❩❣ ❛ ✖ ✄ ❋ ❋ ❣ ✤ ❩ æ ❣ ❛ ❩ æ ❣ ➒ ❣ ❛ r æ ❣ ➒ r æ ❣ ✤ ✩ ❧⑥❣ ✤ æ❩❣ ❛ æ❩❣ ➒ ⑥ ❧ ❣ ❛ ær❣ ➒ ær❣ ✤ ❧⑥❣ ➒ ær❣ ✤ ær❣ ❛ ✙ ✙ ❋ ✠ ✠ ✠ ❋ ✠ ✤ ❛✌❣✂➒②❧ ✤ ➒✂❣✳❛▲❧ ✤✴ ✤ ✌ ➒ ❣✳❛ ❛ ❣✂➒ ❋④✠ ✠ ✤ ❛✌❣✂➒●❧ ✤ ➒✴❣✳❛ ❣ ✤ ➜➀❣✳❛■➜➀❣✌➒ ✳ ✵✴✳ ❣ ➒ æ❩❣ ✤ æ❩❣ ❛ ✴✳ ✳ ✡ ✳ (2-27) which is cumbersome due to the fact that we have to recast exterior products into Clifford products where we can use the (commutator) relations. Finally one has to transform back at the ✙ end into the wedge basis of reduced words of the Graßmann basis.

An arbitrary bilinear form induced by ❛ . ⑨ Since we have no occasion to follow this interesting path, the reader is invited to consult the original work [101]. We will deliver an example which provides some evidence that the above described mechanism works. ✞✦✿✍❇ ✤ ✄✜✆✍✡ ✺ ✸✇✆ ➛ ✆ ✼ ÿ ✄✝✆Ò✡ be defined as ✺ ❛ ✄ ✾Ò➛ ✿❄✡ ✩ ✾❝➛ ✿ë❋✻☞ ÿ and ❛ Example: Let ✾ Þ①✇ ☞ where is an arbitrary bilinear form. We compute the above given commutative diagram on these elements ✺❛ ✄ ✿✴✡ ✾ç➛ ✞ ó ✾t➜ ✿ ✩ ✤ ✄ ✿ ❧ ✿ ➧ ✡ ➛Ô✾ ó ❁ ✿▲❋Ô☞ ✾q➜ ✯✇ ✩ ✤ ✄ ❛ ✾ç➛ ✯✇ Þ ✞ ✿ ❧ (2-47) ✿ ➛Ô✾ Þ ✿ ✿ is interpreted as the action of ✾ on it constitutes a Clifford map ➧ Þ ✩ ❛ ✾t➛ ✿■❋Ô☞ ✾ç➛ ✡▼❋⑥☞ ①✇ ✿②❋❑☞ Þ .

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A treatise on quantum Clifford algebras by Fauser.


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