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In our case ✾ ✤➌➋ ❡ would be the lower outgoing arrow etc. 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➠ .

Since the Clifford algebra can be described using a Graßmann basis, it seems to be possible to introduce a ✩ -grading here also. However, a short calculation shows that the Clifford product does not respect this grading, but only a weaker filtration, see later chapters. ✞✖✪ be extensors of step ➯ and → one obtains Let × ✪ × æ ➟❇ ➁ ➢ ➍✜↕ ✬✫☞➤ ✭ ➍ ✣ ➤ ✭ ÷ ➁ ✆ ✲ (2-20) This is not an accident of the foreign basis, but remains to be true in a Clifford basis also. The terms of lower step emerge from the necessary commutation of some generators to the proper place in a reduced word.

The isomorphisms in ✔ ❛ -graded. and Proof: see [47, 50, 57, 56]. 6 (Chevalley [31]). The opposite Clifford algebra ✂✁☎✄✝✆✟✞ ✡ ❧q✈ . 7. The opposite Clifford algebra ① ✂✁ ä ✄✜✆❫✞✂☞✍✡ of ① ✂✁ ä ✂✁☎✄✝✆✟✞✌☞✍✡ ✂✁☎✄✝✆✟✞✌☞✍✡ ✄✜✆❫✞✜✠☛✡ of are isomorphic as ✌✁☎✄✜✆❫✞✝✠☛✡ is isomorphic to is isomor- ✂✁☎✄✝✆✟✞ ❧ ☞ ● ✡ . Proof: see [60, 50]. One obtains that ✶ ➅ ➢ ➠ ❈ ✆ ✩ ➢ ✆ ✷❞ ➛ ✆ ✌✁☎✄✜✆❫✞✂☞✍✡ ✩ ✪ ➛ ❈ ✌✁☎✄✜✆❫✞ ☞ ❧ ● ✡ ✩ ✂✁☎✄✝✆ ÷ ✆❫✞✌☞ ÷✻❧ ☞ ● ✡ (2-37) where ➛ is a ✩ ❛ -graded tensor product. 6 Clifford algebras of multivectors An intriguing approach to Clifford algebras was developed by Oziewicz and will be called Clifford algebra of multivectors.

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A treatise on quantum Clifford algebras by Fauser.


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