A Tutorial on Quantum Clifford Algebras and Some of Their Applications !

Arbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .qA Tutorial on Quantum Clifford Algebrasand Some of Their Applications!ROLDÃO DA ROCHA1Center of Mathematics - ABC Federal University, São Paulo, BrazilYerevanSupersymmetry in Integrable Systems - SIS’10 InternationalWorkshop, 24-28th August 2010Arbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .qOutline1 Arbitrary Gradings2 Theorem3 (No) Periodicity Theorems4 Hecke Algebras5 Conformal Maps in quantum AC6 U (so(3,2)),-Poincaré algebra,. . .qArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .qV vector space overK, endowed with g2 lin-Hom(VV;K).v2 V, g(v; v) = Q(v),A unital associative algebra. : V!A linear.The pair (A;) is a Clifford algebra with respect to (V;g) ifA is generated byf(v)j v2 Vg efa1 j a2Kg andA(v)(u) +(u)(v) = 2g(u; v)1AIn orthonormal basisfeg2 Vie e + e e = 2g(e; e )1 = 2g 1:i j j i i j ijThe definition of the Clifford algebra reads:C‘(V;g)’ Alg(e )mod e e = 2g 1 e e :i i j ij j iArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .qI =fX2 T (V )jX = L (v v g(v; v)1)M; L;M2 T (V )g’Square law’ of Clifford algebras.T (V )C‘(V;g) :=IArbitrary ...