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

Roldão Da Rocha

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• Publié par: Udru
• Nombre de lectures: 13
• Langue: English

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Arbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
A Tutorial on Quantum Clifford Algebras
and Some of Their Applications!
ROLDÃO DA ROCHA
1Center of Mathematics - ABC Federal University, São Paulo, Brazil
Yerevan
Supersymmetry in Integrable Systems - SIS’10 International
Workshop, 24-28th August 2010Arbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
Outline
1 Arbitrary Gradings
2 Theorem
3 (No) Periodicity Theorems
4 Hecke Algebras
5 Conformal Maps in quantum AC
6 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,. . .q
V 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 by
f(v)j v2 Vg efa1 j a2Kg andA
(v)(u) +(u)(v) = 2g(u; v)1A
In orthonormal basisfeg2 Vi
e e + e e = 2g(e; e )1 = 2g 1:i j j i i j ij
The 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,. . .q
I =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 Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
Classification
C‘ ’C‘
C‘p+1;q+1 1;1 p;q
C‘ ’C‘
C‘q+2;p 2;0 p;q
C‘ ’C‘
C‘q;p+2 0;2 p;q
C‘ ’C‘
C‘ (Atiyah-Bott-Shapiro)p;q+8 p;q 0;8
p q mod 8 0 1 2 3
2[n=2] [n=2] [n=2] [n=2]
C‘ M(2 ;R) M(2 ;R) M(2 ;R) M(2 ;C)p;q
p q mod 8 4 5 6 7
2[n=2] [n=2] 1 [n=2] 1 [n=2]
C‘ M(2 ;H) M(2 ;R) M(2 ;H) M(2 ;C)p;q
C‘(V;g)’M(kk;L), whereL =R;C;H;RR;HHArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
.
TLetB : VV!R a bilinear form, and thenB = g +A; whereA = A; and
Tg = g:
B(u; v) = uy v
B
A(u; v) = uy v
A
g(u; v) = uy v = u v:
gArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
we could define
The pair (A;) is a quantum Clifford algebra with respect to (V;B) ifA is
generated byf(v)j v2 Vg efa1 j a2Kg andA
(v)(u) +(u)(v) = 2B(u; v)1A
it would imply that B = g is a symmetric bilinear form !!!Arbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
B-dependent Clifford product uv of two 1-vectors u and v inC‘(B;V ) can be
B
decomposed in (different ways) into scalar and bi-vector parts as follows
_uv = uy v + u^v Hestenes
gB
uv = uy v + u^ v Oziewicz, Lounesto, Abłamowicz, Fauser;
B B
_where u^v = u^ v +A(u; v) = u^ v + uy v:AArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
Clifford product
Given u2 V and 2 ( V )
u = uy + u^ (Standard Clifford product)
g
u = uy + u^ (arbitrary Clifford product)
BB
_u = uy + u^
gB
_where u^v = u^ v +A(u; v) = u^ v + uy v:AArbitrary Gradings Theorem (No) Periodicity Theorems Hecke Algebras Conformal Maps in quantum AC U (so(3,2)),-Poincaré algebra,. . .q
(u)(v)+(v)(u) = 2g(u; v)1:
In the anticommutation relation only the symmetric part of B occurs.
However, the anticommutators are altered
(u)(v) (v)(u) = 2 u^ v + 2A(u; v)1:
The Z -grading depends directly on the presence of the antisymmetric part.n
Two different Graßmann algebras! One is Z -graded while the other is not!n
_u^v = u^ v +A(u; v)
_ _u^v^w = u^ v^ w +A(u; v)w +A(v; w)u +A(w; u)v
etc.