[assumption]theorem [assumption]corollary [assumption]lemma [assumption]definition. Andrzej Sitarz

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1 [assumption]theorem [assumption]corollary [assumption]lemma [assumption]definition κ-minkowski STAR PRODUCT AND ITS SYMMETRIES Andrzej Sitarz Jagiellonian University, Kraków , Kɛρκυρα Joint work with Bergfinnur Durhuus, Department of Mathematical Sciences, University of Copenhagen. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 1 / 21

2 OUTLINE 1 MOTIVATION 2 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. 3 THE -PRODUCT ON κ-minkowski. 4 THE COMPLETION:κ C -ALGEBRA 5 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS 6 THE COALGEBRA STRUCTURE 7 THE ACTION OF κ-poincaré 8 THE TWISTED TRACE 9 CONCLUSIONS ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 2 / 21

3 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

4 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) κ-deformation of the Poincaré group has a nice bicrossproduct structure ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

5 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) κ-deformation of the Poincaré group has a nice bicrossproduct structure There have been many studies of the algebra, differential calculi, field theory, representations etc. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

6 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) κ-deformation of the Poincaré group has a nice bicrossproduct structure There have been many studies of the algebra, differential calculi, field theory, representations etc. κ-minkowski became popular under the label of Double Special Relativity ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

7 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) κ-deformation of the Poincaré group has a nice bicrossproduct structure There have been many studies of the algebra, differential calculi, field theory, representations etc. κ-minkowski became popular under the label of Double Special Relativity So far: most of the approaches were formal or not manageable. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

8 MOTIVATION THE LANDSCAPE: κ-minkowski was introduced by Lukierski, Nowicki, Ruegg: [x 0, x i ] = i κ x i, [x i, x j ] = 0, i, j = 1,..., d 1, (1) κ-deformation of the Poincaré group has a nice bicrossproduct structure There have been many studies of the algebra, differential calculi, field theory, representations etc. κ-minkowski became popular under the label of Double Special Relativity So far: most of the approaches were formal or not manageable. Earlier attempts: D Andrea, Agostini, Dabrowski-Piacitelli. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 3 / 21

9 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. THE RESPRESENTATION We fix κ: [t, x] = ix. (2) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 4 / 21

10 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. THE RESPRESENTATION We fix κ: [t, x] = ix. (2) this algebra has a faithful 2-dimensional representation ρ given by ρ(it) = ( ) 1 0, ρ(ix) = 0 0 ( ) 0 1, (3) 0 0 ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 4 / 21

11 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. THE RESPRESENTATION We fix κ: [t, x] = ix. (2) this algebra has a faithful 2-dimensional representation ρ given by ρ(it) = ( ) 1 0, ρ(ix) = 0 0 ( ) 0 1, (3) 0 0 the corresponding connected and simply connected Lie group is the group G 2 of 2 2-matrices of the form: ( ) e a b S(a, b) =, a, b R, (4) 0 1 ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 4 / 21

12 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. THE CONVOLUTION ALGEBRA Let A be the the convolution algebra of G 2 with respect to the right invariant measure. Identifying functions on G 2 with functions on R 2 : (f ˆ g)(a, b) = da db f (a a, b e a a b )g(a, b ), (5) f (a, b) = e a f ( a, e a b), (6) where f, g A and f is the complex conjugate of f. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 5 / 21

13 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. THE CONVOLUTION ALGEBRA Let A be the the convolution algebra of G 2 with respect to the right invariant measure. Identifying functions on G 2 with functions on R 2 : (f ˆ g)(a, b) = da db f (a a, b e a a b )g(a, b ), (5) f (a, b) = e a f ( a, e a b), (6) where f, g A and f is the complex conjugate of f. If π is a unitary representation (always assumed to be strongly continuous) of G 2 it is well known that π gives rise to a -representation, also denoted by π, of A by setting π(f ) = dadb f (a, b)π(s(a, b)). (7) Thus, we have π(f ˆ g) = π(f )π(g) and π(f ) = π(f ). (8) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 5 / 21

14 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. WEYL QUANTIZATION Following the same procedure as in the case of the Weyl quantisation we define the Weyl map W π associated with the representation π by W π (f ) = π(ff ) for f L 1 (R 2 ) F 1 (L 1 (R 2 )), where F denotes the Fourier transformation. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 6 / 21

15 QUANTIZATIONS AND STAR PRODUCTS FOR κ-minkowski. WEYL QUANTIZATION Following the same procedure as in the case of the Weyl quantisation we define the Weyl map W π associated with the representation π by W π (f ) = π(ff ) for f L 1 (R 2 ) F 1 (L 1 (R 2 )), where F denotes the Fourier transformation. Finally we define the star-product: DEFINITION and f g = 1 2π F 1 ((Ff )ˆ (Fg)). (9) f = F 1 (F(f ) ), (10) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 6 / 21

16 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE DOMAIN AND THE FORMULA. As in the case of the standard Moyal product, one needs to exercise some care about the domain of definition for the righthand sides of (9) and (10). As a first result in this direction we note the following. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 7 / 21

17 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE DOMAIN AND THE FORMULA. As in the case of the standard Moyal product, one needs to exercise some care about the domain of definition for the righthand sides of (9) and (10). As a first result in this direction we note the following. PROPOSITION Let Cc denote the space of smooth functions on R 2 with compact support. If f, g F 1 (Cc ) then f and f g also belong to F 1 (Cc ) and are given by f g(α, β) = 1 dv du f (α + u, β)g(α, e v β)e iuv (11) 2π and respectively. f (α, β) = 1 2π dv du f (α + u, e v β)e iuv. (12) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 7 / 21

18 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE PROPERTIES REMARK Clearly, the -product defined earlier is well defined as a function on R 2 for a larger class of functions than those discussed above. We note the following: ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 8 / 21

19 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE PROPERTIES REMARK Clearly, the -product defined earlier is well defined as a function on R 2 for a larger class of functions than those discussed above. We note the following: If g(α, β) = g(α) is any function depending only on α and f is, say, a Schwartz function of α for fixed value of β, the f g is well defined and f g(α, β) = f (α, β)g(α). If f (α, β) = f (β) is any function depending only on β and g is e.g. smooth with compact support as a function of β for fixed value of α, then the integral in (11) can be interpreted in a distributional sense and yields f g(α, β) = f (β)g(α, β). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 8 / 21

20 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE PROPERTIES If f (α, β) = α and g(α, β) = g(β) is a smooth function of β of compact support, a distributional interpretation yields (f g)(α, β) = αg(β) + iβg (β), (g f )(α, β) = g(β)α. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 9 / 21

21 THE -PRODUCT ON κ-minkowski. κ- -PRODUCT: THE PROPERTIES If f (α, β) = α and g(α, β) = g(β) is a smooth function of β of compact support, a distributional interpretation yields (f g)(α, β) = αg(β) + iβg (β), (g f )(α, β) = g(β)α. In particular, α g(β) g(β) α = iβg (β). Note that, formally, setting g(β) = β in this relation yields a representation of the defining relation (2) in terms of a -commutator with t, x corresponding to α, β. Note also that, α = α and β = β by (12), suitably interpreted. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 9 / 21

22 THE COMPLETION:κ C -ALGEBRA THE COMPLETION OF THE ALGEBRA B Let B denote the subspace of B consisting of Fourier transforms of derivatives w.r.t. the second variable of functions in C0. B 2 is dense in L 2 (R 2, dµ), where the measure dµ = β 1 dαdβ. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 10 / 21

23 THE COMPLETION:κ C -ALGEBRA THE COMPLETION OF THE ALGEBRA B Let B denote the subspace of B consisting of Fourier transforms of derivatives w.r.t. the second variable of functions in C0. B 2 is dense in L 2 (R 2, dµ), where the measure dµ = β 1 dαdβ. ϕ, W ± (f )ψ = dadbds Ff (a, b) ϕ(s) e ±ibe s ψ(s + a) = dsdudb ϕ(s)ff (u s, b) e ±ibe s ψ(u) = 2π dsdu ϕ(s) f (u s, ±e s )ψ(u). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 10 / 21

24 THE COMPLETION:κ C -ALGEBRA THE COMPLETION OF THE ALGEBRA B Let B denote the subspace of B consisting of Fourier transforms of derivatives w.r.t. the second variable of functions in C0. B 2 is dense in L 2 (R 2, dµ), where the measure dµ = β 1 dαdβ. ϕ, W ± (f )ψ = dadbds Ff (a, b) ϕ(s) e ±ibe s ψ(s + a) = dsdudb ϕ(s)ff (u s, b) e ±ibe s ψ(u) = 2π dsdu ϕ(s) f (u s, ±e s )ψ(u). The map W : f W + (f ) W (f ) is injective from L 2 (R 2, dµ) into H H, where H denotes the space of Hilbert-Schmidt operators on L 2 (R). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 10 / 21

25 THE COMPLETION:κ C -ALGEBRA THE COMPLETION OF THE ALGEBRA B THEOREM Let B and W be as defined above and set B = L 2 (R 2, dµ). Then the -product (11) and involution (12) have unique extensions from B to B, such that B becomes an involutive algebra and W an isomorphism, W (f g) = W (f )W (g) W (f ) = W (f ). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 11 / 21

26 THE COMPLETION:κ C -ALGEBRA THE COMPLETION OF THE ALGEBRA B THEOREM Let B and W be as defined above and set B = L 2 (R 2, dµ). Then the -product (11) and involution (12) have unique extensions from B to B, such that B becomes an involutive algebra and W an isomorphism, COROLLARY W (f g) = W (f )W (g) W (f ) = W (f ). The integrals w.r.t. dµ over R R ± are positive traces on B in the sense that duds(f f )(u, ±e s ) 0, duds(f g)(u, ±e s ) = duds(g f )(u, ±e s ). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 11 / 21

27 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS THE LEFT-INVARIANT STAR-PRODUCT We can apply the same procedure as above using instead the left invariant Haar measure on G 2. It is then convenient to use the parametrisation R(a, c) = S(a, e a c), a, c R, (13) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 12 / 21

28 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS THE LEFT-INVARIANT STAR-PRODUCT We can apply the same procedure as above using instead the left invariant Haar measure on G 2. It is then convenient to use the parametrisation R(a, c) = S(a, e a c), a, c R, (13) The corresponding star-product φ and involution φ are given by: f g(α, β) = 1 dv du f (α, e v β)g(α + u, β)e iuv, (14) 2π and the involution is f (α, β) = 1 2π dv du f (α + u, e v β)e iuv. (15) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 12 / 21

29 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS ALL STAR PRODUCT... PROPOSITION Assume φ is positive and smooth. For f L 1 (R 2 ) F 1 (L 1 (R 2 )) the operators W φ ± (f ) are integral operators on L 2(R) with kernels given by K ± f (s, u) = 2π f (u s, ±φ(u s)e s ) = dvf (v, ±φ(u s)e s )e iv(u s). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 13 / 21

30 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS ALL STAR PRODUCT... PROPOSITION Assume φ is positive and smooth. For f L 1 (R 2 ) F 1 (L 1 (R 2 )) the operators W φ ± (f ) are integral operators on L 2(R) with kernels given by K ± f (s, u) = 2π f (u s, ±φ(u s)e s ) = dvf (v, ±φ(u s)e s )e iv(u s). It can now be seen that the norm and trace formulas hold independently of the choice of φ. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 13 / 21

31 THE LEFT-INVARIANT CASE AND OTHER STAR-PRODUCTS ALL STAR PRODUCT... PROPOSITION Assume φ is positive and smooth. For f L 1 (R 2 ) F 1 (L 1 (R 2 )) the operators W φ ± (f ) are integral operators on L 2(R) with kernels given by K ± f (s, u) = 2π f (u s, ±φ(u s)e s ) = dvf (v, ±φ(u s)e s )e iv(u s). It can now be seen that the norm and trace formulas hold independently of the choice of φ. THEOREM The involutive algebras B φ, resp. B φ, where φ is positive and smooth are isomorphic. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 13 / 21

32 THE COALGEBRA STRUCTURE COPRODUCT, COUNIT AND ANTIPODE The κ-minkowski, as originally defined by LNR has a natural Hopf algebra structure, which arises by dualisation from the momenta subalgebra of the κ-poincaré. The coalgebra structure alone is underformed when compared to the classical case: x 0 = x x 0, x i = x i x i. (16) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 14 / 21

33 THE COALGEBRA STRUCTURE COPRODUCT, COUNIT AND ANTIPODE The κ-minkowski, as originally defined by LNR has a natural Hopf algebra structure, which arises by dualisation from the momenta subalgebra of the κ-poincaré. The coalgebra structure alone is underformed when compared to the classical case: PROPOSITION x 0 = x x 0, x i = x i x i. (16) The usual cocommutative coproduct on the space of functions B, : B B 2 f (α, β; α, β ) = f (α + α ; β + β ). together with the counit map: ε : B C, and the antipode S : B B: ε : B f f (0, 0) C, S(f )(α, β) = dpds f ( α s, e p β)e ips, equip B with a Hopf algebra structure. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 14 / 21

34 THE ACTION OF κ-poincaré THE LORENTZ SYMMETRY [P, E] = 0, [N, E] = P, P = P 1 + e E κ P, E = E E, and for the relations involving the boost, [N, P] = κ 2E 1 (1 e κ ) + 2 2κ P2, N = N 1 + e E κ N. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 15 / 21

35 THE ACTION OF κ-poincaré THE LORENTZ SYMMETRY [P, E] = 0, [N, E] = P, P = P 1 + E P, E = E E, and for the relations involving the boost and E, [N, P] = κ 2 (1 E2 ) + 1 2κ P2, N = N 1 + E N, [P, E] = 0, [E, E] = 0, [N, E] = κ(1 E), E = E E ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 15 / 21

36 THE ACTION OF κ-poincaré THE LORENTZ SYMMETRY [P, E] = 0, [N, E] = P, P = P 1 + E P, E = E E, and for the relations involving the boost and E, [N, P] = κ 2 (1 E2 ) + 1 2κ P2, N = N 1 + E N, [P, E] = 0, [E, E] = 0, [N, E] = κ(1 E), E = E E DEFINITION Let f B so that it is a Fourier transform of ˆf C 0 (R 2 ). We define a one parameter group of linear operations on B in the following way. For any γ R let: T γ (f )(α, β) = 1 2π dudv ˆf (u, v)e γu e i(αu+βv). (17) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 15 / 21

37 THE ACTION OF κ-poincaré THE ACTION OF TRANSLATION The explicit formula for the map T γ : f T g f is: T γ (f )(α, β) = 1 dpds f (s, β)e γp e ip(s α). (18) 2π ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 16 / 21

38 THE ACTION OF κ-poincaré THE ACTION OF TRANSLATION The explicit formula for the map T γ : f T g f is: T γ (f )(α, β) = 1 dpds f (s, β)e γp e ip(s α). (18) 2π PROPOSITION The map T γ : f T γ f is an algebra automorphism. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 16 / 21

39 THE ACTION OF κ-poincaré THE ACTION OF TRANSLATION The explicit formula for the map T γ : f T g f is: T γ (f )(α, β) = 1 dpds f (s, β)e γp e ip(s α). (18) 2π PROPOSITION The map T γ : f T γ f is an algebra automorphism. The T γ automorphism does not preserve the involution in the algebra B, however we have: T γ (f ) = (T γ f ). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 16 / 21

40 THE ACTION OF κ-poincaré THE LORENTZ SYMMETRY PROPOSITION The algebra B is a Hopf module algebra with respect to the following action of the momentum algebra, with the generators E, P, E represented as linear operators on B. (E f )(α, β) = α, (P f )(α, β) = β, (E f )(α, β) = (T 1f )(α, β). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 17 / 21

41 THE ACTION OF κ-poincaré THE LORENTZ SYMMETRY PROPOSITION The algebra B is a Hopf module algebra with respect to the following action of the momentum algebra, with the generators E, P, E represented as linear operators on B. (E f )(α, β) = α, (P f )(α, β) = β, (E f )(α, β) = (T 1f )(α, β). PROPOSITION A linear operator N: acts on B as boost. N = L α P i 2 (1 E2 )L β + i 2 L βp 2, ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 17 / 21

42 THE TWISTED TRACE THE HAAR INTEGRAL REMARK The usual Lebesgue measure on R 2 gives rise to a left (and right) integrals on the Hopf algebra B: Ω H (f ) := dadb f (a, b), Ω H (f ) = da db f (a + a, b + b ) = dadb f (a + a, b + b ). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 18 / 21

43 THE TWISTED TRACE THE HAAR INTEGRAL REMARK The usual Lebesgue measure on R 2 gives rise to a left (and right) integrals on the Hopf algebra B: Ω H (f ) := dadb f (a, b), Ω H (f ) = da db f (a + a, b + b ) = dadb f (a + a, b + b ). The Haar integral Ω H has the following properties: Ω H is compatible with the conjugation in B, for every f B: Ω H (f ) = Ω(f ), ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 18 / 21

44 THE TWISTED TRACE THE HAAR INTEGRAL - PROPERTIES Ω H is a twisted trace, that is for all f, g B: Ω H (f g) = Ω H (T 1 (g) f ), ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 19 / 21

45 THE TWISTED TRACE THE HAAR INTEGRAL - PROPERTIES Ω H is a twisted trace, that is for all f, g B: Ω H (f g) = Ω H (T 1 (g) f ), Ω H is invariant with respect to the action of κ-poincaré, that is, for any h P κ and f B: Ω H (h f ) = ε(h)ω H (f ). ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 19 / 21

46 THE TWISTED TRACE THE HAAR INTEGRAL - PROPERTIES Ω H is a twisted trace, that is for all f, g B: Ω H (f g) = Ω H (T 1 (g) f ), Ω H is invariant with respect to the action of κ-poincaré, that is, for any h P κ and f B: Ω H (h f ) = ε(h)ω H (f ). For every f B: Ω H (f f ) = f 2 L 2, ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 19 / 21

47 CONCLUSIONS REINTRODUCING κ: We can write the product with κ present: f κ g(α, β) = dudv f (α + u, β)g(α, e v κ β)e iuv, (19) and the conjugation: f (α, β) = dudv f (α + u, e v κ β)e iuv. (20) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 20 / 21

48 CONCLUSIONS REINTRODUCING κ: We can write the product with κ present: f κ g(α, β) = dudv f (α + u, β)g(α, e v κ β)e iuv, (19) and the conjugation: f (α, β) = dudv f (α + u, e v κ β)e iuv. (20) The formal powers series expansion of the product: (f κ g)(α, β) = ( i n n ) κ n n! ( n αf (α, β) Bk n βk ( β k g)(α, β). n k=1 where B k are integer coeeficients of the expansion: n (x x ) n = B k x k ( x ) k. k=1 ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 20 / 21

49 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21

50 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. The symmetries are well defined (in terms of Hopf algebras) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21

51 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. The symmetries are well defined (in terms of Hopf algebras) The twisted trace provides an equivariant representation on L 2 (R 2 ) ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21

52 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. The symmetries are well defined (in terms of Hopf algebras) The twisted trace provides an equivariant representation on L 2 (R 2 ) The geometry a la Connes - Dirac operator, spectral triple - no problem. ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21

53 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. The symmetries are well defined (in terms of Hopf algebras) The twisted trace provides an equivariant representation on L 2 (R 2 ) The geometry a la Connes - Dirac operator, spectral triple - no problem. Field theory - like in the Moyal case - is within reach!!! ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21

54 CONCLUSIONS OUTLOOK We have a well-defined algebra with a nice C completion. The symmetries are well defined (in terms of Hopf algebras) The twisted trace provides an equivariant representation on L 2 (R 2 ) The geometry a la Connes - Dirac operator, spectral triple - no problem. Field theory - like in the Moyal case - is within reach!!! THANK YOU! ANDRZEJ SITARZ () κ-minkowski , Kɛρκυρα 21 / 21