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Lorentz type algebras Introduction to the Lorentz algebra Cándido Martı́n González University of Málaga [email protected] October 13, 2015 Lorentz type algebras This talk is part of a research jointly with: Pablo Alberca (U. of Málaga) Dolores Martı́n (U. of Málaga) Daouda Ndoye (U. of Dakar) The master document can be found in: http://arxiv.org/abs/1508.01634 Lorentz type algebras Precedents E. G. Beltrametti and A. A. Blasi, Dirac spinors, covariant currents and the Lorentz group over a finite field. Nuovo Cimento A, http://dx.doi.org/10.1007/BF02759228 55, 301 (1968). E. G. Beltrametti and A. A. Blasi, Rotation and Lorentz groups in a finite geometry. J. Math. Phys. http://dx.doi.org/10.1063/1.1664670 9, 1027 (1968). H. R. Coish, Elementary particles in a finite world geometry. Phys. Rev. http://dx.doi.org/10.1103/PhysRev.114.383 114, 383 (1959). L. E. Dickson, Determination of the structure of all linear homogeneous groups in a Galois field which are defined by a quadratic invariant. Am. J. Math. http://dx.doi.org/10.2307/2369602 21, 193 (1899). Stephan Foldes. The Lorentz group and its finite field analogs: Local isomorphism and approximation. J. Math. Phys. 49, 093512 (2008); http://dx.doi.org/10.1063/1.2982519. Lorentz type algebras From an algebraic view point the more interesting work above is the one of Dickson who, as early as in 1899, makes a detailed study of groups defined by quadratic forms in prime characteristic. As a by product of his research, he identifies the Icosahedral group as a subgroup of a Lorentz group in characteristic 2. Some recent developments on applications of finite Lorentz groups to Signal and Image Processing, seem to be under research. Lorentz type algebras Abstract We start with the Lorentz algebra L = oR (1, 3) over the reals and find a suitable basis B relative to which the structure constants are integers. Thus we consider the Z-algebra LZ which is free as a Z-module and its Z-basis is B. This allows us to define the Lorentz type algebra LK := LZ ⊗Z K over any field K . In this talk we study the ideal structure of Lorentz type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground field has no square root of −1. Thus, they are simple over the reals but not over the complex. Also, if the ground field is of characteristic 2 then Lorentz type algebras are neither simple nor semisimple. We extend the study of simplicity of the Lorentz algebra to the case of a ring of scalars where we have to use the notion of m-simplicity (relative to a maximal ideal m of the ground ring of scalars). Lorentz type algebras Over finite fields.... The Lorentz type algebras over a finite field Fq where q = p n and p is odd, are simple if and only if n is odd and p of the form p = 4k + 3. In case p = 2 then the Lorentz type algebras are not simple. The ideal structure of the algebras, provides some information of their automorphism groups. Lorentz type algebras Classical Lorentz algebra The Lorentz algebra over the reals, denoted by o(1, 3), is the Lie algebra of the orthogonal Lie group O(1, 3): o(1, 3) = Lie(O(1, 3)) = {M ∈ gl4 (R) : MI13 + I13 M t = 0}, where M t denotes matrix transposition of M and I13 = diag(−1, 1, 1, 1) (some authors take I13 = diag(1, 1, 1, −1) which is equivalent). A straightforward computation reveals that a generic element of o(1, 3) is of the form 0 x1 x2 x3 x1 0 x4 x5 x2 −x4 0 x6 x3 −x5 −x6 0 Lorentz type algebras and then denoting by eij the elementary matrix with 1 in the entry (i, j) and 0 elsewhere we have a basis of o(1, 3) given by B = {s12 , s13 , s14 , a23 , a24 , a34 } where sij := eij + eji and aij = eij − eji . [, ] s12 s13 s14 a23 a24 a34 s12 0 −a23 −a24 −s13 −s14 0 s13 a2,3 0 −a34 s12 0 −s14 s14 a2,4 a34 0 0 s12 s13 a23 s1,3 −s12 0 0 a34 −a24 a24 s1,4 0 −s12 −a34 0 a23 Figure : Multiplication table of o(1, 3). a34 0 s14 −s13 a24 −a23 0 Lorentz type algebras A nice basis s12 0 1 = 0 0 a23 0 0 0 0 1 0 0 0 , s13 = 0 0 0 0 , s14 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 −1 0 0 , a24 = 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 a34 = 0 0 0 1 . 0 0 −1 0 1 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 1 , 0 0 0 0 Relative to this basis the structure constants are 0, 1 or −1. 0 0 0 0 1 0 , 0 0 Lorentz type algebras Lorentz algebra over rings Thus we can construct the Z-algebra LZ := Zs12 ⊕ Zs13 ⊕ Zs14 ⊕ Za23 ⊕ Za24 ⊕ Za34 whose multiplication table is given in Figure 1. Fix now an associative, commutative and unital ring Φ and consider the category algΦ of associative commutative and unital Φ-algebras. Then for any object R in algΦ we may define the Lorentz type algebra LR := LZ ⊗Z R. This is nothing but the free R-module with basis s12 , s13 , s14 , a23 , a24 and a34 , enriched with an R-algebra structure by the multiplication table as in Figure 1. As a free R-module we have dim LR = 6. Lorentz type algebras Lorentz functor Of course if we take R = R then LR ∼ = o(1, 3), the Lorentz algebra. If R = C then LR is the complexified Lorentz algebra. If R and S are objects in algΦ and f : R → S a Φ-algebras homomorphism, then we may define a Lie Φ-algebras homomorphism Lf : LR → LS in an obvious way. Thus, if LieΦ denotes the category of Lie algebras over Φ, we have defined a covariant functor L: algΦ −→ LieΦ R 7−→ LR f 7−→ Lf Lorentz type algebras Orthogonal Lie algebra? Not exactly! Let O(n) be the orthogonal Lie group over the reals: the group of all matrices M in GLn (R) such that MM t = 1n . Then, its Lie algebra o(n) consists of all matrices M in gln (R) such that M + M t = 0. This is generated (as a vector space) by the matrices eij − eji where i < j with i, j ∈ {1, . . . , n} and the structure constants relative to the basis of these elements are again 0 or ±1. Thus, we can consider as before the Z-algebra o(n; Z) := ⊕i<j Z(eij − eji ). Fix as before a ring Φ and then, for any algebra R in algΦ , it is tempting to define the scalar extension o(n; R) := o(n; Z) ⊗Z R. So, this is the Lie R-algebra with basis eij − eji as before and multiplication table as the one for o(n) in the corresponding basis. Lorentz type algebras However in characteristic two, the orthogonal Lie algebra is different from this and in order to be coherent with the classical definition we should not use the notation o(n; R). Instead we will use but Γn (R) for the algebraic group of matrices in GLn (R) such that MM t = 1n and γn (R) for its Lie algebra γn (R) := o(n; Z) ⊗ R. We will also use the notation γ(n; R) meaning γn (R), and if the ground ring of scalars is free of 2-torsion, we will write o(n; R) or on (R) instead of γn (R). Lorentz type algebras Summarizing γn (R) = γ(n; R) is the algebra of skew-symmetric n × n matrices with entries in R. It is a free R-module with basis the set of all matrices eij − eji with i < j. Thus we have dimR (γ(n; R)) = n(n − 1)/2 and we have again a functor γ(n) : algΦ → LieΦ such that R 7→ γ(n; R). If f : R → S is a homomorphism of algebras in algΦ then we will denote by γ(n; f ) : γ(n; R) → γ(n; S) the homomorphism of Lie algebras γ(n; f ) := 1 ⊗ f . Lorentz type algebras Remark If Φ is a ring agreeing with its 2-torsion, that is, 1 + 1 = 0, then for any Φ-algebra R in algΦ , the Lie algebra γ(4; R) agrees with the Lorentz type Lie algebra LR . In particular this is the case for a field K of characteristic two: LK = γ(4; K). Lorentz type algebras Lorentz Type algebra over a field of char=2 Definition √ Consider next the full subcategory −1√ Φ of algΦ whose objects are the Φ-algebras √ R such that −1 ∈ R. Denote by I the inclusion functor I : −1Φ → algΦ . This allows a more general result: Lemma √ For any Φ, the functors L ◦ I and γ(4) ◦ I : −1Φ → LieΦ are isomorphic. More precisely (i) for any algebra R in algΦ such that the equation x 2 + 1 = 0 has a solution in R, there is an isomorphism ηR : LR ∼ = γ(4; R);√(ii) If f : R → S is a homomorphism of Φ-algebras and −1 ∈ R, the following diagram commutes: Lorentz type algebras LR Lf LS ηR / γ(4; R) γ(4;f ) / γ(4; S). ηS Proof. Take i ∈ R such that i2 = −1. Starting from the standard basis B of LR , we define a new basis 0 C = {aij0 : i, j ∈ {1, 2, 3, 4}, i < j} where a12 := is12 , 0 0 0 a13 := is13 , a14 := is14 and aij := aij for the remaining elements. Then the isomorphism LR → γ(4; R) is the induced by aij0 7→ eij − eji for i < j. On the other hand, the commutativity of the square above is straightforward. Lorentz type algebras Simplicity We would like to study under what conditions the Lorentz functor L: algΦ → LieΦ produces simple Lie algebras. A second goal would be to describe the algebraic group (in the sense of affine group schemes): Aut(LΦ ) : algΦ −→ Grp R 7−→ AutR (LR ) Lorentz type algebras To shorten the notations, we write b1 := s12 , b2 := s13 , b3 := s14 , b4 := a23 , b5 := a24 , b6 := a34 so that the basis B of LR is now B = {bi }61 and has the multiplication table: [, ] b1 b2 b3 b4 b5 b6 b1 0 −b4 −b5 −b2 −b3 0 b2 b4 0 −b6 b1 0 −b3 b3 b5 b6 0 0 b1 b2 b4 b2 −b1 0 0 b6 −b5 b5 b3 0 −b1 −b6 0 b4 b6 0 b3 −b2 b5 −b4 0 Figure : Second version of table in Figure 1. Lorentz type algebras Simplicity As an primary goal we investigate the simplicity of LΦ when Φ is a field. It is known that LR is simple while LC is not. In fact LC ∼ = sl2 (C) ⊕ sl2 (C). How can we investigate the simplicity of an algebra (just from its multiplication table)? Lorentz type algebras Investigating simplicity If we start with an algebra A over Φ (not necessarily satisfying any particular identity) such that A2 6= 0, the simplicity of A is equivalent to the fact that the ideal generated by any nonzero element is the whole algebra A. Lorentz type algebras Multiplication algebra For any a ∈ A denote La : A → A the left multiplication operator La (x) := ax and Ra : A → A the right multiplication operator such that Ra (x) = xa. Denote by EndΦ (A) the algebra of all linear maps A → A. Denote by M(A) the subalgebra of EndΦ (A) generated by all the left and right multiplication operators. The elements of M(A) are linear combinations of compositions of left and right multiplication operators. Lorentz type algebras A preorder relation in A If x, y ∈ A we say x ` y if there is some T ∈ M(A) such that y = T (x). Roughly speaking: if y is obtained form x after a finite number of multiplications we have x ` y . The relation ` is reflexive and transitive (a preorden in A). Observe that x ` y is equivalent to the assertion that the ideal hy i generated by y is contained in the ideal hxi generated by x. Lorentz type algebras A simplicity condition Theorem Let A be an algebra A such that A2 6= 0 and {bi }i a basis of the algebra. Then a sufficient condition for the simplicity of A is: 1 bi ` bj for any i and j. 2 ∀x ∈ A \ {0}, ∃i : x ` bi . Proof. Let 0 6= I / A and 0 6= x ∈ I . Since ∃i such that x ` bi , and bi ` bj for any j, we have x ` bj for any index j. So hbj i ⊂ hxi for each j. Consequently A ⊂ hxi or hxi = A. hence A is simple. Lorentz type algebras Ground field with no √ −1 √ By a field Φ with −1 6∈ Φ we mean that there is no element x ∈ Φ such that x 2 = −1. This condition implies that Φ is of characteristic other than 2. Also, this condition is equivalent to the assertion that: ∀x, y ∈ Φ, (x 2 + y 2 = 0 implies x = y = 0). If x 2 + y 2 = 0 and (say) x 6= 0, then 1 + ( yx )2 = 0 so that √ −1 exists. Lorentz type algebras Ground field with no Theorem If Φ is a field and √ −1 √ −1 6∈ Φ, then LΦ is simple. Sketch of proof. [, ] b1 b2 b3 b4 b5 b6 b1 0 −b4 −b5 −b2 −b3 0 b2 b4 0 −b6 b1 0 −b3 b3 b5 b6 0 0 b1 b2 b4 b2 −b1 0 0 b6 −b5 b5 b3 0 −b1 −b6 0 b4 b6 0 b3 −b2 b5 −b4 0 Lorentz type algebras bi ` bj By looking at the multiplication table we see that bi ` bj . For instance b1 `1 {b1, b2, b3, b4, b5} and b2 `1 b6 hence b1 ` bi for any i. The same applies to the others so that bi ` bj for any i, j. Lorentz type algebras 0 6= x ` bi for some i P Take 0 6= x = λi bi , if we take T = λ2 Rb2 Rb1 + λ5 Rb6 Rb2 , this is an element of M(LΦ ) and T (x) = −(λ22 + λ25 )b1 hence x ` (λ22 + λ25 )b1 and if λ2 √ or λ5 6= 0 then λ22 + λ25 6= 0 (on the contrary −1 ∈ Φ). Thus, if λ2 or λ5 6= 0 x ` b1 and we are done. So assume λ2 = λ5 = 0. Lorentz type algebras 0 6= x ` bi for some i 0 6= x = λ1 b1 + λ3 b3 + λ4 b4 + λ6 b6 . Things are getting easier because x is “shorter”. Take S := λ4 Rb1 + λ3 Rb6 ∈ M(LΦ ), then it is easy to see that S(x) = −(λ23 + λ24 )b2 . Hence if λ3 or λ4 6= 0 we have as before x ` b2 and we are done again. In the worst case λ3 = λ4 = 0. Lorentz type algebras 0 6= x ` bi for some i So far we have λ2 = λ3 = λ4 = λ5 = 0. Thus x = λ1 b1 + λ6 b6 and λ1 or λ6 6= 0. Thus λ21 + λ26 6= 0. The operator R = λ1 Rb2 + λ6 Rb5 ∈ M(LΦ ) satisfies R(x) = (λ21 + λ26 )b4 hence x ` b4 which finishes the proof. Lorentz type algebras Representation theory language For any Φ-algebra A, the multiplication algebra M(A) is associative and unital. Furthermore A is an M(A)-module and the irreducible M(A)-submodules of A are precisely the ideals of A. Thus the simplicity of LΦ proved above is just the assertion that the M(LΦ )-module LΦ is irreducible. Lorentz type algebras √ When Φ is a field such that −1 ∈ Φ the same relation ` helps to realize that LΦ is not simple. Of course there are elements x such that x ` LΦ , but one realizes immediately that there are other elements x such that x ` I for a certain three-dimensional ideal I of Lφ . Thus we detect proper nonzero ideals inside LΦ . Moreover there are only two nonzero proper ideals I and J and LΦ = I ⊕ J being I ∼ = sl2 (Φ) ∼ = J. Lorentz type algebras Let now Φ be any commutative unital ring and R any algebra in the category algΦ . Let Max(R) denote the maximal spectrum of R (the set of maximal ideals) and m ∈ Max(R). Let V be any R-algebra (not necessarily associative) with V 2 6= 0 and I / V an ideal. Of course the R-submodule mV is an ideal of V , but we would like to exclude these ideals from our study. Also, we want to exclude, those ideals which complement mV . Thus Definition We say that I / V is m-null if I ⊂ mV . Also I is said to be m-total if V = I + mV . The algebra V is said to be m-simple if its unique ideals are the m-null and the m-total ones. Lorentz type algebras Definition We say that a field K is 2-formally real if for any x, y ∈ K, the equality x 2√ + y 2 = 0 implies x = y = 0 (equivalently −1 ∈ / K). More generally for an ideal i of a Φ-algebra R we say that R is i-2-formally real if for any x, y ∈ R, the fact x 2 + y 2 ∈ i implies x, y ∈ i. Lorentz type algebras Theorem Let m ∈ Max(R) be a maximal ideal. Then the Lorentz type algebra LR is m-simple if and only if R is m-2-formally real. In particular, the Lorentz type algebra LK over a field K is simple if and only if K is 2-formally real (equivalently, if and only if √ −1 ∈ K). Theorem Let R be an algebra in algΦ with Jacobson radical rad(R) = 0. ThenQ LR is m-simple for any m ∈ Max(R) if and only if R ⊂ i∈I Ki is a subdirect product of 2-formally real field {Ki }i∈I and for anyQm ∈ Max(R) there is some j ∈ I such that πj (m) = 0, being πj : i∈I Ki → Kj the canonical projection onto the field Kj .