Laudal s Noncommutative Geometry
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1 Laudal s Noncommutative Geometry Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland 2016 Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
2 Laudal s classical moduli theory Moduli and classification Algebraic objects can be classified as points in an algebraic space. The demand for functorial properties gives the spectrum of a k-algebra A as the classifying object of all maximal ideals, or all simple A-modules ρ V : A A/m = V. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
3 Laudal s classical moduli theory Classification of singularities and Lie-algebras Together with Gerhard Pfister, Laudal gave a moduli space of algebraic singularities: This was in need of a stratification because of some jump phenomenon in accordance to non-closed orbits. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
4 Laudal s classical moduli theory Classification of singularities and Lie-algebras Together with Gerhard Pfister, Laudal gave a moduli space of algebraic singularities: This was in need of a stratification because of some jump phenomenon in accordance to non-closed orbits. The root systems of singularities gives a link to classification of Lie algebras. Together with Harald Bjar, Laudal gave a modulispace of Lie algebras. Quite natural, this was in need of a stratification due to the same jump phenomenon. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
5 Laudal s classical moduli theory Classification of singularities and Lie-algebras Together with Gerhard Pfister, Laudal gave a moduli space of algebraic singularities: This was in need of a stratification because of some jump phenomenon in accordance to non-closed orbits. The root systems of singularities gives a link to classification of Lie algebras. Together with Harald Bjar, Laudal gave a modulispace of Lie algebras. Quite natural, this was in need of a stratification due to the same jump phenomenon. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
6 Laudal s classical moduli theory First noncommutative geometry The first attempt to make a noncommutative geometry just tried to make a scheme-theory for noncommutative k-algebras by replacing the localizations. This gave the localization in Ore-sets after Øystein Ore. This theory seems not very suited for moduli problems. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
7 Laudal s classical moduli theory First noncommutative geometry The first attempt to make a noncommutative geometry just tried to make a scheme-theory for noncommutative k-algebras by replacing the localizations. This gave the localization in Ore-sets after Øystein Ore. This theory seems not very suited for moduli problems. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
8 Laudal s classical moduli theory Second noncommutative geometry So if it is a problem with localizations, we might just ignore it and look to the representations of a noncommutative k-algebra A: Spec(A) = mod A. Of course, this is far to easy, we should take the derived category of finitely generated A-modules. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
9 Laudal s classical moduli theory Second noncommutative geometry So if it is a problem with localizations, we might just ignore it and look to the representations of a noncommutative k-algebra A: Spec(A) = mod A. Of course, this is far to easy, we should take the derived category of finitely generated A-modules. In fact, this works nearly very well: For any S = Spec(A), qmod A /S is a stack over nsch. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
10 Laudal s classical moduli theory Second noncommutative geometry So if it is a problem with localizations, we might just ignore it and look to the representations of a noncommutative k-algebra A: Spec(A) = mod A. Of course, this is far to easy, we should take the derived category of finitely generated A-modules. In fact, this works nearly very well: For any S = Spec(A), qmod A /S is a stack over nsch. It might happen in this case, for A B, that Spec(A) Spec(B). That is the reason for the nearly well. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
11 Laudal s classical moduli theory Second noncommutative geometry So if it is a problem with localizations, we might just ignore it and look to the representations of a noncommutative k-algebra A: Spec(A) = mod A. Of course, this is far to easy, we should take the derived category of finitely generated A-modules. In fact, this works nearly very well: For any S = Spec(A), qmod A /S is a stack over nsch. It might happen in this case, for A B, that Spec(A) Spec(B). That is the reason for the nearly well. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
12 Laudal s classical moduli theory Laudal s Noncommutative geometry It is well known that the theory of flat deformations gives the local moduli in the commutative situation. Notice that analytic deformations are interesting, but they do not give the local algebraic moduli. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
13 Laudal s classical moduli theory Laudal s Noncommutative geometry It is well known that the theory of flat deformations gives the local moduli in the commutative situation. Notice that analytic deformations are interesting, but they do not give the local algebraic moduli. To be precise, for A a commutative k-algebra, for the category of local artinian k-algebras l = a 1 with objects the commutative diagrams of local artinian k-algebras k A id π k and commuting homomorphisms, the deformation functor has a prorepresenting hull H. Def M : a 1 Sets Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
14 Laudal s classical moduli theory Laudal s Noncommutative geometry It is well known that the theory of flat deformations gives the local moduli in the commutative situation. Notice that analytic deformations are interesting, but they do not give the local algebraic moduli. To be precise, for A a commutative k-algebra, for the category of local artinian k-algebras l = a 1 with objects the commutative diagrams of local artinian k-algebras k A id π k and commuting homomorphisms, the deformation functor has a prorepresenting hull H. Def M : a 1 Sets Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
15 Laudal s classical moduli theory One slide on hulls The definition of the deformation functor is: Def M (S) = {S-mod M S M S is S-flat, A S M S M}/. The residue π of every S a 1 can be decomposed in small morphisms φ ((ker φ)m = 0), S/m n S/m n 1 S/m k. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
16 Laudal s classical moduli theory One slide on hulls The definition of the deformation functor is: Def M (S) = {S-mod M S M S is S-flat, A S M S M}/. The residue π of every S a 1 can be decomposed in small morphisms φ ((ker φ)m = 0), S/m n S/m n 1 S/m k. Then there is a famous theorem of Laudal stating that Def M has a prorepresenting hull H, which can be computed algorithmically by Generalized Matric Massey products, and that the hull H of Def A/m satisfies H Ô Spec A,m. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
17 Laudal s classical moduli theory One slide on hulls The definition of the deformation functor is: Def M (S) = {S-mod M S M S is S-flat, A S M S M}/. The residue π of every S a 1 can be decomposed in small morphisms φ ((ker φ)m = 0), S/m n S/m n 1 S/m k. Then there is a famous theorem of Laudal stating that Def M has a prorepresenting hull H, which can be computed algorithmically by Generalized Matric Massey products, and that the hull H of Def A/m satisfies H Ô Spec A,m. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
18 Laudal s classical moduli theory Classification of commutative moduli of modules If there is a commutative solution of the moduli of a family of A-modules M = {M i } i I, the space can be defined as the set of all modules (with the Zariski=Jacobson topology), and the local ring in the point M can be defined as H M, the hull of Def M. We notice that commutative schemes can be defined this way, though we have some problems with the classification of prime ideals that are not maximal. The problem turns out to be solved in a natural way. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
19 Laudal s classical moduli theory Classification of commutative moduli of modules If there is a commutative solution of the moduli of a family of A-modules M = {M i } i I, the space can be defined as the set of all modules (with the Zariski=Jacobson topology), and the local ring in the point M can be defined as H M, the hull of Def M. We notice that commutative schemes can be defined this way, though we have some problems with the classification of prime ideals that are not maximal. The problem turns out to be solved in a natural way. Notice that the tangent space in a point is essential for pointing out the directions of deformations. This is, as it has to be, directly involved in the computation of H by the GMMP s. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
20 Laudal s classical moduli theory Classification of commutative moduli of modules If there is a commutative solution of the moduli of a family of A-modules M = {M i } i I, the space can be defined as the set of all modules (with the Zariski=Jacobson topology), and the local ring in the point M can be defined as H M, the hull of Def M. We notice that commutative schemes can be defined this way, though we have some problems with the classification of prime ideals that are not maximal. The problem turns out to be solved in a natural way. Notice that the tangent space in a point is essential for pointing out the directions of deformations. This is, as it has to be, directly involved in the computation of H by the GMMP s. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
21 Laudal s Noncommutative Deformation Theory Computing non-commutatively The computation of the hull works very well even if the k-algebra A is non-commutative. In some way, this solves the problem of a noncommutative scheme, but it does not solve the noncommutative moduli problem. ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
22 Laudal s Noncommutative Deformation Theory Computing non-commutatively The computation of the hull works very well even if the k-algebra A is non-commutative. In some way, this solves the problem of a noncommutative scheme, but it does not solve the noncommutative moduli problem. The new, and as all simple ideas, genius idea is to look at semi-points (=tuples of points) rather than points, and the tangent directions from one point to another. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
23 Laudal s Noncommutative Deformation Theory Computing non-commutatively The computation of the hull works very well even if the k-algebra A is non-commutative. In some way, this solves the problem of a noncommutative scheme, but it does not solve the noncommutative moduli problem. The new, and as all simple ideas, genius idea is to look at semi-points (=tuples of points) rather than points, and the tangent directions from one point to another. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
24 Laudal s Noncommutative Deformation Theory Definition The objects in the category a r are the k-algebras S with morphisms commuting in the diagram k r ι S, id ϱ k r such that ker(ϱ) n = 0. We call ker(ϱ) = I(S) the radical, the morphisms are the morphisms commuting with ι and ϱ. The category a r is called the category of r-pointed Artinian k-algebras. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
25 Laudal s Noncommutative Deformation Theory Definition The objects in the category a r are the k-algebras S with morphisms commuting in the diagram k r ι S, id ϱ k r such that ker(ϱ) n = 0. We call ker(ϱ) = I(S) the radical, the morphisms are the morphisms commuting with ι and ϱ. The category a r is called the category of r-pointed Artinian k-algebras.the notation â r denotes the procategory of a r, the category of objects that are projective limits of objects in a r. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
26 Laudal s Noncommutative Deformation Theory Definition The objects in the category a r are the k-algebras S with morphisms commuting in the diagram k r ι S, id ϱ k r such that ker(ϱ) n = 0. We call ker(ϱ) = I(S) the radical, the morphisms are the morphisms commuting with ι and ϱ. The category a r is called the category of r-pointed Artinian k-algebras.the notation â r denotes the procategory of a r, the category of objects that are projective limits of objects in a r. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
27 Laudal s Noncommutative Deformation Theory Definition Let A be a k-algebra, let M = {M 1,..., M r } be A-modules, and let M = r i=1 M i. The noncommutative deformation functor Def M : a r Sets is given by: Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
28 Laudal s Noncommutative Deformation Theory Definition Let A be a k-algebra, let M = {M 1,..., M r } be A-modules, and let M = r i=1 M i. The noncommutative deformation functor Def M : a r Sets is given by: Def M (S) = {S k A-Mod M S, flat over S : k r S M S M}/ = where the equivalence of M S and M S is given as an isomorphism M S M S. M Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
29 Laudal s Noncommutative Deformation Theory Definition Let A be a k-algebra, let M = {M 1,..., M r } be A-modules, and let M = r i=1 M i. The noncommutative deformation functor Def M : a r Sets is given by: Def M (S) = {S k A-Mod M S, flat over S : k r S M S M}/ = where the equivalence of M S and M S is given as an isomorphism M S M S. M Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
30 Laudal s Noncommutative Deformation Theory Theorem There exists a semi-local formal moduli (Ĥ M, ˆξ M ) for the noncommutative deformation functor Def M. There is a homomorphism ι : A (H ij ) k r Hom k (M i, M j ). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
31 Laudal s Noncommutative Deformation Theory Theorem There exists a semi-local formal moduli (Ĥ M, ˆξ M ) for the noncommutative deformation functor Def M. There is a homomorphism ι : A (H ij ) k r Hom k (M i, M j ). Its kernel is given by ker ι = i,n a n i, where a i = ker ρ i : A End k (M i ) is the kernel of the structure homomorphism. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
32 Laudal s Noncommutative Deformation Theory Theorem There exists a semi-local formal moduli (Ĥ M, ˆξ M ) for the noncommutative deformation functor Def M. There is a homomorphism ι : A (H ij ) k r Hom k (M i, M j ). Its kernel is given by ker ι = i,n a n i, where a i = ker ρ i : A End k (M i ) is the kernel of the structure homomorphism. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
33 Laudal s Noncommutative Deformation Theory The space Simp A Let A be an associative, unital k-algebra, k algebraically closed of characteristic 0. All A-modules V are right A-modules, and the A-module structure on V is given by its structure morphism ρ V : A End k (V ). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
34 Laudal s Noncommutative Deformation Theory The space Simp A Let A be an associative, unital k-algebra, k algebraically closed of characteristic 0. All A-modules V are right A-modules, and the A-module structure on V is given by its structure morphism ρ V : A End k (V ). We Define the topological space X = Simp(A) = Simp < (A) which has as points the set of finite-dimensional A-modules, and the topology is defined by the base of open sets D(f ) = {V X = Simp(A) ρ V : A End k (V ) is invertible }. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
35 Laudal s Noncommutative Deformation Theory The space Simp A Let A be an associative, unital k-algebra, k algebraically closed of characteristic 0. All A-modules V are right A-modules, and the A-module structure on V is given by its structure morphism ρ V : A End k (V ). We Define the topological space X = Simp(A) = Simp < (A) which has as points the set of finite-dimensional A-modules, and the topology is defined by the base of open sets D(f ) = {V X = Simp(A) ρ V : A End k (V ) is invertible }. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
36 Laudal s Noncommutative Deformation Theory A parenthesis When A is commutative, this is nothing but the usual definition of an affine variety. The simple modules are all of dimension one and in one-to-one correspondence with the maximal ideals. Notice that we choose use the term variety when we use only maximal ideals, before we generalize to schemes. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
37 Laudal s Noncommutative Deformation Theory Noncommutative Schemes We define a semi-locally sheaf of rings on X = Simp(A): For each basis open set D(f ), f A we define the ring of regular functions on D(f ) by ivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
38 Laudal s Noncommutative Deformation Theory Noncommutative Schemes We define a semi-locally sheaf of rings on X = Simp(A): For each basis open set D(f ), f A we define the ring of regular functions on D(f ) by A f = {φ f : p(d(f )) c D(f ) O(π, c ) φ f (w) = af n, a A, n N} where p(d(f )) denotes the family of finite subsets of D(f ). This is well-definied because ρ V (f ) is invertible for all V c : Then ρ(η c (f )) = u is a unit and η c (f ) = u + r with u ker ρ. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
39 Laudal s Noncommutative Deformation Theory Noncommutative Schemes We define a semi-locally sheaf of rings on X = Simp(A): For each basis open set D(f ), f A we define the ring of regular functions on D(f ) by A f = {φ f : p(d(f )) c D(f ) O(π, c ) φ f (w) = af n, a A, n N} where p(d(f )) denotes the family of finite subsets of D(f ). This is well-definied because ρ V (f ) is invertible for all V c : Then ρ(η c (f )) = u is a unit and η c (f ) = u + r with u ker ρ. Because D(f ) is a base for the topology, we can define the semi-local sheaf of rings on X = Simp(A) by O X (U) = lim D(f ) U A f. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
40 Laudal s Noncommutative Deformation Theory Noncommutative Schemes We define a semi-locally sheaf of rings on X = Simp(A): For each basis open set D(f ), f A we define the ring of regular functions on D(f ) by A f = {φ f : p(d(f )) c D(f ) O(π, c ) φ f (w) = af n, a A, n N} where p(d(f )) denotes the family of finite subsets of D(f ). This is well-definied because ρ V (f ) is invertible for all V c : Then ρ(η c (f )) = u is a unit and η c (f ) = u + r with u ker ρ. Because D(f ) is a base for the topology, we can define the semi-local sheaf of rings on X = Simp(A) by O X (U) = lim D(f ) U A f. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
41 Laudal s Noncommutative Deformation Theory The Serre s theorem We have to do some adjustments: Simp(A) Spec(A) in the sense that we have to remove all eventual tangent directions from one (quotient of) a prime ideal to another. With this adjustmnent we have Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
42 Laudal s Noncommutative Deformation Theory The Serre s theorem We have to do some adjustments: Simp(A) Spec(A) in the sense that we have to remove all eventual tangent directions from one (quotient of) a prime ideal to another. With this adjustmnent we have A = O Spec A (Spec A), and so the noncommutative algebraic geometry is a true generalization of the commutative. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
43 Laudal s Noncommutative Deformation Theory The Serre s theorem We have to do some adjustments: Simp(A) Spec(A) in the sense that we have to remove all eventual tangent directions from one (quotient of) a prime ideal to another. With this adjustmnent we have A = O Spec A (Spec A), and so the noncommutative algebraic geometry is a true generalization of the commutative. Laudal s noncommutative theory is a solution of noncommutative moduli problems (e.g. not good quotients of groups). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
44 Laudal s Noncommutative Deformation Theory The Serre s theorem We have to do some adjustments: Simp(A) Spec(A) in the sense that we have to remove all eventual tangent directions from one (quotient of) a prime ideal to another. With this adjustmnent we have A = O Spec A (Spec A), and so the noncommutative algebraic geometry is a true generalization of the commutative. Laudal s noncommutative theory is a solution of noncommutative moduli problems (e.g. not good quotients of groups). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
45 Application of noncommutative algebraic geometry Dynamics In the following, we will assume that all k-algebras are finitely generated. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
46 Application of noncommutative algebraic geometry Dynamics In the following, we will assume that all k-algebras are finitely generated.let alg k /A be the category where the objects are the k-algebra homomorphisms κ : A R, and the morphisms ψ Mor(κ, κ ) are the commutative diagrams R κ A ψ κ R. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
47 Application of noncommutative algebraic geometry Dynamics In the following, we will assume that all k-algebras are finitely generated.let alg k /A be the category where the objects are the k-algebra homomorphisms κ : A R, and the morphisms ψ Mor(κ, κ ) are the commutative diagrams R κ A ψ κ R. We then have the functor defined by Der k (A, κ) = Der k (A, R). Der k (A, ) : alg k /A sets Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
48 Application of noncommutative algebraic geometry Dynamics In the following, we will assume that all k-algebras are finitely generated.let alg k /A be the category where the objects are the k-algebra homomorphisms κ : A R, and the morphisms ψ Mor(κ, κ ) are the commutative diagrams R κ A ψ κ R. We then have the functor defined by Der k (A, κ) = Der k (A, R). Der k (A, ) : alg k /A sets Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
49 Application of noncommutative algebraic geometry Lemma The functor Der k (A, ) : alg k /A sets is represented by ι : A Ph(A) with the universal derivation d : A Ph(A). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
50 Application of noncommutative algebraic geometry Lemma The functor Der k (A, ) : alg k /A sets is represented by ι : A Ph(A) with the universal derivation d : A Ph(A). Proof. Let F = k t 1, t 2,..., t r be the free noncommutative k-algebra in r variables mapping surjectively onto A, and let I = ker π. Then we put π : F A, Ph(A) = k t 1, t 2,..., t r, dt 1, dt 2,..., dt r /(I, di ). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
51 Application of noncommutative algebraic geometry Lemma The functor Der k (A, ) : alg k /A sets is represented by ι : A Ph(A) with the universal derivation d : A Ph(A). Proof. Let F = k t 1, t 2,..., t r be the free noncommutative k-algebra in r variables mapping surjectively onto A, and let I = ker π. Then we put π : F A, Ph(A) = k t 1, t 2,..., t r, dt 1, dt 2,..., dt r /(I, di ). Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
52 Application of noncommutative algebraic geometry Geometry of dynamics We see that Ph(A) gives a point with a tangent directions, and as the process can be repeated, Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
53 Application of noncommutative algebraic geometry Geometry of dynamics We see that Ph(A) gives a point with a tangent directions, and as the process can be repeated, Ph (A) = Ph n (A), Ph(A) represents a point together with the momenta of all higher orders. This is the dynamics of an algebraic system. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
54 Application of noncommutative algebraic geometry Geometry of dynamics We see that Ph(A) gives a point with a tangent directions, and as the process can be repeated, Ph (A) = Ph n (A), Ph(A) represents a point together with the momenta of all higher orders. This is the dynamics of an algebraic system. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
55 Application of noncommutative algebraic geometry Physics As Platon claimed, we can only observe change (shadows). So physics is classification of change, i.e. dynamics. This means that the strange physics of Arnfinn is not strange at all: Start with the definition of a particle by letting it have the observed invariants as tangent directions in a dynamic system, and find its moduli. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
56 Application of noncommutative algebraic geometry Physics As Platon claimed, we can only observe change (shadows). So physics is classification of change, i.e. dynamics. This means that the strange physics of Arnfinn is not strange at all: Start with the definition of a particle by letting it have the observed invariants as tangent directions in a dynamic system, and find its moduli. Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
57 Application of noncommutative algebraic geometry Conclusion Thank You! Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
58 Application of noncommutative algebraic geometry E. Eriksen. An Introduction to noncommutative Deformations of Modules. Lect. Notes Pure Appl. Math., 243 (2005), R. Hartshorne. Algebraic geometry Graduate Texts in Mathematics, No. 52 New York 1977 ISBN O.A. Laudal. Formal moduli of algebraic structures. Lecture Notes in Mathematics, Springer Verlag, 754 (1979). O. A. Laudal. Noncommutative Algebraic Geometry. Rev. Mat. Iberoamericana, 19 (2) (2003), M. Schlessinger Functors of Artin rings. Trans-Amer.Math.soc., 130 (1968), A. Siqveland. Geometry of noncommutative k-algebras. J. Gen. Lie Theory Appl., 5 (2011), Eivind Eriksen, O. Arnfinn Laudal, and Arvid Siqveland Laudal s Noncommutative Geometry / 21
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