- Non-commutative Deformations and the Structure of the. - H is the blow-up of the diagonal, and Z 2 is the obvious group-action. - Together this define a decomposition of the tangent space of H. - and we obtain a canonical decomposition of the complexified tangent bundle. - will be called a solution of the Lagrange equation. - for the particles of the system.. - A section φ of the bundle ˜ V , is now a function on the moduli space Simp(A), not just a function on the configuration space, Simp 1 (A), nor Simp 1 (A(σ. - V ˜ (v) of φ, at some point v ∈ Simp n (A), will be called a state of the particle, at the event v.. - a tangent of the deformation functor of V 0. - a tangent of the deformation functor of V 1. - a tangent of the deformation functor of V 2. - An integral curve γ of ξ is a solution of the equations of motion. - V 0 , of the particle V. - l , as the generator(s) of the generalized monoid,. - This is the subject of the next section.. - It is then easy to see that the kernel of the natural map,. - is defined in terms of the matrix Y ∈ M 2 (k), then the P h(A)-module V δ is the k <. - t, dt >-module defined by the action of the two matrices X, Y ∈ M 2 (k), and we find. - and we notice that this generator is of the type δ(d. - Clearly the exis- tence of the canonical homomorphism, i : M 2 (k. - A simple consequence of the definition of P h ∞ (A), that we identify all i n q , q = 1. - and via the composition of the canonical linear maps,. - Non-commutative Deformations and the Structure of the Moduli Space of. - an object H , of the category of pro-objects ˆ. - is the k-algebra of the quiver associated to c 0 , where { V i } r i=1. - inducing a corresponding composition of homomorphisms of the centers, Z(α. - Let us denote by σ the cyclical permutation of the integers { 1, 2. - We shall return to the study of the (notion of) completion, together with the degeneration pro- cesses that occur, at infinity in Simp n (A).. - locally in a Zariski neighborhood of the origin. - untill we have reached the endpoint of the last arrow. - is is an interval of the ordered graph, then γ i,j (n − 1).γ j,k (n + 1. - be the formal moduli of the family | Γ. - above, that any iterated extension of the { V i } r i=1 with extension type, i.e. - by a quotient space of the affine scheme,. - Now, repeat the basics of the construction of Spec(C(n)) above. - For any g-string, consider the non-commutative tangent space of the the pair of points,. - We shall call it the space of tensions, between the two points of the string.. - ∂σ = ∂y ∂σ = 0 at one of the points p i . - Spec(C(n)) is an ´etale covering of the moduli space Simp n (A(σ. - This Q, the Hamiltonian of the system, is in the singular case, when [δ. - of γ is in Simp n (A(σ)) should be called the lifetime of the particle. - are solutions of the differential equation,. - where τ 1 − τ 0 is the length of the integral curve γ connecting the two points v 0 and v 1 , i.e. - γ is the non-commutative version of the ordinary action integral, essentially defined by the equation,. - This is a well known consequence of the Schr¨ odinger equation above. - Above, Simp n (A(σ)) is our time-space , and Simp 1 (A) or Simp 1 (A(σ)) are the analogues of the classical configuration space. - This is the purpose of the next examples. - Look at the singularities of the fundamental vectorfield ξ ∈ Der k (C(n)).. - see (4.4), where we have treated the simple case of the harmonic oscillator.. - These relations induce a split-up of the representation V , i.e.. - t r ] is the commutative affine algebra of the configuration variety, X. - Here σ means spin, n the particle species, and n c the antiparticle of the species n. - This is the analogy of the Heisenberg uncertainty relation of the classical quantum theory.. - on the right hand side, and see that we have got a solution of the Lagrange equation,. - F i,j being the curvature tensor, of the connection. - where q (by definition) is the charge of the field. - are endomorphisms of the bundle. - the curvature of the connection, F(ξ i , ξ j. - We claimed that the integral curves of the vector field. - The fundamental equation of the dynamical system is,. - In the case of the potential, V = 1/2x 2 , we get the equations, x 2 =[δ](x. - In the case of the oscillator, V. - locally, in a Zariski neighborhood of the origin. - On the right hand side of the equations we have the terms, [δ](x. - The integral curves are therefore intersections of the form, C(c 1 , c 2. - where the u-coordinates are those of the trace ring, see Example (4.2). - of the form T rξ. - P h(A) leaves the dynamical structure of the versal family ˜ ρ-invariant, i.e. - 0, this proves that Q ξ is a constant of the theory.. - ∂τ ∂ as the generator of the vector fields on the τ -line. - It is, however, a limit of the finite representations, V n. - where M is the moduli space of the complete algebraic curves. - In fact, consider the restriction of the versal family. - H/Z 2 , classifying the family of pairs of points (o, x) of the Euclidean 3-space, E 3 . - Recall that ˜ H → H , is the (real) blow up of the diagonal. - of the projective 2-space, and of the covering 2-sphere. - space, of the vector (o, x). - Consider any metric on ˜ H , of the form,. - then K is the kinetic energy of the system. - In fact, this suggests that mass, is a property of the space ˜ H . - In this case it is a function of the surface of the exceptional fiber, i.e. - of the tangent bundle Θ S(l). - acts on the restriction of the tangent bundle Θ H to H(l). - where U(g) is the universal enveloping algebra of the Lie algebra g.. - Fix now a representation of su(2), given in terms of the generators, α 1. - the analogues of the Pauli matrices, and put, L = (α 1 , α 2 , α 3. - the decomposition of the tangent bundle of S(l) is different. - is a candidate for mass-density of the Universe. - H , is created by the (non-commutative) deformations of the obvious singularity in 3-dimensions, U. - Notice that we are working with the non-commutative model of the 3-dimensional space. - The tangent space of the versal base H (U. - of the deformation functor of U , as an algebra, is given, see e.g. - is the tangent space of the mini-versal deformation space, W (U. - of the imbedded singularity.. - the subspace of the tangent space T H(U. - This shows that the part of the modular substratum of H (U), sitting in T (2) is reduced to. - This is the purpose of the following, tentative, definition,. - just compare the coefficients of the resulting power series. - of the Lie-algebra su(3). - See also the part of the paper published in: