Classification problems in symplectic linear algebra Jonathan Lorand Institute of Mathematics, University of Zurich UC Riverside, 15 January, 2019 Thanks to my collaborators: Christian Herrmann (Darmstadt) Alan Weinstein (Berkeley) Alessandro Valentino (Zurich) Introduction... Plan: 1. Introduction 2. Symplectic vectors spaces 3. Why symplectic? Connection to dynamical systems 4. More symplectic linear algebra 5. Some classification problems 6. Poset representations 7. A more general picture Goals: I Basic introduction to linear symplectic geometry I Poset representations as a tool for classification problems I Hint at a category-theoretic picture A theme: Connection between symplectic geometry and (twisted) involutions: symplectic structures as fixed points in an appropriate sense Context I Baez & team: black-box functors often land in categories where I objects: symplectic vector spaces I morphisms: lagrangian relations I Weinstein: the \symplectic category" I Scharlau & Co.: developed a category-theoretic framework in late 70's with focus on quadratic forms I School of Kiev (Navarova & Roiter): representations of posets, quivers, algebras; Sergeichuk: applications to linear algebra I Representations of quivers I Involutions / duality involutions in categories Symplectic geometry?? A first explanation via (anti)analogy... A Euclidean structure on V = Rn is a bilinear form B : V × V −! R which is I non-degenerate: if B(v; w) = 0 8w 2 V , then v = 0 I symmetric: B(v; w) = B(w; v) 8v; w 2 V I positive definite: I B(v; v) ≥ 0 8v 2 V I If B(v; v) = 0, then v = 0 A Euclidean structure B on V gives us: I lengths: kvk := B(v; v) B(v;w) I angles: cos(θ) := kvkkwk for θ = v\w 2 [0; π] More generally: a metric structure on V = Rn is a bilinear form B : V × V −! R which is non-degenerate and symmetric (but not necessarily positive definite). From this one can define a \length", but it might be zero or negative for non-zero vectors. [E.g.: Lorentzian geometry, as in Einstein's theories of relativity] Note: this definition works for a vector space V over any field k. A symplectic structure on V (over k) is a bilinear form ! : V × V −! k which is non-degenerate and antisymmetric: !(v; w) = −!(w; v) 8v; w 2 V : Note: if char(k) 6= 2, then !(v; v) = 0 8v 2 V : We'll stick mostly with k = R (and always char(k) 6= 2). A symplectic vector space is (V ;!), where ! is a symplectic form on V . Given (V ;!) and (V 0;!0), a linear map f : V ! V 0 is a (linear) symplectomorphism if !0(fv; fw) = !(v; w) 8v; w 2 V : One might also say \isometry" (even though we don't have a \metric"). Fact: Every symplectic vector space is necessarily even dimensional. Fact: Any two symplectic vector spaces of the same (finite) dimension are symplectomorphic. Fact: Given any vector space U, the space U∗ ⊕ U carries a canonical symplectic structure, which I'll usually denote by Ω: Ω((ξ; v); (η; w)) = ξ(w) − η(v) for ξ; η 2 U∗; v; w 2 U: Let (V ;!) be symplectic, with dim V = 2n. A basis (q1; :::; qn; p1; :::; pn) of V is a symplectic basis if !(qi ; qj ) = 0 8i; j = 1; ::; n !(pi ; pj ) = 0 8i; j = 1; ::; n ( 1 if i = j !(qi ; pj ) = 0 else. Every (V ;!) admits a symplectic basis (many, actually). Given a symplectic basis, the associated coordinate matrix of ! is a block matrix of the form 0 I : −I 0 Any symplectic form ! on V induces an isomorphism !~ : V ! V ∗; v 7! !(v; −): Note: f symplecto , f ∗!~f =! ~. Note: if (q1; ::; qn; p1; ::; pn) is a symplectic basis, and ∗ ∗ ∗ ∗ ∗ (q1 ; ::; qn ; p1 ; ::; pn ) the dual basis in V , the coordinate matrix of! ~ is 0 − I ; I 0 the inverse of which is 0 I : −I 0 Why symplectic?? Origins of symplectic geometry: classical mechanics (planetary motion, projectiles, etc.). More precisely: origins are in Hamiltonian mechanics I Newton's mechanics: from ca. 1687 I Lagrange's mechanics: from ca. 1788 I Hamilton's mechanics: from ca. 1833 Very quick sketch: from Newtonian to Hamiltonian Example: Harmonic oscillator (e.g. a mass attached to a coil spring). Newton: (\F = ma") mx¨ = −Cx: We can rewrite as a system of 1st order ODEs. Set: q(t) := x(t) p(t) := mx_(t); and get 1 q_(t) = p(t) m p_(t) = −Cq(t) Reformulate the equations as: q_ 0 1 Cq(t) = 1 p_ −1 0 m p(t) @ ! 0 1 @q H(q; p) = @ −1 0 @p H(q; p) 1 2 1 1 2 where H(p; q) := 2 Cq + 2 m p . The function H is called the Hamiltonian of the dynamical system, and Hamilton's equations are @ ! q_ @p H(q; p) = @ =: XH (q; p): p_ − @q H(q; p) XH (q; p) is called the hamiltonian vector field associated to H. The set of all possible (generalized) positions q and (generalized) momenta p in a dynamical system is called phase space. In general: phase space modelled as a symplectic manifold (M;!), or Poisson manifold; we'll stick with (V ;!). A hamiltonian vector field XH : V ! V is related to the function H by @ −1 0 1 @q H(q; p) XH (v) =! ~ ◦ dH(v)= −1 0 @ ; @p H(q; p) thinking of dH(v)(−): V ! R as a 1-form. Equivalently: !~ ◦ XH (v) = dH(v) i.e. !(XH (v); − ) = dH(v)(−): Role of symplectomorphisms: I Symmetries of phase space: solutions of Ham. equations are mapped to solutions. I Time-evolution/flow of a Ham. system (V ; !; H): Given a time interval [t0; t1], we have a symplectomorphism V −! V ; (q0; p0) 7! (q(t1); p(t1)) where c(t) = (q(t); p(t)) is the solution to the Ham. initial value problem c_(t) = XH (c(t)) c(0) = (q0; p0) Upshots of Hamiltonian mechanics: (compared to Newtonian; comparing with Lagrangian is more complicated!) I a framework which is more general/abstract/conceptual/geometric I has a variational formulation (\principle of stationary action") I beautiful interplay between geometry and physics; e.g. symmetries $ conserved quantities Example benefit: even if one can't \solve" a Hamiltonian system, one can often prove qualitative aspects. More symplectic linear algebra... (V ;!) symplectic. Given a subspace U ⊆ V , its (symplectic) orthogonal is the subspace U! = fv 2 V j !(v; u) = 0 8u 2 Ug: Special subspaces: ! I symplectic U \ U = 0 ! I isotropic U ⊆ U ! I coisotropic U ⊆ U ! I lagrangian U = U . The operation (−)! defines an order-reversing involution on the poset Σ(V ) of subspaces of V . 0 0 0 0 Given (V ;!) and (V ;! ) symplectic (V ⊕ V ;! ⊕ ! ). Given a (linear) symplectomorphism f : V ! V 0, its graph Γ(f ) ⊆ V ⊕ V 0 is a lagrangian subspace of (V ⊕ V 0; (−!) ⊕ !0): Notation: for V with symplectic !, V := same vector space but with \ − !": A (linear) lagrangian relation V ! V 0 is a lagrangian subspace L ⊆ V ⊕ V 0: Note: these form the morphisms of a category; composition is the same as for set-relations. (V ;!) symplectic. Def: A vector field X : V ! V is hamiltonian if! ~ ◦ X (v) = dH(v) for some function H. Call it linear when X is a linear map. Fact: X lin. ham. , !~X = −X ∗!~. Symplectomorphisms V ! V form the symplectic group Sp(V ;!); it's a Lie group. Fact: The set sp(V ;!) of linear hamiltonian vector fields on V corresponds to the Lie algebra of Sp(V ;!). Def: denote by Lag(V ;!) the set of lagrangian relations L : V ! V . Some classification problems... Sp(V ;!) acts on itself, Lag(V ;!), and sp(V ;!) by conjugation: Sp(V ;!) × Sp(V ;!) ! Sp(V ;!); (f ; g) 7! fgf −1: Sp(V ;!) × Lag(V ;!) ! Lag(V ;!); (f ; L) 7! fLf −1: Sp(V ;!) × sp(V ;!) ! sp(V ;!); (f ; X ) 7! fXf −1: Compare with: GL(V ) × End(V ) ! End(V ); (f ; η) 7! f ηf −1. Typical questions: I what are the orbits? I can we find representatives given in a normal form? Common theme in algebra: I objects of study (often) decompose into basic building blocks, and this decomposition is sometimes essentially unique I strategy: classify the indecomposable building blocks. Example: GL(V ) × End(V ) ! End(V ); (f ; η) 7! f ηf −1. Consider a category we'll call Endk: I Objects:(U; η), with η 2 End(U) 0 0 0 I Morphisms: a map f :(U; η) ! (U ; η ) is a linear map f : U ! U such that U f U0 η η0 commutes. U f U0 In particular: (U; η) and (U0; η0) are isomorphic if there exists f 2 GL(V ) such that f ηf −1 = η0. Direct sums:(U; η) ⊕ (U0; η0) := (U ⊕ U; η ⊕ η0). Indecomposable = not isomorphic to some direct sum with (at least) two non-zero summands. Fact: (Krull-Schmidt holds) Every (U; η) is isomorphic to a direct sum of indecomposable pieces, and such a decomposition is essentially unique. For general k, the indecomposable objects are (up to iso): m (k[X ]=(p ); µX ) p 2 k[X ] monic irreducible; m 2 N; where the endomorphism µX is \multiplication by X ". For k = C: monic irreducibles p are p(X ) = X − λ for any λ 2 C. For k = R: ( p(X ) = X − λ, λ 2 R; or p(X ) = X 2 − 2<(λ)X + jλj2 λ 2 CnR: Normal forms: e.g. Jordan canonical form. For Sp(V ;!), Lag(V ;!) and sp(V ;!): I Direct sums: are orthogonal direct sums E.g. (V ; !; g) ⊕ (V 0;!0; g 0) := (V ⊕ V 0;! ⊕ !0; g ⊕ g 0).