Classification problems in symplectic linear algebra

Transcription

1 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)

2 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

3 Goals: Basic introduction to linear symplectic geometry Poset representations as a tool for classification problems Hint at a category-theoretic picture A theme: Connection between symplectic geometry and (twisted) involutions: symplectic structures as fixed points in an appropriate sense

4 Context Baez & team: black-box functors often land in categories where objects: symplectic vector spaces morphisms: lagrangian relations Weinstein: the symplectic category Scharlau & Co.: developed a category-theoretic framework in late 70 s with focus on quadratic forms School of Kiev (Navarova & Roiter): representations of posets, quivers, algebras; Sergeichuk: applications to linear algebra Representations of quivers Involutions / duality involutions in categories

5 Symplectic geometry?? A first explanation via (anti)analogy... A Euclidean structure on V = R n is a bilinear form which is B : V V R non-degenerate: if B(v, w) = 0 w V, then v = 0 symmetric: B(v, w) = B(w, v) v, w V positive definite: B(v, v) 0 v V If B(v, v) = 0, then v = 0

6 A Euclidean structure B on V gives us: lengths: v := B(v, v) angles: cos(θ) := B(v,w) v w for θ = v w [0, π]

7 More generally: a metric structure on V = R n 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.

8 A symplectic structure on V (over k) is a bilinear form ω : V V k which is non-degenerate and antisymmetric: ω(v, w) = ω(w, v) v, w V. Note: if char(k) 2, then ω(v, v) = 0 v V. We ll stick mostly with k = R (and always char(k) 2).

9 A symplectic vector space is (V, ω), where ω is a symplectic form on V. Given (V, ω) and (V, ω ), a linear map f : V V is a (linear) symplectomorphism if ω (fv, fw) = ω(v, w) v, w V. One might also say isometry (even though we don t have a metric ).

10 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 U, v, w U.

11 Let (V, ω) be symplectic, with dim V = 2n. A basis (q 1,..., q n, p 1,..., p n ) of V is a symplectic basis if ω(q i, q j ) = 0 i, j = 1,.., n ω(p i, p j ) = 0 i, j = 1,.., n { 1 if i = j ω(q i, p j ) = 0 else. Every (V, ω) admits a symplectic basis (many, actually). Given a symplectic basis, the associated coordinate matrix of ω is a block matrix of the form ( ) 0 I. I 0

12 Any symplectic form ω on V induces an isomorphism ω : V V, v ω(v, ). Note: f symplecto f ωf = ω. Note: if (q 1,.., q n, p 1,.., p n ) is a symplectic basis, and (q1,.., q n, p1,.., p n) the dual basis in V, the coordinate matrix of ω is ( ) 0 I, I 0 the inverse of which is ( 0 I I 0 ).

13 Why symplectic?? Origins of symplectic geometry: classical mechanics (planetary motion, projectiles, etc.). More precisely: origins are in Hamiltonian mechanics Newton s mechanics: from ca Lagrange s mechanics: from ca Hamilton s mechanics: from ca. 1833

14 Very quick sketch: from Newtonian to Hamiltonian Example: Harmonic oscillator (e.g. a mass attached to a coil spring). Newton: ( F = ma ) mẍ = Cx. We can rewrite as a system of 1st order ODEs. Set: q(t) := x(t) p(t) := mẋ(t), and get q(t) = 1 m p(t) ṗ(t) = Cq(t)

15 Reformulate the equations as: ( ) ( ) ( ) q 0 1 Cq(t) = 1 ṗ 1 0 m p(t) ( ) ( 0 1 q H(q, p) = 1 0 H(q, p) p ) where H(p, q) := 1 2 Cq m p2. The function H is called the Hamiltonian of the dynamical system, and Hamilton s equations are ( ) ( ) q p H(q, p) = =: X ṗ H(q, p) H (q, p). q X H (q, p) is called the hamiltonian vector field associated to H.

16 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, ω), or Poisson manifold; we ll stick with (V, ω). A hamiltonian vector field X H : V V is related to the function H by ) ( ) X H (v) = ω 1 H(q, p) dh(v) =, ( q H(q, p) p thinking of dh(v)( ) : V R as a 1-form. Equivalently: ω X H (v) = dh(v) i.e. ω(x H (v), ) = dh(v)( ).

17 Role of symplectomorphisms: Symmetries of phase space: solutions of Ham. equations are mapped to solutions. Time-evolution/flow of a Ham. system (V, ω, H): Given a time interval [t 0, t 1 ], we have a symplectomorphism V V, (q 0, p 0 ) (q(t 1 ), p(t 1 )) where c(t) = (q(t), p(t)) is the solution to the Ham. initial value problem { ċ(t) = XH (c(t)) c(0) = (q 0, p 0 )

18 Upshots of Hamiltonian mechanics: (compared to Newtonian; comparing with Lagrangian is more complicated!) a framework which is more general/abstract/conceptual/geometric has a variational formulation ( principle of stationary action ) 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.

19 More symplectic linear algebra... (V, ω) symplectic. Given a subspace U V, its (symplectic) orthogonal is the subspace U ω = {v V ω(v, u) = 0 u U}. Special subspaces: symplectic U ω U = 0 isotropic U U ω coisotropic U ω U lagrangian U = U ω. The operation ( ) ω defines an order-reversing involution on the poset Σ(V ) of subspaces of V.

20 Given (V, ω) and (V, ω ) symplectic (V V, ω ω ). Given a (linear) symplectomorphism f : V V, its graph Γ(f ) V V is a lagrangian subspace of Notation: for V with symplectic ω, (V V, ( ω) ω ). V := same vector space but with ω. A (linear) lagrangian relation V V is a lagrangian subspace L V V. Note: these form the morphisms of a category; composition is the same as for set-relations.

21 (V, ω) symplectic. Def: A vector field X : V V is hamiltonian if ω X (v) = dh(v) for some function H. Call it linear when X is a linear map. Fact: X lin. ham. ωx = X ω. Symplectomorphisms V V form the symplectic group Sp(V, ω); it s a Lie group. Fact: The set sp(v, ω) of linear hamiltonian vector fields on V corresponds to the Lie algebra of Sp(V, ω). Def: denote by Lag(V, ω) the set of lagrangian relations L : V V.

22 Some classification problems... Sp(V, ω) acts on itself, Lag(V, ω), and sp(v, ω) by conjugation: Sp(V, ω) Sp(V, ω) Sp(V, ω), (f, g) fgf 1. Sp(V, ω) Lag(V, ω) Lag(V, ω), (f, L) flf 1. Sp(V, ω) sp(v, ω) sp(v, ω), (f, X ) fxf 1. Compare with: GL(V ) End(V ) End(V ), (f, η) f ηf 1. Typical questions: what are the orbits? can we find representatives given in a normal form? Common theme in algebra: objects of study (often) decompose into basic building blocks, and this decomposition is sometimes essentially unique strategy: classify the indecomposable building blocks.

23 Example: GL(V ) End(V ) End(V ), (f, η) f ηf 1. Consider a category we ll call End k : Objects: (U, η), with η End(U) Morphisms: a map f : (U, η) (U, η ) is a linear map f : U U such that f U U η f U U η commutes. In particular: (U, η) and (U, η ) are isomorphic if there exists f GL(V ) such that f ηf 1 = η. Direct sums: (U, η) (U, η ) := (U U, η η ). Indecomposable = not isomorphic to some direct sum with (at least) two non-zero summands.

24 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): (k[x ]/(p m ), µ X ) p k[x ] monic irreducible, m N, where the endomorphism µ X is multiplication by X. For k = C: monic irreducibles p are p(x ) = X λ for any λ C. For k = R: { p(x ) = X λ, p(x ) = X 2 2R(λ)X + λ 2 λ R, or λ C\R. Normal forms: e.g. Jordan canonical form.

25 For Sp(V, ω), Lag(V, ω) and sp(v, ω): Direct sums: are orthogonal direct sums E.g. (V, ω, g) (V, ω, g ) := (V V, ω ω, g g ). Indecomposability: analogously Define classes of objects as (V, ω, g), (V, ω, L), (V, ω, X ), respectively For morphisms: want isomorphisms to be symplectomorphisms Krull-Schmidt: objects decompose into indecomposables; essential uniqueness depends on further hypotheses. For C (and R?) we have essentially uniqueness.

26 Poset representations... Let (P, ) be a finite poset (with elements labeled 1 through n) A representation of P is a vector space V and subspaces {U i } n i=1 of V such that if i j in P, then U i U j. So: a representation is a monotone map ψ : P Σ(V ). Two representations (V ; U 1,..., U n ) and (V ; U 1,..., U n) of P are isomorphic if there exists a linear isomorphism f : V V such that f (U i ) = U i (for all i = 1,..., n).

27 Representations of a fixed poset P form a category, Rep k (P). Direct sums of poset reps: defined in the obvious way Krull-Schmidt holds: any ψ Rep k (P) is isomorphic to a direct sum of indecomposable poset reps, and such a decomposition is essentially unique.

28 Many classification problems of linear algebra can be encoded using poset representations. Example: Given an endomorphism (U, η), consider the poset P = {1, 2, 3, 4} with empty ordering and associate to (U, η) the following poset representation in V = U U: (U U; U 0, 0 U, Γ(Id), Γ(η)). Fact: objects (U, η) and (U, η ) are isomorphic iff their associated poset reps are isomorphic; and indecomposables correspond to indecomposables

29 Symplectic poset representations: Start with a poset P equipped with an order-reversing ( twisted ) involution ( ) : P P op. Def: a symplectic poset rep of (P, ) on a symplectic space (V, ω) is a monotone map such that ϕ : P Σ(V ), ϕ(i ) = ϕ(i) ω i P. Example: If P = {1 2}, with 1 = 2, then a symplectic poset rep ϕ of (P, ) corresponds to an isotropic subspace of (V, ω): ϕ(1) ϕ(2) = ϕ(1 ) = ϕ(1) ω.

30 Objects such as (V, ω, g), where g Sp(V, ω), can be encoded in symplectic poset reps: To (V, ω, g), associate the system of subspaces Note: (V V ; V 0, 0 V, Γ(Id), Γ(g)). V 0 and 0 V are symplectic subspaces of V V, Γ(Id) and Γ(g) are lagrangian subspace of V V. This is a symplectic poset rep of P = {1, 2, 3, 4}, with empty order, and 1 = 2 2 = 1 3 = 3 4 = 4. We can also treat Lag(V, ω) and sp(v, ω) with symplectic poset reps.

31 Symplectic reps of a fixed (P, ) form a category, SRep k (P, ). Direct sums: again, orthogonal Krull-Schmidt?: any ϕ SRep k (P, ) is isomorphic to a direct sum of indecomposable poset reps; essential uniqueness depends on further hypotheses. A basic task: classify indecomposables! Strategy: relate SRep k (P, ) and Rep k (P). Caveat: depending on P, it can be that Rep k (P) is not well-understood.

32 Given: (P, ), (V, ω). Def: A linear (ordinary) representation of (P, ) on V is a monotone map ψ : P Σ(V ). Any symplectic poset rep ϕ has an underlying linear rep ˆϕ. Given a linear rep ψ of (P, ) on V, define dual representation on V by ψ (i) = ψ(i ) = {ξ V ξ ψ(i ) 0}.

33 Symplectification: building symplectic reps from linear reps. Given a linear rep of (P, ), its symplectification is ψ : P Σ(V V, Ω) ψ (x) := ψ (x) ψ(x). Fact: ψ is a symplectic representation. We call an indecomposable symplectic rep split if it is (isomorphic to) a symplectification. Some indecomposable symplectic reps are non-split: they come from an ordinary indecomposable rep ψ : P Σ(V ) such that V happens to admit a symplectic form which is compatible with ψ (making ψ symplectic). We call such an ω a compatible form.

34 Magic Lemma (Sergeichuk / Scharlau et. al): Let ϕ be an indecomposable symplectic representation. Then ϕ is either split or non-split (but not both): 1. ϕ ψ, the symplectification of some indecomposable linear rep ψ. 2. ˆϕ is linearly indecomposable. Consequence: we can classify indecomposables of SRep k P using indecomposables of Rep k P, by 1. identifying which linear indecomposables admit compatible symplectic structures, and classifying these. Tricky part: a given linear indecomposable ψ might admit multiple non-equivalent compatible forms! 2. For those that don t admit compatible symplectic forms: symplectify! Current work (Hermann, L., Weinstein): Classification of triples of isotropic subspaces.

35 A more general picture... Def: A category with twisted involution (a tcat) is (C, δ, η), where δ δ op : C C op is an adjoint equivalence, with unit η. Example: C = FinVect k, with δ(v ) = V, δ(f ) = f and η V = ι : V V the canonical isomorphism. A variant: take η V = 1 ι. Example: (C, δ, id) where C is a poset with twisted involution δ.

36 Def: A fixed point in a tcat (C, δ, η) is (x, h) where h : x δ(x) op is an isomorphism in C such that x h (δx) op op η x (δh) commutes. δ op δx Def: A morphism of fixed points (x, h) (x, h ) is f : x x in C such that f x h δx δf commutes. x h δx

37 Example: Take C = FinVect k, with δ(v ) = V, δ(f ) = f, η V = 1 ι. Fixed points are (V, ω) with ω : V V such that ω = ω ι encodes symplectic spaces (V, ω). Morphisms of fixed points encode symplectomorphisms (isometries): V ω V f ω V V Example: C = Rep k (P, ) = [(P, ), FinVect k ], with δψ = ψ and η ψ = ι : ψ ψ. Fixed points encode symplectic poset representations Morphisms of fixed points = morphisms of symplectic poset reps f

38 Example: Take C = Aut k (objects are (V, g) with g Aut(V )); set δ(v, g) := (V, (g ) 1 ) and η (V,g) := ι : V V. Fixed points are (V, g, ω) with ω : V V such that g V ω V (g ) 1 commutes. V ω V this encodes symplectomorphisms g Sp(V, ω). Morphisms of fixed points are symplectomorphisms f : (V, g, ω) (V, g, ω ) such that fgf 1 = g.

39 Example: Take C = End k (objects are (V, X ) with X End(V )); set δ(v, X ) := (V, X ) and η (V,X ) := ι : V V. Fixed points are (V, X, ω) with ω : V V such that X V ω V X commutes. V ω V this encodes lin. ham. vector fields X sp(v, ω). Morphisms of fixed points are symplectomorphisms f : (V, X, ω) (V, X, ω ) such that fxf 1 = X.

40 Summary of patterns and themes: symplectic (and metric) geometry is linked with (twisted) involutions where there are involutions, there are split and non-split things non-split things can be built by doubling ( symplectification) beautiful category theory is also lurking

41 Thanks for listening!