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Home , Symplectic vector space

Classification problems in symplectic linear algebra

Jonathan Lorand Institute of , 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 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

B : V × V −→ R which is

I non-degenerate: if B(v, w) = 0 ∀w ∈ V , then v = 0

I symmetric: B(v, w) = B(w, v) ∀v, w ∈ V

I positive definite:

I B(v, v) ≥ 0 ∀v ∈ 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 ∈ [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 V over any field k. 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) 6= 2, then ω(v, v) = 0 ∀v ∈ V .

We’ll stick mostly with k = R (and always char(k) 6= 2). A symplectic vector space is (V , ω), where ω is a symplectic form on V .

Given (V , ω) and (V 0, ω0), a f : V → V 0 is a (linear) if

ω0(fv, fw) = ω(v, w) ∀v, w ∈ 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) 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. Let (V , ω) be symplectic, with dim V = 2n. A (q1, ..., qn, p1, ..., pn) of V is a symplectic basis if

ω(qi , qj ) = 0 ∀i, j = 1, .., n

ω(pi , pj ) = 0 ∀i, j = 1, .., n ( 1 if i = j ω(qi , pj ) = 0 else.

Every (V , ω) admits a symplectic basis (many, actually).

Given a symplectic basis, the associated coordinate of ω is a of the form  0  I . −I 0 Any symplectic form ω on V induces an isomorphism

ω˜ : V → V ∗, v 7→ ω(v, −).

Note: f symplecto ⇔ f ∗ω˜f =ω ˜.

Note: if (q1, .., qn, p1, .., pn) is a symplectic basis, and ∗ ∗ ∗ ∗ ∗ (q1 , .., qn , p1 , .., pn ) the dual basis in V , the coordinate matrix ofω ˜ 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

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 (M, ω), or Poisson manifold; we’ll stick with (V , ω).

A hamiltonian vector field XH : V → V is related to the function H by

 ∂  −1  0 1  ∂q H(q, p) XH (v) =ω ˜ ◦ dH(v)= −1 0 ∂ , ∂p H(q, p) thinking of dH(v)(−): V → R as a 1-form. Equivalently:

ω˜ ◦ XH (v) = dH(v) i.e. ω(XH (v), − ) = dH(v)(−). Role of :

I Symmetries of phase space: solutions of Ham. equations are mapped to solutions.

I Time-evolution/flow of a Ham. system (V , ω, 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 , ω) symplectic.

Given a subspace U ⊆ V , its (symplectic) orthogonal is the subspace

Uω = {v ∈ V | ω(v, u) = 0 ∀u ∈ U}.

Special subspaces: ω I symplectic U ∩ U = 0 ω I isotropic U ⊆ U ω I coisotropic U ⊆ U ω I lagrangian U = U .

The operation (−)ω defines an order-reversing involution on the poset Σ(V ) of subspaces of V . 0 0 0 0 Given (V , ω) and (V , ω ) symplectic (V ⊕ V , ω ⊕ ω ).

Given a (linear) symplectomorphism f : V → V 0, its graph Γ(f ) ⊆ V ⊕ V 0 is a lagrangian subspace of

(V ⊕ V 0, (−ω) ⊕ ω0).

Notation: for V with symplectic ω,

V := same vector space but with “ − ω”.

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 , ω) symplectic.

Def: A vector field X : V → V is hamiltonian ifω ˜ ◦ X (v) = dH(v) for some function H. Call it linear when X is a linear map.

Fact: X lin. ham. ⇔ ω˜X = −X ∗ω˜.

Symplectomorphisms V → V form the symplectic Sp(V , ω); it’s a .

Fact: The set sp(V , ω) of linear hamiltonian vector fields on V corresponds to the of Sp(V , ω).

Def: denote by Lag(V , ω) the set of lagrangian relations L : V → V . 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) 7→ fgf −1. Sp(V , ω) × Lag(V , ω) → Lag(V , ω), (f , L) 7→ fLf −1. Sp(V , ω) × sp(V , ω) → sp(V , ω), (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 ∈ 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 ∈ 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 ∈ k[X ] monic irreducible, m ∈ N, where the endomorphism µX 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, λ ∈ R, or p(X ) = X 2 − 2<(λ)X + |λ|2 λ ∈ C\R.

Normal forms: e.g. Jordan canonical form. For Sp(V , ω), Lag(V , ω) and sp(V , ω):

I Direct sums: are orthogonal direct sums E.g. (V , ω, g) ⊕ (V 0, ω0, g 0) := (V ⊕ V 0, ω ⊕ ω0, g ⊕ g 0).

I Indecomposability: analogously

I Define classes of objects as (V , ω, g), (V , ω, L), (V , ω, X ), respectively

I For morphisms: want isomorphisms to be symplectomorphisms

I Krull-Schmidt: objects decompose into indecomposables; essential uniqueness depends on further hypotheses. For C (and R?) we have essentially uniqueness. Poset representations...

Let (P, ≤) be a finite poset (with elements labeled 1 through n)

n A representation of P is a vector space V and subspaces {Ui }i=1 of V such that if i ≤ j in P, then Ui ⊆ Uj . So: a representation is a monotone map

ψ : P → Σ(V ).

0 0 0 Two representations (V ; U1, ..., Un) and (V ; U1, ..., Un) of P are isomorphic if there exists a linear isomorphism f : V → V 0 such that 0 f (Ui ) = Ui (for all i = 1, ..., n). Representations of a fixed poset P form a category, Repk(P).

Direct sums of poset reps: defined in the obvious way

Krull-Schmidt holds: any ψ ∈ Repk(P) is isomorphic to a direct sum of indecomposable poset reps, and such a decomposition is essentially unique. 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 (U0, η0) are isomorphic iff their associated poset reps are isomorphic; and indecomposables correspond to indecomposables Symplectic poset representations:

Start with a poset P equipped with an order-reversing (“twisted”) involution (−)⊥ : P → Pop.

Def: a symplectic poset rep of (P, ⊥) on a symplectic space (V , ω) is a monotone map

ϕ : P → Σ(V ), such that 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. 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

(V ⊕ V ; V ⊕ 0, 0 ⊕ V , Γ(Id), Γ(g)).

Note:

I V ⊕ 0 and 0 ⊕ V are symplectic subspaces of V ⊕ V ,

I 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. Symplectic reps of a fixed (P, ⊥) form a category, SRepk(P, ⊥).

Direct sums: again, orthogonal

Krull-Schmidt?: any ϕ ∈ SRepk(P, ⊥) is isomorphic to a direct sum of indecomposable poset reps; essential uniqueness depends on further hypotheses.

A basic task: classify indecomposables!

Strategy: relate SRepk(P, ⊥) and Repk(P).

Caveat: depending on P, it can be that Repk(P) is not well-understood. 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 repϕ ˆ.

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}. 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. 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 SRepk P using indecomposables of Repk 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. A more general picture...

Def: A category with twisted involution (a tCat) is (C, δ, η), where

δ a δop : C → Cop

is an adjoint equivalence, with unit η.

∗ ∗ Example: C = FinVectk, with δ(V ) = V , δ(f ) = f and ∗∗ ηV = ι : V → V the canonical isomorphism. A variant: take ηV = −1 · ι.

Example:(C, δ, id) where C is a poset with twisted involution δ. 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 commutes. ηx (δh) δopδx

Def:A morphism of fixed points (x, h) → (x 0, h0) is

f : x → x 0 in C such that

x h δx

f δf commutes. 0 x 0 h δx 0 ∗ ∗ Example: Take C = FinVectk, with δ(V ) = V , δ(f ) = f , ηV = −1 · ι. ∗ I Fixed points are (V , ω˜) withω ˜ : V → V such that

∗ ω˜ = −ω˜ ◦ ι encodes symplectic spaces (V , ω).

I Morphisms of fixed points encode symplectomorphisms (isometries):

V ω˜ V ∗

f f ∗ 0 V 0 ω˜ V 0∗

∗ Example: C = Repk(P, ⊥) = [(P, ⊥), FinVectk], with δψ = ψ and ∗∗ ηψ = −ι : ψ → ψ .

I Fixed points encode symplectic poset representations

I Morphisms of fixed points = morphisms of symplectic poset reps Example: Take C = Autk (objects are (V , g) with g ∈ Aut(V )); set

∗ ∗ −1 ∗∗ δ(V , g) := (V , (g ) ) and η(V ,g) := −ι : V → V .

∗ I Fixed points are (V , g, ω˜) withω ˜ : V → V such that

V ω˜ V ∗

g (g ∗)−1 commutes. V ω˜ V ∗

this encodes symplectomorphisms g ∈ Sp(V , ω).

I Morphisms of fixed points are symplectomorphisms f :(V , g, ω) → (V 0, g 0, ω0) such that fgf −1 = g 0. Example: Take C = Endk (objects are (V , X ) with X ∈ End(V )); set

∗ ∗ ∗∗ δ(V , X ) := (V , −X ) and η(V ,X ) := −ι : V → V .

∗ I Fixed points are (V , X , ω˜) withω ˜ : V → V such that

V ω˜ V ∗

X −X ∗ commutes. V ω˜ V ∗

this encodes lin. ham. vector fields X ∈ sp(V , ω).

I Morphisms of fixed points are symplectomorphisms f :(V , X , ω) → (V 0, X 0, ω0) such that fXf −1 = X 0. Summary of patterns and themes:

I symplectic (and metric) geometry is linked with (twisted) involutions

I where there are involutions, there are “split” and “non-split” things

I “non-split” things can be built by “doubling” ( symplectification)

I beautiful category theory is also lurking Thanks for listening!