Classification problems in symplectic linear algebra

Jonathan LorandInstitute 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 whereI objects: symplectic vector spacesI morphisms: lagrangian relations

I Weinstein: the “symplectic category”

I Scharlau & Co.: developed a category-theoretic framework in late70’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 ∀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 ∈ VI If B(v , v) = 0, then v = 0

A Euclidean structure B on V gives us:

I lengths: ‖v‖ := B(v , v)

I angles: cos(θ) := B(v ,w)‖v‖‖w‖ 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 positivedefinite).

From this one can define a “length”, but it might be zero or negative fornon-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

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 onV .

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”).

Fact: Every symplectic vector space is necessarily even dimensional.

Fact: Any two symplectic vector spaces of the same (finite) dimensionare symplectomorphic.

Fact: Given any vector space U, the space U∗ ⊕ U carries a canonicalsymplectic 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 dimV = 2n. A basis (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

ω(qi , pj) =

1 if i = j

0 else.

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

Given a symplectic basis, the associated coordinate matrix of ω is a blockmatrix 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 ∗ωf = ω.

Note: if (q1, .., qn, p1, .., pn) is a symplectic basis, and(q∗1 , .., q

∗n , p∗1 , .., p

∗n) the dual basis in V ∗, the coordinate matrix of ω is(

0 −II 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

q(t) =1

mp(t)

p(t) = −Cq(t)

Reformulate the equations as:(qp

)=

(0 1−1 0

)(Cq(t)1mp(t)

)=

(0 1−1 0

)( ∂∂qH(q, p)∂∂pH(q, p)

)

where H(p, q) := 12Cq

2 + 12

1mp2.

The function H is called the Hamiltonian of the dynamical system, andHamilton’s equations are(

qp

)=

(∂∂pH(q, p)

− ∂∂qH(q, p)

)=: XH(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, ω), orPoisson manifold; we’ll stick with (V , ω).

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

XH(v) = ω−1 dH(v) =(

0 1−1 0

)( ∂∂q

H(q, p)∂∂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 symplectomorphisms:

I Symmetries of phase space: solutions of Ham. equations are mappedto 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 valueproblem

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, onecan 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:

I symplectic Uω ∩ U = 0

I isotropic U ⊆ Uω

I coisotropic Uω ⊆ U

I lagrangian U = Uω.

The operation (−)ω defines an order-reversing involution on the posetΣ(V ) of subspaces of V .

Given (V , ω) and (V ′, ω′) symplectic (V ⊕ V ′, ω ⊕ ω′).

Given a (linear) symplectomorphism f : V → V ′, its graphΓ(f ) ⊆ V ⊕ V ′ is a lagrangian subspace of

(V ⊕ V ′, (−ω)⊕ ω′).

Notation: for V with symplectic 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 sameas for set-relations.

(V , ω) symplectic.

Def: A vector field X : V → V is hamiltonian if ω X (v) = dH(v) forsome 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’sa Lie group.

Fact: The set sp(V , ω) of linear hamiltonian vector fields on Vcorresponds to the Lie algebra 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, andthis 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)

I Morphisms: a map f : (U, η)→ (U ′, η′) is a linear map f : U → U ′

such that

U U ′

U U ′

η

f

η′

f

commutes.

In particular: (U, η) and (U ′, η′) are isomorphic if there existsf ∈ 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 (atleast) two non-zero summands.

Fact: (Krull-Schmidt holds) Every (U, η) is isomorphic to a direct sum ofindecomposable pieces, and such a decomposition is essentially unique.

For general k, the indecomposable objects are (up to iso):

(k[X ]/(pm), 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 sumsE.g. (V , ω, g)⊕ (V ′, ω′, g ′) := (V ⊕ V ′, ω ⊕ ω′, g ⊕ g ′).

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; essentialuniqueness depends on further hypotheses. For C (and R?) we haveessentially uniqueness.

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 Uini=1 of Vsuch that

if i ≤ j in P, then Ui ⊆ Uj .

So: a representation is a monotone map

ψ : P → Σ(V ).

Two representations (V ;U1, ...,Un) and (V ′;U ′1, ...,U′n) of P are

isomorphic if there exists a linear isomorphism f : V → V ′ such thatf (Ui ) = U ′i (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 ofindecomposable poset reps, and such a decomposition is essentiallyunique.

Many classification problems of linear algebra can be encoded using posetrepresentations .

Example: Given an endomorphism (U, η), consider the posetP = 1, 2, 3, 4 with empty ordering and associate to (U, η) the followingposet 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 posetreps 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 amonotone map

ϕ : P → Σ(V ),

such thatϕ(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 insymplectic 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 ofindecomposable poset reps; essential uniqueness depends on furtherhypotheses.

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 monotonemap

ψ : P → Σ(V ).

Any symplectic poset rep ϕ has an underlying linear rep ϕ.

Given a linear rep ψ of (P,⊥) on V , define dual representation on V ∗

byψ∗(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 indecomposablesymplectic rep split if it is (isomorphic to) a symplectification.

Some indecomposable symplectic reps are non-split: they come from anordinary indecomposable rep

ψ : P → Σ(V )

such that V happens to admit a symplectic form which is compatiblewith ψ (making ψ symplectic). We call such an ω a compatible form.

Magic Lemma (Sergeichuk / Scharlau et. al): Let ϕ be anindecomposable symplectic representation. Then ϕ is either split ornon-split (but not both):

1. ϕ ' ψ−, the symplectification of some indecomposable linear rep ψ.

2. ϕ is linearly indecomposable.

Consequence: we can classify indecomposables of SRepkP usingindecomposables of RepkP, by

1. identifying which linear indecomposables admit compatiblesymplectic 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 ofisotropic 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 ∗ andηV = ι : V → V ∗∗ the canonical isomorphism. A variant: takeηV = −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 (δx)op

δopδx

h

ηx (δh)op commutes.

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

f : x → x ′ in C such that

x δx

x ′ δx ′

h

f

h′

δf commutes.

Example: Take C = FinVectk, with δ(V ) = V ∗, δ(f ) = f ∗, ηV = −1 · ι.I Fixed points are (V , ω) with ω : V → V ∗ such that

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

I Morphisms of fixed points encode symplectomorphisms (isometries):

V V ∗

V ′ V ′∗

ω

f

ω′

f ∗

Example: C = Repk(P,⊥) = [(P,⊥),FinVectk], with δψ = ψ∗ andηψ = −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

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

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

V V ∗

V V ∗

ω

g (g∗)−1

ω

commutes.

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

I Morphisms of fixed points are symplectomorphismsf : (V , g , ω)→ (V ′, g ′, ω′) such that fgf −1 = g ′.

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) with ω : V → V ∗ such that

V V ∗

V V ∗

ω

X −X∗

ω

commutes.

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

I Morphisms of fixed points are symplectomorphismsf : (V ,X , ω)→ (V ′,X ′, ω′) such that fXf −1 = X ′.

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!