Classi cation problems in symplectic linear algebra
Transcript of Classi cation problems in symplectic linear algebra
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!