Quadratic Poisson structures on representations and double brackets - - PowerPoint PPT Presentation

Quadratic Poisson structures on representations and double brackets

Vladimir Rubtsov, ITEP, Moscow and LAREMA, Universit´ e d’Angers, (joint work with A.Odesskii and V. Sokolov,

• Theor. Math. Phys. 2012, 171, 442447 , arXiv:1208.2935v1

and work in progress) Talk at "Quantum Integrable Systems and Geometry". Faro-Olh˜ ao, Portugal, Septembre 3, 2012

Plan:

◮ Motivations; ◮ Representation functor, double and H−Poisson structures; ◮ Reminder of matrix case; ◮ AYBE and Compatibility of non-abelian Poisson brackets; ◮ Double Poisson brackets depending on a parameter ◮ Perspectives and output;

◮ A, an associative unital algebra over k ◮

V − a finite-dimensional k-vector space.

◮ The space RepV (A) of all representations of A in V is an

affine k-scheme; it comes equipped with a natural GL(V )-action whose orbits correspond to the isomorphism classes of representations.

◮ Many important varieties arise as (quotients of) representation

• spaces. For example, the moduli spaces of flat connections on

a closed manifold M can be interpreted as the space of isomorphisms classes of representations of the group algebra A = C[π1(M)] for the fundamental group of M.

Noncommutatitve Algebraic Geometry

◮ Varying A (while keeping V fixed) defines a contravariant

functor RepV : Algk → Schk

◮ Heuristic Principle (Kontsevich): A structure on a

noncommutative algebra A has a geometric meaning if it naturally induces standard geometric structures on all representation spaces {RepV (A)}.

◮ Applications: Cuntz-Quillen (1995), Kontsevich-Rosenberg

(2000), Ginzburg, Etingof, Schedler (2001-12), Van den Bergh (2004-2012), Le Bruyn (2008), Crawley-Boevey (2011), Berest with coauthors (2012) etc‘;

◮ In practice, the above principle works well only when A is a

formally smooth (= quasi-free) algebra (since in that case all RepV (A)’s are nonsingular).

◮ Example

Consider A/[A, A] . There is a canonical trace map TrV (A) : A/[A, A] → k[RepV (A)] which transforms elements of A/[A, A] to functions on RepV (A) for all V .

◮ Thus, by Kontsevich, A/[A, A] should be thought of as a

space of functions on the noncommutative (and probably non-existent!) ‘Spec(A)’.

◮ The schemes {RepV (A)} are supposed to give a good

approximation to Spec(A) that becomes ‘better and better’ as dimkV → ∞.

◮ Remark

There are interesting algebras for which all RepV (A) are trivial (e.g. the Weyl algebras).

Procesi Theorem

◮ To put a multiplicative structure in the game , we extend

Sym TrV (A) : Symk(A/[A, A]) → k[RepV (A)] .

◮ Theorem (Procesi). The image of Sym TrV (A) is

k[RepV (A)]GL(V ).

Van den Bergh’s Double Poisson Structures-1

◮ Let A be a f.g. associative C− algebra; ◮ Repm(A) := Hom(A, Matm(C)); ◮ GLm(C) acts by conjugation on Matm(C); ◮ Question (M.Van den Bergh): "What kind of structures we

need on A in order that Repm(A) and Repm(A)GLm(C) possess a Poisson structure?"

Van den Bergh’s Double Poisson Structures-2

Definition

A double Poisson bracket on A is a C−bilinear map { {−, −} } : A × A → A ⊗ A such that:

◮

{ {a, b} } = −{ {b, a} }◦, (a ⊗ b)◦ = b ⊗ a; (1)

◮ {

{−, −} } is a derivation in its second argument (wrt the outer bimodule structure on A): { {a, bc} } = (b ⊗ 1){ {a, c} } + { {a, b} }(1 ⊗ c); (2)

◮ {

{−, −} } satisfies a sort of Jacobi identity: { {a, { {b, c} }} } + σ{ {b, { {c, a} }} } + σ2{ {c, { {a, c} }} } = 0, (3) where { {a, b ⊗ c} } := { {a, b} } ⊗ c and σ(a ⊗ b ⊗ c) := c ⊗ a ⊗ b.

• M. Van den Bergh’s Double Poisson Structures-3

Define {−, −} : A × A → A by {−, −} := µ({ {a, b} }) = { {a, b} }′{ {a, b} }”

Theorem

(M.Van den Bergh) Let A, { {−, −} } be a double Poisson algebra. Then,

• 1. {−, −} is a derivation in its second argument and vanishes on

commutators in its first argument;

• 2. {−, −} induces an anti-symmetric bracket on A/[A, A];
• 3. {−, −} makes A into a left Loday algebra ({−, −} satisfies

the following version of the Jacobi identity: {a, {b, c}} = {{a, b}, c} + {b, {a, c}});

• 4. {−, −} makes A/[A, A] into a Lie algebra.

Crawley-Boevey H−structure-1

◮ Let A be a f.g. associative C− algebra, Der(A) its derivation

algebra and HH0(A) = A/[A, A];

◮ Any ∂ ∈ Der(A) - "descends" under the projection

p : A → HH0(A) to the map p(∂) : HH0(A) → HH0(A) such that p(∂)(p(a)) = p(∂(a));

◮ Definition

H− Poisson structure on A is a Lie bracket [−, −] on HH0(A) such that the map [p(a), −] ∈ EndHH0(A) is induced by some derivation ∂a : p(∂a) = [p(a), −].

Crawley-Boevey H−structure-2

◮ Example

Any double Poisson structure on A induces on H− structure on A via the multiplication µ: [p(a), p(b)] := p(µ({ {a, b} })).

◮ Theorem

(W.Crawley-Boevey) Each H− Poisson structure on A defines a unique Poisson structure on C[Repm(A)GLm] such that {Tr(a), Tr(b)} = Tr([p(a), p(b)]). .

The coordinate ring k[Repm(A)] is generated by symbols xj

i where

1 ≤ i, j ≤ m for all x ∈ A with the relations (xy)j

i =

• l

xl

i yj l ,

1j

i = δj i .

Example

Let A = k < x1, . . . , xN > then k[Repm(A)] = k[(xj

i,α)] where

1 ≤ α ≤ N The trace Tr(a) :=

i ai i defines GLm−invariant polynomial on

k[Repm(A)]

Non-abelian Poisson brackets : matrix case.

Suppose now k = C

◮ Matm(C) - matrix algebra . ◮ Take N exemplairs of Matm(C) and let xj i,α be a matrix

element of the matrix with a number 1 ≤ α ≤ N and 1 ≤ i, j ≤ m.

◮ A Poisson bracket on Matm(C) × . . . × Matm(C) is called a

non-abelian Poisson bracket iff the bracket between traces of any two matrix polynomials Pi(x1, ..., xN), i = 1, 2 is the trace of a matrix polynomial P3.

Theorem

Theorem

(Odesskii, R., Sokolov) Any linear non-abelian Poisson bracket has the form {xj

i,α, xj′ i′,β} = bγ α,βxj′ i,γδj i′ − bγ β,αxj i′,γδj′ i ;

(4) Any quadratic non-abelian Poisson bracket is given by {xj

i,α, xj′ i′,β} = rγǫ αβxj′ i,γxj i′,ǫ + aγǫ αβxk i,γxj′ k,ǫδj i′ − aγǫ βαxk i′,γxj k,ǫδj′ i ,

(5) Formula (4) defines a Poisson bracket iff bµ

αβbσ µγ = bσ αµbµ βγ;

(6)

Quadratic Non-Abelian brackets conditions

Formula (5) defines a Poisson bracket iff the following relations hold: rσǫ

αβ = −rǫσ βα,

(7) rλσ

αβrµν στ + rµσ βτ rνλ σα + rνσ ταrλµ σβ = 0,

(8) aσλ

αβaµν τσ = aµσ ταaνλ σβ,

(9) aσλ

αβaµν στ = aµσ αβrλν τσ + aµν ασrσλ βτ

(10) and aλσ

αβaµν τσ = aσν αβrλµ στ + aµν σβrσλ τα.

(11)

Poisson conditions: three ways of presentation

We may regard the tensors r and a as:

◮ operators on V ⊗ V , where V is an m-dimensional C−vector

space;

◮ elements of Matm(C) ⊗ Matm(C); ◮ operators on Matm(C).

First interpretation: AYBE

◮ The identities (7-11) can be written as

r12 = −r21, r23r12 + r31r23 + r12r31 = 0, a12a31 = a31a12, σ23a13a12 = a12r23 − r23a12, a32a12 = r13a12 − a32r13. (12)

◮ Here all operators act in V ⊗ V ⊗ V , σij means the

transposition of i-th and j-th components of the tensor product, and aij, rij mean operators a, r acting in the product

• f the i-th and j-th components.

Second interpretation

◮ In the second interpretation we consider the following elements

from Matm(C) ⊗ Matm(C): r = rkm

ij ei k ⊗ ej m,

a = akm

ij ei k ⊗ ej m, where ei j are the matrix

unities:

◮ ej i em k = δj kem i . Then (7), (8)-(11) are equivalent to (12), where

tensors belong to Matm(C) ⊗ Matm(C) ⊗ Matm(C).

◮ Namely, r12 = rmk ij

ei

k ⊗ ej m ⊗ 1 and so on. ◮ The element σ is given by σ = ej i ⊗ ei j . .

Third interpretation: Rota-Baxter operators

In the third interpretation one can define operators r, a,¯ r, a∗ : Matm(C) → Matm(C) by r(x)p

q = rlp nqxn l ,

a(x)p

q = alp nqxn l ,

¯ r(x)p

q = rpl nqxn l ,

a∗(x)p

q = apl qnxn l . Then for any x, y ∈ Matm(C)

a(x)a∗(y) = a∗(y)a(x), r(x) = −r∗(x), r(x)r(y) = r(xr(y)) + r(x)y), ¯ r(x) = −¯ r∗(x), ¯ r(x)¯ r(y) = ¯ r(x¯ r(y)) + ¯ r(x)y), a∗(ya(x)) = r(xa∗(y)) − r(x)a∗(y), a(x)a(y) = −a(r(y)x) − a(yr(x)), a∗(a(x)y) = r(a∗(y)x) − a∗(y)r(x), a(ya∗(x)) = −¯ r(xa(y)) + ¯ r(x)a(y), a∗(x)a∗(y) = a∗(¯ r(y)x) + a∗(y¯ r(x)), a(a∗(x)y) = −¯ r(a(y)x) + a(y)¯ r(x)

Relation between Non-Abelian and Double Poisson brackets

◮ Let µ : A ⊗ A → A be the multiplication µ(u ⊗ v) = uv; ◮ Let {−, −} := µ({

{−, −} }) be the Lie algebra structure on A/[A, A] (from (3) p.4);

◮ Lemma

C[Repm(A)] is a Poisson algebra, with Poisson bracket given by {xj

i,α, xl k,β} = {

{xα, xβ} }

′j

k {

{xα, xβ} }”l

i

(13) where (by the Sweedler convention) we write an element x of A ⊗ A as x′ ⊗ x′′ (here we drop the sign of sum).

Linear and quadratic double Poisson brackets

◮ Linear and quadratic double Poisson brackets on the free

associative algebra A = C < x1, . . . , xN > are given by

◮

{ {xi, xj} } = bk

ijxk ⊗ 1 − bk ji1 ⊗ xk.

and by

◮

{ {xα, xβ} } = ruv

αβ xu ⊗ xv + avu αβ xuxv ⊗ 1 − auv βα 1 ⊗ xvxu. (14)

AYBE and double Poisson structures

Theorem

Let V be a f.d vector space, A = T(V )−the tensor algebra. Suppose R ∈ End(A ⊗ A) defines on the basis elements of A by R(xα ⊗ xβ) = ruv

αβ xu ⊗ xv + avu αβ xuxv ⊗ 1 − auv βα 1 ⊗ xvxu.

(15) skew and satisfies the associative Yang-Baxter equation on A: R12 = −R21, R23R12 + R31R23 + R12R31 = 0. iff identities (7)-(11) hold. In this case { {xα, xβ} }R = R(xα ⊗ xβ) is a double Lie bracket on A. If, in addition, we extend R on A by the appropriate Leibniz rule (2) the bracket { {xα, xβ} }R is a quadratic double Poisson structure on A.

Examples and classification of low dimensional quadratic double Poisson brackets

It is easy to see that for m = 1 non-zero quadratic double Poisson brackets does not exist. In the simplest non-trivial case m = 2 the system of algebraic equations (7), (8)-(11) can be solved straightforwardly.

Theorem

Let m = 2. Then the following Cases 1-7 form a complete list of quadratic double Poisson brackets up to equivalence.

Case 1 One have r21

22 = −r12 22 = 1. The corresponding (non-zero)

double brackets read { {v, v} } = v ⊗ u − u ⊗ v; Case 2 One have r21

22 = −r12 22 = 1, a11 21 = a12 22 = 1. The

corresponding (non-zero) double brackets: { {v, v} } = v ⊗ u − u ⊗ v + vu ⊗ 1 − 1 ⊗ vu, { {v, u} } = u2 ⊗ 1, { {u, v} } = −1 ⊗ u2; Case 3 One have r21

22 = −r12 22 = 1,

a11

12 = a21 22 = 1.

The corresponding (non-zero) double brackets: { {v, v} } = v ⊗ u − u ⊗ v + uv ⊗ 1 − 1 ⊗ uv, { {u, v} } = u2 ⊗ 1, { {v, u} } = −1 ⊗ u2;

Examples and classification of low dimensional quadratic double Poisson brackets-2

Case 4 r22

21 = −r22 12 = 1. The corresponding (non-zero) double

brackets: { {v, u} } = v ⊗ v, { {u, v} } = −v ⊗ v; Case 5 r22

21 = −r22 12 = 1,; a21 11 = a22 12 = 1. The corresponding

(non-zero) double brackets: { {v, u} } = v ⊗ v − 1 ⊗ v2, { {u, v} } = −v ⊗ v + v2 ⊗ 1, { {u, u} } = uv ⊗ 1 − 1 ⊗ uv; Case 6 r22

21 = −r22 12 = 1,; a12 11 = a22 21 = −1. The corresponding

(non-zero) double brackets: { {v, u} } = v ⊗ v − v2 ⊗ 1, { {u, v} } = −v ⊗ v + 1 ⊗ v2, { {u, u} } = −vu ⊗ 1 + 1 ⊗ vu; Case 7 a11

22 = 1. The corresponding (non-zero) double brackets:

{ {v, v} } = u2 ⊗ 1 − 1 ⊗ u2.

Matrix examples

The non-abelian Poisson brackets for the Case 2 and for the Case 4 are non-degenerate. The corresponding symplectic forms on Repm(C < u, v >) = Matm(C) ⊕ Matm(C) can be written as Ω4 = Tr(d(U−1) ∧ dV ). This is an extension of the Poisson structure induced by this symplectic form on GLn(C) × gln(C) to Matm(C) ⊕ Matm(C). Ω2 = Tr(d(U−1) ∧ d(U−1VU)) (second is due to R. Bielawski).

Pairs of compatible double Poisson brackets

◮ Definition

Double Poisson brackets { {a, b} }1 and { {a, b} }2 are called compatible if { {a, b} }1 + λ{ {a, b} }2 is a double Poisson bracket for any λ ∈ C.

◮ Let A be the m-dimensional associative algebra with the

multiplication law eiej =

• k

bk

ijek. ◮ Define linear operators r, a on the space A ⊗ A by

r(eα ⊗ eβ) = rσǫ

αβeσ ⊗ eǫ,

a(eα ⊗ eβ) = aσǫ

αβeσ ⊗ eǫ.

Compatibility conditions for Non-abelian Poisson brackets: corrdinates

The compatibility conditions for linear and quadratic brackets have the following form: bs

αγavu sβ − bs γβavu αs + bu sβavs αγ − bv αsasu γβ = 0

(16) bs

βαruv sγ − bu βsrsv αγ − bv sαrus βγ − bv γsaus βα + bu sγasv βα = 0

(17)

Compatibility conditions for Non-abelian Poisson brackets: cocycle conditions

The compatibility conditions can be written as

◮

a(xz, y) − a(x, zy) + a(x, z)(1 ⊗ y) − (x ⊗ 1)a(z, y) = 0, and

◮

r(yx, z) − (y ⊗ 1)r(x, z) − r(y, z)(1 ⊗ x)− (1 ⊗ z)a(y, x) + a(y, x)(z ⊗ 1) = 0.

◮ If H2(A, A ⊗ A) = 0, then

a(x, y) = φ(xy) − (x ⊗ 1)φ(y) − φ(x)(1 ⊗ y) for some φ : A → A ⊗ A.

◮ Proposition

If H1(A, A ⊗ A) = 0, then without loss of generality r(x, y) = σφ(y)(1 ⊗ x) − (x ⊗ 1)σφ(y)+ (1 ⊗ y)φ(x) − φ(x)(y ⊗ 1), where σ(x ⊗ y) = y ⊗ x.

◮ The case a(x, y) = 0 corresponds to

φ : x → (x ⊗ 1) b − b (1 ⊗ x), where b ∈ A ⊗ A is any fixed element.

AYBE and compatibility of double Poisson structures

Theorem

Let r be as above and a = 0. Suppose b ∈ A ⊗ A satisfies the associative Yang-Baxter equation on A: b12 = −b21, b23b12 + b31b23 + b12b31 = 0. Then { {xα, xβ} } = ruv

αβ xu ⊗ xv

is a quadratic double Poisson bracket on T(A) compatible with the linear bracket { {xi, xj} } = bk

ijxk ⊗ 1 − bk ji1 ⊗ xk

corresponding to A.

Example of a solution for the relations-1

The example of this situation is given by:

Definition

An infinitesimal bialgebra (ǫ−bialgebra) is a triple (A, µ, ∆) where µ : A⊗2 → A is an associative multiplication and ∆ : A → A⊗2 is a coassociative comultiplication together with the following compatibility condition: ∆ ◦ µ = (µ ⊗ Id) ◦ (Id ⊗ ∆) + (Id ⊗ µ) ◦ (∆ ⊗ Id). In terms of elements b, c ∈ A the last condition can be rewritten as ∆(bc) =

• (c)

bc1 ⊗ c2 +

• (b)

b1 ⊗ b2c. (18)

Example of a solution for the relations-2

Definition

A coboundary ǫ− bialgebra is a quadruple (A, µ, ∆b, b) where b ∈ A⊗2 such that b =

i ui ⊗ vi and

∆b(a) = a ◦ b − b ◦ a = (a ⊗ 1)b − b(1 ⊗ a) =

• i

aui ⊗ vi − u⊗via. In other words the comultiplication ∆b in a coboundary ǫ− bialgebra is the principal derivation given by b.

Example of a solution for the relations-3

Definition

A quasi-triangular coboundary ǫ− bialgebra is a quadruple (A, µ, ∆b, b) where b ∈ A⊗2 satisfies AYBE: AYBE(b) := b13b12 − b12b23 + b23b13 = 0 ∈ A⊗3.

Example

An example of the tensor r above for a quasi-triangular coboundary ǫ− bialgebra with σb = −b, AYBE(b) = 0 such that H1(A, A⊗2) = 0. Then r(x, y) = 2σ[y, ∆op

b (x)]◦ = 2σ[(y ⊗1)σ(∆b(x))−σ(∆b(x))(1⊗y)].

Parameter-dependent double Poisson brackets

◮ Example

Define an associative multiplication on a vector space with a basis e1, ..., em by the formula E(u) E(v) = uq(v) u − v E(u) + vq(u) v − u E(v), (19) where E(u) = e1 + u e2 + · · · + um−1 em is a generating function, q(u) = q0 + u q1 + · · · + um qm is an arbitrary polynomial. The commutative associative algebra A is semi-simple for generic values

• f the parameters qi.

◮ Then the formula

{ {E(u), E(v)} } = uv u − v

• E(u) ⊗ E(v) − E(v) ⊗ E(u)
• ,

defines a double Poisson bracket compatible with linear bracket corresponding to A.

Parameter-dependent quadratic double Poisson brackets: general solution-1

◮ Consider double Poisson brackets of the form

{ {E(u), E(v)} } = α(u, v)E(u) ⊗ E(v) + β(u, v)E(v) ⊗ E(u)+ γ(u, v)E(u)E(v) ⊗ 1 + δ(u, v)1 ⊗ E(u)E(v)+ ǫ(u, v)E(v)E(u) ⊗ 1 + µ(u, v)1 ⊗ E(v)E(u).

◮ It follows from (1) that α(u, v) = −α(v, u),

β(u, v) = −β(v, u), µ(u, v) = −γ(v, u), ǫ(u, v) = −δ(v, u).

Parameter-dependent quadratic double Poisson brackets: general solution-2

◮ Condition (3) is equivalent to γ(u, v) = 0,

α(v, w)α(u, v) + α(w, u)α(v, w) + α(u, v)α(w, u) = 0, β(v, w)β(u, v) + β(w, u)β(v, w) + β(u, v)β(w, u) = 0, δ(v, u)δ(v, w) = α(w, u)

• δ(v, u) − δ(v, w)
• ,

δ(w, v)δ(u, v) = β(w, u)

• δ(w, v) − δ(u, v)
• .

◮ The general solution is given by

α(u, v) = 1 g(u) − g(v), β(u, v) = 1 h(u) − h(v), δ(u, v) = 1 g(v) − h(u).

Links and open problems

◮ There are various links to symplectic and Poisson geometry of

Quiver Path algebras (examples of R.Bielawski);

◮ Relation with "Derived Poisson structures" of Berest and his

co-authors;

◮ Hamiltonian formalism with non-abelian brackets and

integarble equations;

◮ Deformation theory, Rota-Baxter and Loday algebras. ◮ Interesting and intriguing problems are related to a

quantization of the non-abelian Poisson brackets, double Poisson brackets and the corresponding quantization of the non-abelian integrable equations.

THANKS FOR YOUR ATTENTION!