Integrable Systems and Quantum Deformations

Peter Koroteev

University of Minnesota

In collaboration with N. Beisert, F. Spill, A. Rej

0802.0777, work in progress

MG12, July 13th 2009 Outline

Introduction. Integrability and Symmetry What is Integrability Coordinate Bethe Ansatz AdS/CFT Correspondence

Quantum Deformation of 1D Hubbard Model 2 Uq(su(2|2) n R ) algebra Hubbard-like Models

Universal R-matrix Yangian Universal R-matrix I “Complicated”, Chaotic models. Almost untractable

I Integrable models: not necessrily easy/complicated

Integrable models can be completely solved

Why Integrability?

I Easy models: Classical Mechanics (ossilator, free point particle) I Integrable models: not necessrily easy/complicated

Integrable models can be completely solved

Why Integrability?

I Easy models: Classical Mechanics (ossilator, free point particle)

I “Complicated”, Chaotic models. Almost untractable Integrable models can be completely solved

Why Integrability?

I Easy models: Classical Mechanics (ossilator, free point particle)

I “Complicated”, Chaotic models. Almost untractable

I Integrable models: not necessrily easy/complicated Why Integrability?

I Easy models: Classical Mechanics (ossilator, free point particle)

I “Complicated”, Chaotic models. Almost untractable

I Integrable models: not necessrily easy/complicated

Integrable models can be completely solved I Finite dimensional: mechanical with n d.o.f.s – exist n-1 local integrals in involution

I Field Theory: Inﬁnite dimensional symmetry, S–matrix satisﬁes integrability constraints. We are focused on QFTs and spin chains

What is Integrability

Exists an inﬁnite set of independent commuting charges

{Qα}, [Qα, Qβ] = 0 I Field Theory: Inﬁnite dimensional symmetry, S–matrix satisﬁes integrability constraints. We are focused on QFTs and spin chains

What is Integrability

Exists an inﬁnite set of independent commuting charges

{Qα}, [Qα, Qβ] = 0

I Finite dimensional: mechanical with n d.o.f.s – exist n-1 local integrals in involution We are focused on QFTs and spin chains

What is Integrability

Exists an inﬁnite set of independent commuting charges

{Qα}, [Qα, Qβ] = 0

I Finite dimensional: mechanical with n d.o.f.s – exist n-1 local integrals in involution

I Field Theory: Inﬁnite dimensional symmetry, S–matrix satisﬁes integrability constraints. What is Integrability

Exists an inﬁnite set of independent commuting charges

{Qα}, [Qα, Qβ] = 0

I Finite dimensional: mechanical with n d.o.f.s – exist n-1 local integrals in involution

I Field Theory: Infinite dimensional symmetry, S–matrix satisfies integrability constraints. We are focused on QFTs and spin chains Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions

Some methods

I Analytic Bethe Ansatz. [Leningrad school, 70-80s]

I Coordinate Bethe Ansatz. Mostly for spin chains

I Algebraic integrability

Signatures of Integrability

Explicitly ﬁnd all commuting charges Some methods

I Analytic Bethe Ansatz. [Leningrad school, 70-80s]

I Coordinate Bethe Ansatz. Mostly for spin chains

I Algebraic integrability

Signatures of Integrability

Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions I Coordinate Bethe Ansatz. Mostly for spin chains

I Algebraic integrability

Signatures of Integrability

Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions

Some methods

I Analytic Bethe Ansatz. [Leningrad school, 70-80s] I Algebraic integrability

Signatures of Integrability

Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions

Some methods

I Analytic Bethe Ansatz. [Leningrad school, 70-80s]

I Coordinate Bethe Ansatz. Mostly for spin chains Signatures of Integrability

Explicitly find all commuting charges Investigate S-matrix (field theory, spin chains) S-matrix satisfies Yang-Baxter equation, Unitarity conditions

Some methods

I Analytic Bethe Ansatz. [Leningrad school, 70-80s]

I Coordinate Bethe Ansatz. Mostly for spin chains

I Algebraic integrability Hamiltonian

L X H = Hk,k+1, k=1 1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 1 − ~σk ⊗ ~σk+1

XXX Spin Chain

Hilbert space |ψi = |↑↓↓↑ ...i Symmetry su(2) interchanges |↑i and |↓i. H commutes with all generators What is the spectrum of the linear operator H? Brute force: list all states with given n↑, n↓, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000)

XXX Spin Chain

Hilbert space |ψi = |↑↓↓↑ ...i Hamiltonian

L X H = Hk,k+1, k=1 1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 1 − ~σk ⊗ ~σk+1 What is the spectrum of the linear operator H? Brute force: list all states with given n↑, n↓, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000)

XXX Spin Chain

Hilbert space |ψi = |↑↓↓↑ ...i Hamiltonian

L X H = Hk,k+1, k=1 1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 1 − ~σk ⊗ ~σk+1

Symmetry su(2) interchanges |↑i and |↓i. H commutes with all generators Brute force: list all states with given n↑, n↓, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000)

XXX Spin Chain

Hilbert space |ψi = |↑↓↓↑ ...i Hamiltonian

L X H = Hk,k+1, k=1 1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 1 − ~σk ⊗ ~σk+1

Symmetry su(2) interchanges |↑i and |↓i. H commutes with all generators What is the spectrum of the linear operator H? XXX Spin Chain

Hilbert space |ψi = |↑↓↓↑ ...i Hamiltonian

L X H = Hk,k+1, k=1 1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 1 − ~σk ⊗ ~σk+1

Symmetry su(2) interchanges |↑i and |↓i. H commutes with all generators What is the spectrum of the linear operator H? Brute force: list all states with given n↑, n↓, evaluate H in this basis, diagonalize H. Straightforward but hard (L=20, basis about 10000) Find H12|↓↓i = 0 hence H|0i = 0 Spin ﬂip |ki = |↓↓↓↑k ↓↓ ...i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X |pi = eipk |ki

Act with H to ﬁnd eigenvalue and dispersion relation

+∞ X H|pi = eipk (|ki − |k − 1i + |ki − |k + 1i) −∞ = 2(1 − cos p)|pi =: e(p)|pi

Bethe Ansatz Consider inﬁnite chain. Vacuum (ferromagnetic)

|0i = |↓↓↓↓↓ ...i Spin ﬂip |ki = |↓↓↓↑k ↓↓ ...i Hamiltonian homogeneous, eigenvectors are plane waves Momentum eigenstate X |pi = eipk |ki

Act with H to ﬁnd eigenvalue and dispersion relation

+∞ X H|pi = eipk (|ki − |k − 1i + |ki − |k + 1i) −∞ = 2(1 − cos p)|pi =: e(p)|pi

Bethe Ansatz Consider inﬁnite chain. Vacuum (ferromagnetic)

|0i = |↓↓↓↓↓ ...i

Find H12|↓↓i = 0 hence H|0i = 0 Momentum eigenstate X |pi = eipk |ki

Act with H to ﬁnd eigenvalue and dispersion relation

+∞ X H|pi = eipk (|ki − |k − 1i + |ki − |k + 1i) −∞ = 2(1 − cos p)|pi =: e(p)|pi

Bethe Ansatz Consider inﬁnite chain. Vacuum (ferromagnetic)

|0i = |↓↓↓↓↓ ...i

Find H12|↓↓i = 0 hence H|0i = 0 Spin ﬂip |ki = |↓↓↓↑k ↓↓ ...i Hamiltonian homogeneous, eigenvectors are plane waves Act with H to ﬁnd eigenvalue and dispersion relation

+∞ X H|pi = eipk (|ki − |k − 1i + |ki − |k + 1i) −∞ = 2(1 − cos p)|pi =: e(p)|pi

Bethe Ansatz Consider inﬁnite chain. Vacuum (ferromagnetic)

|0i = |↓↓↓↓↓ ...i

|0i = |↓↓↓↓↓ ...i

Act with H to ﬁnd eigenvalue and dispersion relation

+∞ X H|pi = eipk (|ki − |k − 1i + |ki − |k + 1i) −∞ = 2(1 − cos p)|pi =: e(p)|pi Two excitations

P ipk+iql Position State |p < qi = e |... ↑k ... ↑l ...i k

6=3! asymptotic regions Match up regions at contact terms, ﬁnd eigenstate

|p1, p2, p3i = |p1 < p2 < p3i + S12|p2 < p1 < p3i + S23|p1 < p3 < p2i,

+S13S12|p2 < p3 < p1i + S13S23|p3 < p1 < p2i + S12S13S23|p3 < p2 < p1i

eigenvalue e(p1) + e(p2) + e(p3) Integrability: Scattering factorizes for any number of particles Two particles scattering phase enough to construct any eigenstate on inﬁnite chain Move one ip L excitation pk past L sites e k of the chain and k − 1 other Q particles Skj . Should end up with the same state Bethe equations k Y eipk +ipj − 2eipk + 1 1 = e−ipk L − eipk +ipj − 2eipj + 1 j=1

Reparametrise pk = 2arccot2uk via rapidity k uk − i/2L Y uk − uj + i 1 = − uk + i/2 uk − uj + 1 j=1 Total energy

k k k X X 2 X i i E = e(pj ) = 4 sin pj /2 = 4 − uj + i/2 uj − i/2 j=1 j=1 j=1

u +i/2 Total momentum eip = Q eipj = Q j uj −i/2

Bethe Equations We have: inﬁnite chain. We want: ﬁnite periodic chain Bethe equations k Y eipk +ipj − 2eipk + 1 1 = e−ipk L − eipk +ipj − 2eipj + 1 j=1

Reparametrise pk = 2arccot2uk via rapidity k uk − i/2L Y uk − uj + i 1 = − uk + i/2 uk − uj + 1 j=1 Total energy

k k k X X 2 X i i E = e(pj ) = 4 sin pj /2 = 4 − uj + i/2 uj − i/2 j=1 j=1 j=1

u +i/2 Total momentum eip = Q eipj = Q j uj −i/2

k k k X X 2 X i i E = e(pj ) = 4 sin pj /2 = 4 − uj + i/2 uj − i/2 j=1 j=1 j=1

u +i/2 Total momentum eip = Q eipj = Q j uj −i/2

Bethe Equations We have: infinite chain. We want: finite periodic chain Move one ip L excitation pk past L sites e k of the chain and k − 1 other Q particles Skj . Should end up with the same state Bethe equations k Y eipk +ipj − 2eipk + 1 1 = e−ipk L − eipk +ipj − 2eipj + 1 j=1 Bethe Equations We have: infinite chain. We want: finite periodic chain Move one ip L excitation pk past L sites e k of the chain and k − 1 other Q particles Skj . Should end up with the same state Bethe equations k Y eipk +ipj − 2eipk + 1 1 = e−ipk L − eipk +ipj − 2eipj + 1 j=1

Reparametrise pk = 2arccot2uk via rapidity k uk − i/2L Y uk − uj + i 1 = − uk + i/2 uk − uj + 1 j=1 Total energy

k k k X X 2 X i i E = e(pj ) = 4 sin pj /2 = 4 − uj + i/2 uj − i/2 j=1 j=1 j=1

u +i/2 Total momentum eip = Q eipj = Q j uj −i/2 I Completely√ ﬁxed by√ supersymmetry – two parameters g and N g ∼ λ ∼ gYM N

I All ﬁleds massless

I Finiteness: beta function is exactly zero, no running

I Unbroken conformal symmetry psu(2, 2|4)

N = 4 4D Super Yang Mills

su(N) gauge connection A, 4 Majorana gluinos as 16 component 10d Majorana-Weyl spinor χ, 6 scalars Φi . All adjoint! Z 2 4 n1 2 1 2 1 2 1 i o S = 2 d x Tr F + (∇Φi ) − [Φi ,Φj ] + χ¯∇/ χ− χ¯ Γi [φi , χ] gYM 4 2 4 2 2 I All ﬁleds massless

I Finiteness: beta function is exactly zero, no running

I Unbroken conformal symmetry psu(2, 2|4)

N = 4 4D Super Yang Mills

I Completely√ ﬁxed by√ supersymmetry – two parameters g and N g ∼ λ ∼ gYM N I Finiteness: beta function is exactly zero, no running

I Unbroken conformal symmetry psu(2, 2|4)

N = 4 4D Super Yang Mills

I Completely√ ﬁxed by√ supersymmetry – two parameters g and N g ∼ λ ∼ gYM N

I All ﬁleds massless I Unbroken conformal symmetry psu(2, 2|4)

N = 4 4D Super Yang Mills

I Completely√ ﬁxed by√ supersymmetry – two parameters g and N g ∼ λ ∼ gYM N

I All ﬁleds massless

I Finiteness: beta function is exactly zero, no running N = 4 4D Super Yang Mills

I Completely√ ﬁxed by√ supersymmetry – two parameters g and N g ∼ λ ∼ gYM N

I All ﬁleds massless

I Finiteness: beta function is exactly zero, no running

I Unbroken conformal symmetry psu(2, 2|4) need renormalization O → ZO

hO(x)O(y)i = α|x − y|−2(D0+D(λ))

No mixing in scalar sector at 1 loop [Minahan, Zarembo] d log Z Anomalous dimension matrix (dilatation generator) D = d log Λ interpreted as a spin chain Hamiltonian

D(g)O = g 2HO = D(g)O

Anomalous Dimensions

Composite gauge invariant operators

O(x) = Tr(Φi (x)Φj (y) ... ) No mixing in scalar sector at 1 loop [Minahan, Zarembo] d log Z Anomalous dimension matrix (dilatation generator) D = d log Λ interpreted as a spin chain Hamiltonian

D(g)O = g 2HO = D(g)O

Anomalous Dimensions

Composite gauge invariant operators

O(x) = Tr(Φi (x)Φj (y) ... )

need renormalization O → ZO

hO(x)O(y)i = α|x − y|−2(D0+D(λ)) d log Z Anomalous dimension matrix (dilatation generator) D = d log Λ interpreted as a spin chain Hamiltonian

D(g)O = g 2HO = D(g)O

Anomalous Dimensions

Composite gauge invariant operators

O(x) = Tr(Φi (x)Φj (y) ... )

need renormalization O → ZO

hO(x)O(y)i = α|x − y|−2(D0+D(λ))

No mixing in scalar sector at 1 loop [Minahan, Zarembo] Anomalous Dimensions

Composite gauge invariant operators

O(x) = Tr(Φi (x)Φj (y) ... )

need renormalization O → ZO

hO(x)O(y)i = α|x − y|−2(D0+D(λ))

No mixing in scalar sector at 1 loop [Minahan, Zarembo] d log Z Anomalous dimension matrix (dilatation generator) D = d log Λ interpreted as a spin chain Hamiltonian

D(g)O = g 2HO = D(g)O Hibert space 6 6 H = R ⊗ · · · ⊗ R States – single trace operators

|Φi Φj Φk Φl ...i = TrΦi Φj Φk Φl ... P Hamiltonian H = Hk,k+1. Boiled down to su(2) sector XXX 1 2 spin chain

1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 (1 − ~σk~σk+1) Integrable !

Scalar Sector

[Φ]=1. Length of spin chain = bare scaling dimension. Spectrum of Hamiltonian – anomalous dimension States – single trace operators

|Φi Φj Φk Φl ...i = TrΦi Φj Φk Φl ... P Hamiltonian H = Hk,k+1. Boiled down to su(2) sector XXX 1 2 spin chain

1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 (1 − ~σk~σk+1) Integrable !

Scalar Sector

[Φ]=1. Length of spin chain = bare scaling dimension. Spectrum of Hamiltonian – anomalous dimension Hibert space 6 6 H = R ⊗ · · · ⊗ R P Hamiltonian H = Hk,k+1. Boiled down to su(2) sector XXX 1 2 spin chain

1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 (1 − ~σk~σk+1) Integrable !

Scalar Sector

[Φ]=1. Length of spin chain = bare scaling dimension. Spectrum of Hamiltonian – anomalous dimension Hibert space 6 6 H = R ⊗ · · · ⊗ R States – single trace operators

|Φi Φj Φk Φl ...i = TrΦi Φj Φk Φl ... Integrable !

Scalar Sector

[Φ]=1. Length of spin chain = bare scaling dimension. Spectrum of Hamiltonian – anomalous dimension Hibert space 6 6 H = R ⊗ · · · ⊗ R States – single trace operators

|Φi Φj Φk Φl ...i = TrΦi Φj Φk Φl ... P Hamiltonian H = Hk,k+1. Boiled down to su(2) sector XXX 1 2 spin chain

1 Hk,k+1 = Ik,k+1 − Pk,k+1 = 2 (1 − ~σk~σk+1) Scalar Sector

[Φ]=1. Length of spin chain = bare scaling dimension. Spectrum of Hamiltonian – anomalous dimension Hibert space 6 6 H = R ⊗ · · · ⊗ R States – single trace operators

|Φi Φj Φk Φl ...i = TrΦi Φj Φk Φl ... P Hamiltonian H = Hk,k+1. Boiled down to su(2) sector XXX 1 2 spin chain

SYM Spin Chain Vacuum

SYM Spin Chain Vacuum

Three complex scalars X = Φ1 + iΦ2, Y = Φ3 + iΦ4, Z = Φ5 + iΦ6 transform under su(4) ' so(6) Vacuum: |X X X ...i breaks superconformal symmetry Excitations transform in reps of psu(2|2) × psu(2|2) Can work with one copy of psu(2|2). Leads us to psu(2|2) spin chain

SYM Spin Chain Vacuum

Three complex scalars X = Φ1 + iΦ2, Y = Φ3 + iΦ4, Z = Φ5 + iΦ6 transform under su(4) ' so(6) Vacuum: |X X X ...i breaks superconformal symmetry Residual symmetry u(1) n psu(2|2) × psu(2|2) n R Excitations transform in reps of psu(2|2) × psu(2|2) Can work with one copy of psu(2|2). Leads us to psu(2|2) spin chain 0 0 I S-matrix: V1 ⊗ V2 → V2 ⊗ V1 is fully constrained by symmetry up to overall scalar factor (peculiarity of representation theory)

I Central extension to su(2|2) n R makes spectrum continuous I Magnon dispersion relation purely from symmetry [Beisert] q 2 2 1 E(p) = 1 + g sin ( 2 p)

su(2|2) Spin Chain

1 2 1 2 I d.o.f: 2 bosons φ , φ , 2 fermions ψ , ψ I Central extension to su(2|2) n R makes spectrum continuous I Magnon dispersion relation purely from symmetry [Beisert] q 2 2 1 E(p) = 1 + g sin ( 2 p)

su(2|2) Spin Chain

1 2 1 2 I d.o.f: 2 bosons φ , φ , 2 fermions ψ , ψ 0 0 I S-matrix: V1 ⊗ V2 → V2 ⊗ V1 is fully constrained by symmetry up to overall scalar factor (peculiarity of representation theory) I Magnon dispersion relation purely from symmetry [Beisert] q 2 2 1 E(p) = 1 + g sin ( 2 p)

su(2|2) Spin Chain

1 2 1 2 I d.o.f: 2 bosons φ , φ , 2 fermions ψ , ψ 0 0 I S-matrix: V1 ⊗ V2 → V2 ⊗ V1 is fully constrained by symmetry up to overall scalar factor (peculiarity of representation theory)

I Central extension to su(2|2) n R makes spectrum continuous su(2|2) Spin Chain

1 2 1 2 I d.o.f: 2 bosons φ , φ , 2 fermions ψ , ψ 0 0 I S-matrix: V1 ⊗ V2 → V2 ⊗ V1 is fully constrained by symmetry up to overall scalar factor (peculiarity of representation theory)

I Central extension to su(2|2) n R makes spectrum continuous I Magnon dispersion relation purely from symmetry [Beisert] q 2 2 1 E(p) = 1 + g sin ( 2 p) Quantum Deformations The Lie superalgebra su(2|2) is generated by the su(2) × su(2) a α α a generators R b, L β, the supercharges Q b, S β and the central charge C. The Lie brackets

a c c a a c α γ γ α α γ [R b, R d ] = δbR d − δd R b, [L β, L δ] = δβL δ − δδ L β, a γ a γ 1 a γ α γ γ α 1 α γ [R b, Q d ] = −δd Q b + 2 δbQ d , [L β, Q d ] = δβQ d − 2 δβ Q d , a c c a 1 a c α c α c 1 α c [R b, S δ] = δbS δ − 2 δbS δ, [L β, S δ] = −δδ S β + 2 δβ S δ.

α c c α α c c α {Q b, S δ} = δbL δ + δδ R b + δbδδ C. Central extension

α γ αγ a c ac {Q b, Q d } = ε εbd P, {S β, S δ} = ε εβδK. denote 2 3 h := su(2|2) n R = psu(2|2) n R . Quantum Deformation

Q-number (q ∈ C) qA − q−A [A] = q q − q−1 Commutators

[E1, F1] = [H1]q , {E2, F2} = −[H2]q , [E3, F3] = −[H3]q ,

Serre relations

0 = [E1, E3] = [F1, F3] = E2E2 = F2F2 −1 −1 = E1E1E2 − (q + q )E1E2E1 + E2E1E1 = E3E3E2 − (q + q )E3E2E3 + E2E3E3 −1 −1 = F1F1F2 − (q + q )F1F2F1 + F2F1F1 = F3F3F2 − (q + q )F3F2F3 + F2F3F3

Central charges... Antipode S : Uq(h) → Uq(h) is uniquely ﬁxed by the compatibility condition

∇ ◦ (S ⊗ 1) ◦ ∆(J) = ∇ ◦ (1 ⊗ S) ◦ ∆(J) = η ◦ ε(J)

2 For Uq(su(2|2) n R )

Hj S(1) = 1, S(Hj ) = −Hj , S(Ej ) = −q Ej ,

for central charges

S(C) = −C, S(P) = −q−2CP, S(K) = −q2CK

Hopf Algebra

Unit element η(1) = 1 Counit ε : Uq(h) → C takes the form

ε(1) = 1, ε(Hj ) = ε(Ej ) = ε(Fj ) = 0. 2 For Uq(su(2|2) n R )

Hj S(1) = 1, S(Hj ) = −Hj , S(Ej ) = −q Ej ,

for central charges

S(C) = −C, S(P) = −q−2CP, S(K) = −q2CK

Hopf Algebra

Unit element η(1) = 1 Counit ε : Uq(h) → C takes the form

ε(1) = 1, ε(Hj ) = ε(Ej ) = ε(Fj ) = 0.

Antipode S : Uq(h) → Uq(h) is uniquely ﬁxed by the compatibility condition

∇ ◦ (S ⊗ 1) ◦ ∆(J) = ∇ ◦ (1 ⊗ S) ◦ ∆(J) = η ◦ ε(J) Hopf Algebra

Unit element η(1) = 1 Counit ε : Uq(h) → C takes the form

ε(1) = 1, ε(Hj ) = ε(Ej ) = ε(Fj ) = 0.

Antipode S : Uq(h) → Uq(h) is uniquely ﬁxed by the compatibility condition

∇ ◦ (S ⊗ 1) ◦ ∆(J) = ∇ ◦ (1 ⊗ S) ◦ ∆(J) = η ◦ ε(J)

2 For Uq(su(2|2) n R )

Hj S(1) = 1, S(Hj ) = −Hj , S(Ej ) = −q Ej ,

for central charges

S(C) = −C, S(P) = −q−2CP, S(K) = −q2CK Coproduct

−H2 ∆(E2) = E2 ⊗ 1 + q U ⊗ E2, H2 −1 ∆(F2) = F2 ⊗ q + U ⊗ F2, ∆(P) = P ⊗ 1 + q2CU2 ⊗ P, ∆(K) = K ⊗ q−2C + U−2 ⊗ K, ∆(U) = U ⊗ U.

Unbraided for the rest generators (i=1,2,3, j=1,3)

∆(1) = 1 ⊗ 1,

∆(Hi ) = Hi ⊗ 1 + 1 ⊗ Hi , −Hj ∆(Ej ) = Ej ⊗ 1 + q ⊗ Ej , Hj ∆(Fj ) = Fj ⊗ q + 1 ⊗ Fj , Co-commutativity and R-matrix

Hopf algebra quasi-cocommutative if

∆op(J)R = R∆(J), ∆op = P∆P Fundamental Reresentation

1 1 1 1 +1/2 2 1 1 H2|φ i = −(C − 2 )|φ i, E1|φ i = q |φ i, F2|φ i = c|ψ i, 2 1 2 2 2 2 −1/2 1 H2|φ i = −(C + 2 )|φ i, E2|φ i = a|ψ i, F1|φ i = q |φ i, 2 1 2 2 −1/2 1 2 2 H2|ψ i = −(C + 2 )|ψ i, E3|ψ i = q |ψ i, F2|ψ i = d|φ i, 1 1 1 1 1 1 +1/2 2 H2|ψ i = −(C − 2 )|ψ i, E2|ψ i = b|φ i, F3|ψ i = q |ψ i. Constraints

1 1 ad = [C + 2 ]q, bc = [C − 2 ]q, ab = P, cd = K. (ad − qbc)(ad − q−1bc) = 1. Parametrization √ √ g α 1 a = g γ, b = x− − q2C−1x+, γ x− √ √ i gγ q−C+1/2 i g c = , d = qC+1/2x− − q−2C−1x+. α x+ γ Fundamental R-matrix

1 1 1 1 R|φ φ i = A12|φ φ i qA + q−1B A − B q−1C C R|φ1φ2i = 12 12 |φ2φ1i + 12 12 |φ1φ2i − 12 |ψ2ψ1i + 12 |ψ1ψ2i q + q−1 q + q−1 q + q−1 q + q−1 A − B q−1A + qB C qC R|φ2φ1i = 12 12 |φ2φ1i + 12 12 |φ1φ2i + 12 |ψ2ψ1i − 12 |ψ1ψ2i q + q−1 q + q−1 q + q−1 q + q−1 2 2 2 2 R|φ φ i = A12|φ φ i

1 1 1 1 R|ψ ψ i = −D12|ψ ψ i qD + q−1E D − E q−1F F R|ψ1ψ2i = − 12 12 |ψ2ψ1i − 12 12 |ψ1ψ2i + 12 |φ2φ1i− 12 |φ1φ2i q + q−1 q + q−1 q + q−1 q + q−1 D − E q−1D + qE F qF R|ψ2ψ1i = − 12 12 |ψ2ψ1i − 12 12 |ψ1ψ2i − 12 |φ2φ1i+ 12 |φ1φ2i q + q−1 q + q−1 q + q−1 q + q−1 ... R-matrix Coeﬃcients

C1 + − 0 q U1 x2 − x1 A12 = R 12 C2 − + q U2 x2 − x1 C1 + − + + − + 0 q U1 x2 − x1 −1 −1 x2 − x1 x2 − s(x1 ) B12 = R 1 − (q + q )q 12 C2 − + + − − − q U2 x2 − x1 x2 − x1 x2 − s(x1 ) −1 C1 −1 + −1 + + 0 −1 igα γ2γ1q U1 ig x2 − (q − q ) s(x2 ) − s(x1 ) C12 = R (q + q ) 12 2C2+3/2 2 − − − + q U2 x2 − s(x1 ) x2 − x1 0 D12 = −R12 x+ − x+ x+ − s(x−) 0 −1 −2C2−1 −2 2 1 2 1 E12 = −R12 1 − (q + q )q U2 − + − − x2 − x1 x2 − s(x1 ) −1 C1 −1 + −1 + + 0 −1 igα γ2γ1q U1 ig x2 − (q − q ) s(x2 ) − s(x1 ) F12 = −R (q + q ) 12 2C2+3/2 2 − − − + q U2 x2 − s(x1 ) x2 − x1 2 C2+1/2 + − C1+1/2 + − α U2q (x2 − x2 ) U1q (x1 − x1 ) · 2 −1 2 2 2 1 − g (q − q ) γ2 γ1 + + + − 0 1 x2 − x1 0 γ1 x2 − x2 G12 = R , H12 = R 12 C2+1/2 − + 12 − + q U2 x2 − x1 γ2 x2 − x1 qC1 U γ x+ − x− x− − x− 0 1 2 1 1 0 C1+1/2 2 1 K12 = R , L12 = R q U1 12 C2 − + 12 − + q U2 γ1 x2 − x1 x2 − x1 Discrete Symmetries of R-matrix

Braiding unitarity R12R21 = 1 ⊗ 1 entails

A12A21 = B12B21 + C12F21 = G12L21 + H12H21 = 1,

A12D12 = B12E12 − C12F12 = H12K12 − G12L12.

Yang–Baxter equation

R12R13R23 = R23R13R12.

Matrix Unitarity † (R12) R12 = 1 ⊗ 1. Crossing Symmetry

(C−1 ⊗ 1)RST⊗1(C ⊗ 1)R = 1 ⊗ 1. 12¯ 12

0 imposes relations on scalar factor R12 Near Neighbor Hamiltonian

Homogeneous Hamiltonian

L X H = Hk,k+1. k=1

The pairwise interaction H12 is the following logarithmic derivative of the R-matrix

+ + − − ∗ −1/2 x − s(x ) x − s(x ) du −1 d H12 = −i −1 + + R12 R12 . q x s(x ) du du1 ± ± x12=x

± The spectral parameters uk are deﬁned via xk i i u = q−1u(x+) − = qu(x−) + . k k 2g k 2g For our case

+ K M i −1 −C−1/2 −1 yk − x Y −1 qu(yk ) − wj + 2 g 1 = q U − q i , yk − x q−1u(y ) − w − g −1 j=1 k j 2 N −1 i −1 M −1 i −1 −1 Y wk − q u(yj ) + g Y q wk − qwj − (q + q )g 1 = q 2 2 . w − qu(y ) − i g −1 qw − q−1w + i (q + q−1)g −1 j=1 k j 2 j=1 k j 2 j6=k

Bethe Equations and Spectrum Generic for rank 3 algebra

K N M Y I,II Y II,II Y III,II 1 = R (xj , yk ) R (yj , yk ) R (wj , yk ), j=1 j=1 j=1 j6=k N N M Y I,III Y II,III Y III,III 1 = R (xj , wk ) R (yj , wk ) R (wj , yk ), j=1 j=1 j=1 j6=k Bethe Equations and Spectrum Generic for rank 3 algebra

K N M Y I,II Y II,II Y III,II 1 = R (xj , yk ) R (yj , yk ) R (wj , yk ), j=1 j=1 j=1 j6=k N N M Y I,III Y II,III Y III,III 1 = R (xj , wk ) R (yj , wk ) R (wj , yk ), j=1 j=1 j=1 j6=k For our case

Energy N X E = E0K + E(yj ). j=1

ipk −ipk E0 = A, E(yk ) = H + K − 2A + Ge + Le , exhibits su(2) × su(2) ∈ su(2|2) symmetry

1 2 † † |φk i = |◦i, |φk i = κc1,k c2,k |◦i, 1 † 2 † |ψk i = c1,k |◦i, |ψk i = c2,k |◦i

1D Hubbard Model

Hamiltonian

Hub X † † Hj,k = cα,j cα,k + cα,k cα,j + Un1,j n2,j . α=1,2 1 2 † † |φk i = |◦i, |φk i = κc1,k c2,k |◦i, 1 † 2 † |ψk i = c1,k |◦i, |ψk i = c2,k |◦i

1D Hubbard Model

Hamiltonian

Hub X † † Hj,k = cα,j cα,k + cα,k cα,j + Un1,j n2,j . α=1,2

exhibits su(2) × su(2) ∈ su(2|2) symmetry 1D Hubbard Model

Hamiltonian

Hub X † † Hj,k = cα,j cα,k + cα,k cα,j + Un1,j n2,j . α=1,2

exhibits su(2) × su(2) ∈ su(2|2) symmetry

1 2 † † |φk i = |◦i, |φk i = κc1,k c2,k |◦i, 1 † 2 † |ψk i = c1,k |◦i, |ψk i = c2,k |◦i Alcaraz and Bariev Chain

AB † † 0 Hj,k = (c1,j c1,k + c1,k c1,j )(1 + t11n2,j + t12n2,k + t1n2,j n2,k ) † † 0 + (c2,j c2,k + c2,k c2,j )(1 + t21n1,j + t22n1,k + t2n1,j n1,k ) † † † † + J(c1,j c2,k c2,j c1,k + c1,k c2,j c2,k c1,j ) † † † † + tp(c1,j c2,j c2,k c1,k + c1,j c2,j c2,k c1,k ) + V11n1,j n1,k + V12n1,j n2,k + V21n2,j n1,k + V22n2,j n2,k + Un1,j n2,j (1) (2) + V3 n2,j n1,k n2,k + V3 n1,j n1,k n2,k (3) (4) + V3 n1,j n2,j n2,k + V3 n1,j n2,j n1,k + V4n1,j n2,j n1,k n2,k ,

where

0 t11 = t4 − 1, t12 = t3 − 1, t1 = t5 − t3 − t4 + 1, 0 t21 = t1 − 1, t22 = t2 − 1, t2 = t5 − t1 − t2 + 1. Relationship to Condensed Matter Notation

Four d.o.f. for each site

† † † † |◦i, |↑i ∼ c1|◦i, |↓i ∼ c2|◦i, |li ∼ c1c2|◦i

1 2 † † 1 † 2 † |φk i = |◦i, |φk i = κc1,k c2,k |◦i, |ψk i = c1,k |◦i, |ψk i = c2,k |◦i. anticommutators

† † † {cα,k , cβ,l } = δαβδkl , {cα,k , cβ,l } = {cα,k , cβ,l } = 0. number operators † nα,k = cα,k cα,k Possible transformations: twist, add central elements... and change spectrum in controllable way

0 −1 1 1 H12 = a0 TH12T + 2 a1∆(H1) + a2∆(1) + 2 a3∆(H3) 1 + 2 b1(H1 ⊗ 1 − 1 ⊗ H1) + b2(H1H1 ⊗ 1 − 1 ⊗ H1H1) 1 + 2 b3(H3 ⊗ 1 − 1 ⊗ H3) Twist [Reshetikhin]

K K X i X T = exp if1 (j − 1)H1,j + 2 f2 (H1,j H3,k − H3,j H1,k ) j=1 j

It is a subsector of Uq(su(2|2)) Hamiltonian! Twist [Reshetikhin]

K K X i X T = exp if1 (j − 1)H1,j + 2 f2 (H1,j H3,k − H3,j H1,k ) j=1 j

It is a subsector of Uq(su(2|2)) Hamiltonian!

Possible transformations: twist, add central elements... and change spectrum in controllable way

0 −1 1 1 H12 = a0 TH12T + 2 a1∆(H1) + a2∆(1) + 2 a3∆(H3) 1 + 2 b1(H1 ⊗ 1 − 1 ⊗ H1) + b2(H1H1 ⊗ 1 − 1 ⊗ H1H1) 1 + 2 b3(H3 ⊗ 1 − 1 ⊗ H3) It is a subsector of Uq(su(2|2)) Hamiltonian!

Possible transformations: twist, add central elements... and change spectrum in controllable way

0 −1 1 1 H12 = a0 TH12T + 2 a1∆(H1) + a2∆(1) + 2 a3∆(H3) 1 + 2 b1(H1 ⊗ 1 − 1 ⊗ H1) + b2(H1H1 ⊗ 1 − 1 ⊗ H1H1) 1 + 2 b3(H3 ⊗ 1 − 1 ⊗ H3) Twist [Reshetikhin]

K K X i X T = exp if1 (j − 1)H1,j + 2 f2 (H1,j H3,k − H3,j H1,k ) j=1 j

Q-deformation of the Hubbard model limit [Beisert PK]

0 0 X 1 Hj,k = A (1 − n1,`)(1 − n2,`) + n1,`n2,` − 2 `=j,k +1/2 † +1/2 −3/2 + iq c1,j c1,k 1 − (1 − q )n2,j 1 − (1 − q )n2,k +1/2 † −1/2 −1/2 + iq c2,j c2,k 1 − (1 − q )n1,j 1 − (1 − q )n1,k −1/2 † +3/2 −1/2 − iq c1,k c1,j 1 − (1 − q )n2,j 1 − (1 − q )n2,k −1/2 † +1/2 +1/2 − iq c2,k c2,j 1 − (1 − q )n1,j 1 − (1 − q )n1,k . I All known deformations of 1D Hubbard model are shown to be embedded into quantum Super Yang Mill chain

I Quantum deformations uncover hidden symmetries

I Quantum Deformed gauge theory is not known

I Gravity dual theory is not known

Outlook so far

I We found embedding of su(2|2) in quantum group setup (some issue are not clear though) I Quantum deformations uncover hidden symmetries

I Quantum Deformed gauge theory is not known

I Gravity dual theory is not known

Outlook so far

I We found embedding of su(2|2) in quantum group setup (some issue are not clear though)

I All known deformations of 1D Hubbard model are shown to be embedded into quantum Super Yang Mill chain I Quantum Deformed gauge theory is not known

I Gravity dual theory is not known

Outlook so far

I We found embedding of su(2|2) in quantum group setup (some issue are not clear though)

I All known deformations of 1D Hubbard model are shown to be embedded into quantum Super Yang Mill chain

I Quantum deformations uncover hidden symmetries I Gravity dual theory is not known

Outlook so far

I We found embedding of su(2|2) in quantum group setup (some issue are not clear though)

I All known deformations of 1D Hubbard model are shown to be embedded into quantum Super Yang Mill chain

I Quantum deformations uncover hidden symmetries

I Quantum Deformed gauge theory is not known Outlook so far

I We found embedding of su(2|2) in quantum group setup (some issue are not clear though)

I All known deformations of 1D Hubbard model are shown to be embedded into quantum Super Yang Mill chain

I Quantum deformations uncover hidden symmetries

I Quantum Deformed gauge theory is not known

I Gravity dual theory is not known Universal R-matrix I Yangian Y (gl(n|m)) is isomorphic to associative algebra U(R) generated by 1 and the matrices (k) Tij , i, j = 1, n + m, k ∈ Z≥0 It is convenient to gather them in the formal series n +∞ X X (n) −n T (λ) = Tij λ Ei,j , i,j=1 n=0 T (λ) satisfy the so-called RTT relations R(n)(λ − µ)(T (λ) ⊗ 1)(1 ⊗ T (µ)) = (1 ⊗ T (µ))(T (λ) ⊗ 1)R(n)(λ − µ) , qdet(T (λ)) = 1 , where qdet is the quantum determinant and the Yang matrix is given by (n) X −1 R (λ) = 1 ⊗ 1 + λ Ei,j ⊗ Ej,i 1≤i,j≤n

Yangian in RTT realization

I Consider Lie (super)algebra gl(n|m) and its vector representation It is convenient to gather them in the formal series n +∞ X X (n) −n T (λ) = Tij λ Ei,j , i,j=1 n=0 T (λ) satisfy the so-called RTT relations R(n)(λ − µ)(T (λ) ⊗ 1)(1 ⊗ T (µ)) = (1 ⊗ T (µ))(T (λ) ⊗ 1)R(n)(λ − µ) , qdet(T (λ)) = 1 , where qdet is the quantum determinant and the Yang matrix is given by (n) X −1 R (λ) = 1 ⊗ 1 + λ Ei,j ⊗ Ej,i 1≤i,j≤n

Yangian in RTT realization

I Consider Lie (super)algebra gl(n|m) and its vector representation I Yangian Y (gl(n|m)) is isomorphic to associative algebra U(R) generated by 1 and the matrices (k) Tij , i, j = 1, n + m, k ∈ Z≥0 T (λ) satisfy the so-called RTT relations R(n)(λ − µ)(T (λ) ⊗ 1)(1 ⊗ T (µ)) = (1 ⊗ T (µ))(T (λ) ⊗ 1)R(n)(λ − µ) , qdet(T (λ)) = 1 , where qdet is the quantum determinant and the Yang matrix is given by (n) X −1 R (λ) = 1 ⊗ 1 + λ Ei,j ⊗ Ej,i 1≤i,j≤n

Yangian in RTT realization

I Consider Lie (super)algebra gl(n|m) and its vector representation I Yangian Y (gl(n|m)) is isomorphic to associative algebra U(R) generated by 1 and the matrices (k) Tij , i, j = 1, n + m, k ∈ Z≥0 It is convenient to gather them in the formal series n +∞ X X (n) −n T (λ) = Tij λ Ei,j , i,j=1 n=0 T (λ) satisfy the so-called RTT relations R(n)(λ − µ)(T (λ) ⊗ 1)(1 ⊗ T (µ)) = (1 ⊗ T (µ))(T (λ) ⊗ 1)R(n)(λ − µ) , qdet(T (λ)) = 1 , where qdet is the quantum determinant and the Yang matrix is given by (n) X −1 R (λ) = 1 ⊗ 1 + λ Ei,j ⊗ Ej,i 1≤i,j≤n I Tij (λ) is a generating function for the Yangian Y (gl(n|m)) generators. Expansion around λ = ∞ gives these generators and commutation relations on Tij (λ) give deﬁning relations on Yangian generators as well as Serre relations. Coproduct for Yangian generators follow from coproduct of Tij (λ). (k) I Call the diagonal and upper/lower triangular part of Tij (k) (k) (k) Hi , Ei , Fi |i = 1, n + m − 1, k ∈ Z>, then from RTT deﬁning relations it follows

(0) (l) (l) (0) ()l (l) [Hi , Ej ] = Aij Ej , [Hi , Fj ] = −Aij Fj ...

I Commutation relations for T (λ)

(λ − µ)[Tij (λ), Tkl (µ)] = Tkj (µ)Til (λ) − Tkj (λ)Til (µ) (k) I Call the diagonal and upper/lower triangular part of Tij (k) (k) (k) Hi , Ei , Fi |i = 1, n + m − 1, k ∈ Z>, then from RTT deﬁning relations it follows

(0) (l) (l) (0) ()l (l) [Hi , Ej ] = Aij Ej , [Hi , Fj ] = −Aij Fj ...

I Commutation relations for T (λ)

(λ − µ)[Tij (λ), Tkl (µ)] = Tkj (µ)Til (λ) − Tkj (λ)Til (µ)

I Tij (λ) is a generating function for the Yangian Y (gl(n|m)) generators. Expansion around λ = ∞ gives these generators and commutation relations on Tij (λ) give deﬁning relations on Yangian generators as well as Serre relations. Coproduct for Yangian generators follow from coproduct of Tij (λ). I Commutation relations for T (λ)

(λ − µ)[Tij (λ), Tkl (µ)] = Tkj (µ)Til (λ) − Tkj (λ)Til (µ)

(0) (l) (l) (0) ()l (l) [Hi , Ej ] = Aij Ej , [Hi , Fj ] = −Aij Fj ... I Classical analogy: Lie algebra g with generators a b ab c [J , J ] = fc J extends to loop algebra (Kac-Moody algebra without central charge) g[λ, λ−1] with generators a b ab c a n a [Jn, Jm] = fc Jn+m, i.e. Jn = λ J . Then Killing form ab a b a b ab κ ∝ str(J , J ) is extended by (Jn, Jm) = κ δn,−m−1. This form splits g[λ, λ−1] = g[λ] + λ−1g[λ−1] into positive and negative degrees.

Quantum Double

I Algebraically, R-matrix is the canonical element of the Hopf Algebra tensored with its dual (similar to a Casimir) Quantum Double

I Algebraically, R-matrix is the canonical element of the Hopf Algebra tensored with its dual (similar to a Casimir)

I Classical analogy: Lie algebra g with generators a b ab c [J , J ] = fc J extends to loop algebra (Kac-Moody algebra without central charge) g[λ, λ−1] with generators a b ab c a n a [Jn, Jm] = fc Jn+m, i.e. Jn = λ J . Then Killing form ab a b a b ab κ ∝ str(J , J ) is extended by (Jn, Jm) = κ δn,−m−1. This form splits g[λ, λ−1] = g[λ] + λ−1g[λ−1] into positive and negative degrees. P ∗ I Quantum R-matrix of Yangian: R = J∈Y(g) J ⊗ J , where J∗ is the dual of J

I Invariant form for Yangian:

+ − hEi,k , Ej,l i = −δij δk,−l−1 − + |i| hEi,k , Ej,l i = −(−1) δij δk,−l−1 A n−m n hH , H i = −2 ij , n ≥ m, i,k j,−l−1 2 m

I For explicit form of R-matrix need to diagonalize this form

I Classical r-matrix: ∞ X a b r = κabJn ⊗ J−n−1 n=0 I Invariant form for Yangian:

+ − hEi,k , Ej,l i = −δij δk,−l−1 − + |i| hEi,k , Ej,l i = −(−1) δij δk,−l−1 A n−m n hH , H i = −2 ij , n ≥ m, i,k j,−l−1 2 m

I For explicit form of R-matrix need to diagonalize this form

I Classical r-matrix: ∞ X a b r = κabJn ⊗ J−n−1 n=0

P ∗ I Quantum R-matrix of Yangian: R = J∈Y(g) J ⊗ J , where J∗ is the dual of J I For explicit form of R-matrix need to diagonalize this form

I Classical r-matrix: ∞ X a b r = κabJn ⊗ J−n−1 n=0

P ∗ I Quantum R-matrix of Yangian: R = J∈Y(g) J ⊗ J , where J∗ is the dual of J

I Invariant form for Yangian:

+ − hEi,k , Ej,l i = −δij δk,−l−1 − + |i| hEi,k , Ej,l i = −(−1) δij δk,−l−1 A n−m n hH , H i = −2 ij , n ≥ m, i,k j,−l−1 2 m I Classical r-matrix: ∞ X a b r = κabJn ⊗ J−n−1 n=0

P ∗ I Quantum R-matrix of Yangian: R = J∈Y(g) J ⊗ J , where J∗ is the dual of J

I Invariant form for Yangian:

+ − hEi,k , Ej,l i = −δij δk,−l−1 − + |i| hEi,k , Ej,l i = −(−1) δij δk,−l−1 A n−m n hH , H i = −2 ij , n ≥ m, i,k j,−l−1 2 m

I For explicit form of R-matrix need to diagonalize this form I Construct a matrix

g g g −1 Cij (q) = ` (q)(A (q))ij

g I The constant ` (q) is deﬁned as the minimal proportionality factor that makes C g(q) polynomial in q and q−1. It is usually proportional to the dual Coxeter number.

R-matrix

I For a simple Lie superalgebra g with symmetrized Cartan matrix Ag deﬁne its quantum counterpart

g g h g i Aij → Aij (q) := Aij q g I The constant ` (q) is deﬁned as the minimal proportionality factor that makes C g(q) polynomial in q and q−1. It is usually proportional to the dual Coxeter number.

R-matrix

I For a simple Lie superalgebra g with symmetrized Cartan matrix Ag deﬁne its quantum counterpart

g g h g i Aij → Aij (q) := Aij q

I Construct a matrix

g g g −1 Cij (q) = ` (q)(A (q))ij R-matrix

I For a simple Lie superalgebra g with symmetrized Cartan matrix Ag deﬁne its quantum counterpart

g g h g i Aij → Aij (q) := Aij q

I Construct a matrix

g g g −1 Cij (q) = ` (q)(A (q))ij

g I The constant ` (q) is deﬁned as the minimal proportionality factor that makes C g(q) polynomial in q and q−1. It is usually proportional to the dual Coxeter number. I induces triangular decomposition of R-matrix

R12 = R+RH R−.

→ Y θ(α) + − R+ = exp(−(−1) a(α)Eα ⊗ Eα ), α∈Ξ+ ← Y θ(α) − + R− = exp(−(−1) a(α)Eα ⊗ Eα ) , α∈Ξ+

± θ(α) is parity of Eα .

I Triangular decomposition of g into subalgebras of positive roots, Cartan and negative roots

g = e+ ⊕ h ⊕ e− ,

one has [e±, h] ⊂ e± → Y θ(α) + − R+ = exp(−(−1) a(α)Eα ⊗ Eα ), α∈Ξ+ ← Y θ(α) − + R− = exp(−(−1) a(α)Eα ⊗ Eα ) , α∈Ξ+

± θ(α) is parity of Eα .

I Triangular decomposition of g into subalgebras of positive roots, Cartan and negative roots

g = e+ ⊕ h ⊕ e− ,

one has [e±, h] ⊂ e±

I induces triangular decomposition of R-matrix

R12 = R+RH R−. I Triangular decomposition of g into subalgebras of positive roots, Cartan and negative roots

g = e+ ⊕ h ⊕ e− ,

one has [e±, h] ⊂ e±

I induces triangular decomposition of R-matrix

R12 = R+RH R−.

→ Y θ(α) + − R+ = exp(−(−1) a(α)Eα ⊗ Eα ), α∈Ξ+ ← Y θ(α) − + R− = exp(−(−1) a(α)Eα ⊗ Eα ) , α∈Ξ+

± θ(α) is parity of Eα . + − −1 [Eα , Eα ] = a(α) Hγ, α = γ + nδ, γ ∈ ∆+(g)

Cartan part of the Yangain RH ∞ Y 0 g 1/2 ˜ g exp Ki,+(λ) ⊗ Ci,j (T )Kj,−(λ + ` (n + 1)) m m+1 n=0

I The set of positive roots

Ξ+ := {γ + nδ|γ ∈ ∆+} ,

δ is aﬃne root Cartan part of the Yangain RH ∞ Y 0 g 1/2 ˜ g exp Ki,+(λ) ⊗ Ci,j (T )Kj,−(λ + ` (n + 1)) m m+1 n=0

I The set of positive roots

Ξ+ := {γ + nδ|γ ∈ ∆+} ,

δ is aﬃne root

+ − −1 [Eα , Eα ] = a(α) Hγ, α = γ + nδ, γ ∈ ∆+(g) I The set of positive roots

Ξ+ := {γ + nδ|γ ∈ ∆+} ,

δ is aﬃne root

+ − −1 [Eα , Eα ] = a(α) Hγ, α = γ + nδ, γ ∈ ∆+(g)

Cartan part of the Yangain RH ∞ Y 0 g 1/2 ˜ g exp Ki,+(λ) ⊗ Ci,j (T )Kj,−(λ + ` (n + 1)) m m+1 n=0 gl(n|m) R-matrix −1 Inverse of q-Cartan matrix Agl(n|m)(q) [PK Rej Spill to appear]   an+m−1,1 ...... upper elements are  . ..   ...... obtained by     am+1,1 ... am+1,n−1 ...... i ↔ j     bm,1 ...... bm,n ......     . .. .   ...... cm−1,n+1 ......     . . ..   b2,1 b2,2 ......  b1,1 b1,2 ... b1,n c1,n+1 ... c1,n+m−1 with [2m − i]q[j]q ai,j = − [n − m]q [i]q[j]q bi,j = − [n − m]q [i]q[2n − j]q ci,j = − [n − m]q I Study diﬀerent representations (not necessarily highest or lowest weight) May help to understand how more than one spectral parameter may appear in R (S)-matrix

I Use it for amplitudes in N = 4

Outlook

I Calculate Universal R-matrix for other superalgebras I Use it for amplitudes in N = 4

Outlook

I Calculate Universal R-matrix for other superalgebras

I Study diﬀerent representations (not necessarily highest or lowest weight) May help to understand how more than one spectral parameter may appear in R (S)-matrix Outlook

I Calculate Universal R-matrix for other superalgebras

I Study diﬀerent representations (not necessarily highest or lowest weight) May help to understand how more than one spectral parameter may appear in R (S)-matrix

I Use it for amplitudes in N = 4