Quantum loop algebras and functional relations: Oscillator vs. - PowerPoint PPT Presentation

Quantum loop algebras and functional relations: Oscillator vs. prefundamental representations Khazret S. Nirov U NI W UPPERTAL & INR RAS, M OSCOW RAQIS’16, Geneva, August 23, 2016

Introduction 1 Quantum groups 2 Universal integrability objects 3 U q ( gl l + 1 ) - and U q ( L ( sl l + 1 )) -modules 4 U q ( b + ) -submodules 5 Universal functional relations 6 Highest ℓ -weight modules 7 Appendix 8

Hidden Grassmann structure Boos, Jimbo, Miwa, Smirnov, Takeyama, 2006 – 2016 Quantum group approach to functional relations Bazhanov, Lukyanov, Zamolodchikov, 1994 – 1999 Bazhanov, Hibberd, Khoroshkin, 2001 Group-theoretic / Algebraic approach revisited Boos, G¨ ohmann, Kl¨ umper, NX, Razumov, 2010 – 2016 Prefundamental representations and functional relations Hernandez, Jimbo, 2011 Frenkel, Hernandez, 2014

Quantum groups: V. Drinfeld & M. Jimbo (1985 - 1986) A is a Hopf algebra with respect to ∆ , S , ε A is a Hopf algebra with ∆ op = Π ◦ ∆ Π ( a ⊗ b ) = b ⊗ a , a , b ∈ A The universal R -matrix ∆ op ( a ) = R ∆ ( a ) R − 1 , R ∈ B + ⊗ B − ⊂ A ⊗ A ( ∆ ⊗ id )( R ) = R 13 R 23 , ( id ⊗ ∆ )( R ) = R 13 R 12 The master equation R 12 R 13 R 23 = R 23 R 13 R 12 B + ⊗ A ⊗ B − ⊂ A ⊗ A ⊗ A defined in

Universal integrability objects Yang-Baxter equation with ∆ ( t ) = t ⊗ t ( R 13 t 1 )( R 23 t 2 ) = ( R 12 ) − 1 ( R 23 t 2 )( R 13 t 1 ) R 12 ( ∗ ) Monodromy operator: ϕ : A → End ( V ) M ϕ ( ζ ) = ( ϕ ζ ⊗ id )( R ) ∈ End ( V ) ⊗ A Transfer operator T ϕ ( ζ ) = ( tr V ⊗ id )( M ϕ ( ζ )( ϕ ζ ( t ) ⊗ 1 )) = (( tr V ◦ ϕ ζ ) ⊗ id )( R ( t ⊗ 1 )) L -operator: ρ : B + → End ( W ) L ρ ( ζ ) = ( ρ ζ ⊗ id )( R ) ∈ End ( W ) ⊗ A Q -operator Q ρ ( ζ ) = ( tr W ⊗ id )( L ρ ( ζ )( ρ ζ ( t ) ⊗ 1 )) = (( tr W ◦ ρ ζ ) ⊗ id )( R ( t ⊗ 1 ))

First universal functional relations Directly from the Yang-Baxter equation ( tr ◦ ϕ 1 ζ 1 ) ⊗ ( tr ◦ ϕ 2 ζ 2 )( ∗ ) ⇒ T ϕ 1 ( ζ 1 ) T ϕ 2 ( ζ 2 ) = T ϕ 2 ( ζ 2 ) T ϕ 1 ( ζ 1 ) ( tr ◦ ρ ζ 1 ) ⊗ ( tr ◦ ϕ ζ 2 )( ∗ ) ⇒ Q ρ ( ζ 1 ) T ϕ ( ζ 2 ) = T ϕ ( ζ 2 ) Q ρ ( ζ 1 ) The rest is more tricky: � R 13 t 1 R 23 t 2 � Q ρ 1 ( ζ 1 ) Q ρ 2 ( ζ 2 ) = (( tr W 1 ⊗ W 2 ◦ ( ρ 1 ζ 1 ⊗ ρ 2 ζ 2 )) ⊗ id ) R 13 t 1 R 23 t 2 = [( ∆ ⊗ id )( R )] [( ∆ ⊗ id )( t ⊗ 1 )] = ( ∆ ⊗ id )( R ( t ⊗ 1 )) Q ρ 1 ( ζ 1 ) Q ρ 2 ( ζ 2 ) = (( tr W 1 ⊗ W 2 ◦ ( ρ 1 ζ 1 ⊗ ∆ ρ 2 ζ 2 )) ⊗ id )( R ( t ⊗ 1 )) Cipher key T λ ( ζ ) = Q 1 ( q 2 ( λ + ρ ) 1 / s ζ ) Q 2 ( q 2 ( λ + ρ ) 2 / s ζ ) Q 3 ( q 2 ( λ + ρ ) 3 / s ζ ) C � C � T λ ( ζ ) = Q 1 ( q − 2 ( λ + ρ ) 1 / s ζ ) Q 2 ( q − 2 ( λ + ρ ) 2 / s ζ ) Q 3 ( q − 2 ( λ + ρ ) 3 / s ζ )

Quantum group U q ( gl l + 1 ) Roots : normal ordering of △ + [ l ( l + 1 ) /2 positive roots ] α 1 , α 1 + α 2 , α 2 , α 1 + α 2 + α 3 , α 2 + α 3 , α 3 , . . . . . . α 1 + α 2 + . . . + α i , α 2 + . . . + α i , . . . , α i , . . . . . . α 1 + α 2 + . . . + α l , α 2 + . . . + α l , . . . , α l Equivalent to the total co-lexicographic order ≤ on the set N × N Generators and Cartan subalgebra of gl l + 1 i = 1, . . . , l E i , F i , l + 1 � K i , i = 1, . . . , l + 1, k l + 1 = C K i i = 1 Quantum group U q ( gl l + 1 ) generated by q X , i = 1, . . . , l , X ∈ k l + 1 E i , F i , Representations of U q ( gl l + 1 ) π λ with the highest weight vector v λ Infinite dimensional representation � E i v λ = 0, q X v λ = q � λ , X � v λ , λ ∈ k ∗ i = 1, . . . , l , X ∈ k l + 1 , l + 1 Finite dimensional representation π λ arises as a quotient sub-representation π λ when λ i − λ i + 1 ∈ Z + for all i = 1, . . . , l from �

Higher root vectors and q -commutation relations Root vectors (M. Jimbo, 1986) E i , i + 1 = E i , i = 1, . . . , l E ij = E i , j − 1 E j − 1, j − q E j − 1, j E i , j − 1 , j − i > 1 and F i , i + 1 = F i , i = 1, . . . , l F ij = F j − 1, j F i , j − 1 − q − 1 F i , j − 1 F j − 1, j , j − i > 1 Simplest commutations q ν K i E mn q − ν K i = q ν ∑ n − 1 q ν K i F mn q − ν K i = q − ν ∑ n − 1 j = m c ij E mn , j = m c ij F mn q ν H i = q ν ( K i − K i + 1 ) , i = 1, . . . , l q ν H i E mn q − ν H i = q ν ∑ n − 1 q ν H i F mn q − ν H i = q − ν ∑ n − 1 j = m a ij E mn , j = m a ij F mn where � c ij � : ( l + 1 ) × l matrix with c ii = 1, c i + 1, i = − 1, c i , i + 1 = 0, i = 1, . . . , l , c ij = 0 for | j − i | ≥ 2 α j ( K i ) = c ij , α j ( K i ) − α j ( K i + 1 ) = α j ( H i ) = a ij , i , j = 1, . . . , l Six different branches for { ( ij ) , ( mn ) } ∈ N 2 × N 2 (H. Yamane, 1989) C I : i = m < j < n , C II : m < i < j < n , C III : i < m < j = n C V : i < j = m < n , C IV : i < m < j < n , C VI : i < j < m < n

Further commutation relations E ij with E mn E ij E mn = q − 1 E mn E ij , { ( ij ) , ( mn ) } ∈ C I ∪ C III E ij E mn = E mn E ij , { ( ij ) , ( mn ) } ∈ C II ∪ C VI E ij E mn − q E mn E ij = E in , { ( ij ) , ( mn ) } ∈ C V E ij E mn − E mn E ij = − κ q E in E mj , { ( ij ) , ( mn ) } ∈ C IV F ij with F mn F ij F mn = q − 1 F mn F ij , { ( ij ) , ( mn ) } ∈ C I ∪ C III F ij F mn = F mn F ij , { ( ij ) , ( mn ) } ∈ C II ∪ C VI F ij F mn − q F mn F ij = − q F in , { ( ij ) , ( mn ) } ∈ C V F ij F mn − F mn F ij = − κ q F in F mj , { ( ij ) , ( mn ) } ∈ C IV

Further commutation relations E ij with F mn � � j − 1 j − 1 [ E ij , F ij ] = κ − 1 q ∑ k = i H k − q − ∑ k = i H k q j − 1 [ E ij , F mn ] = − q − 1 F jn q − ∑ k = i H k , { ( ij ) , ( mn ) } ∈ C I [ E ij , F mn ] = q E im q − ∑ n − 1 k = m H k = q − ∑ n − 1 k = m H k E im , { ( ij ) , ( mn ) } ∈ C III [ E ij , F mn ] = 0, { ( ij ) , ( mn ) } ∈ C II ∪ C V ∪ C VI j − 1 [ E ij , F mn ] = κ q E im F jn q − ∑ k = m H k , { ( ij ) , ( mn ) } ∈ C IV In the last relation [ E im , F jn ] = 0 and j − 1 j − 1 j − 1 j − 1 q − ∑ k = m H k F jn = q − 1 F jn q − ∑ q − ∑ k = m H k E im = q E im q − ∑ k = m H k , k = m H k

Further commutation relations E mn with F ij j − 1 j − 1 k = i H k E jn = − E jn q ∑ [ E mn , F ij ] = − q q ∑ k = i H k , { ( ij ) , ( mn ) } ∈ C I [ E mn , F ij ] = q − 1 q ∑ n − 1 k = m H k F im = F im q ∑ n − 1 k = m H k , { ( ij ) , ( mn ) } ∈ C III [ E mn , F ij ] = 0, { ( ij ) , ( mn ) } ∈ C II ∪ C V ∪ C VI j − 1 [ E mn , F ij ] = − κ q F im E jn q ∑ k = m H k , { ( ij ) , ( mn ) } ∈ C IV In the last relation [ E jn , F im ] = 0 and j − 1 j − 1 j − 1 j − 1 k = m H k E jn = q − 1 E jn q ∑ k = m H k F im = q F im q ∑ q ∑ k = m H k , q ∑ k = m H k The elements q ν K i , i = 1, . . . , l + 1, E ij , F ij , i = 1, . . . , l , j = 1, . . . , l + 1, i < j generate a P. B. W. basis of U q ( gl l + 1 ) as { F i 1 j 1 . . . F i a j a q ν 1 K 1 . . . q ν c K c E m 1 n 1 . . . E m b n b | a , b , c ≥ 0 } where ( i 1 j 1 ) ≤ . . . ≤ ( i a j a ) and ( m 1 n 1 ) ≤ . . . ≤ ( m a n a )

U q ( gl l + 1 ) -module � V λ Basis E i v λ = 0, q X v λ = q λ ( X ) v λ , λ ∈ k ∗ i = 1, . . . , l , X ∈ k l + 1 , l + 1 � � V λ ∼ 23 · · · F m 1, l + 1 1, l + 1 · · · F m l , l + 1 q ν K i v m , F i v m , E i v m | v m = F m 12 12 F m 13 13 F m 23 � l , l + 1 v 0 � � ∈ Z ⊗ ( l + 1 ) l /2 m = m 12 , m 13 , m 23 , . . . , m 1, l + 1 , . . . , m l , l + 1 + Acting by q ν K i and q ν H i on v m q ν K i v m = q ν ( λ i + ∑ i − 1 k = 1 m ki − ∑ l + 1 k = i + 1 m ik ) v m , i = 1, . . . , l + 1 q ν H i v m = q ν [ λ i − λ i + 1 + ∑ i − 1 k = 1 ( m ki − m k , i + 1 ) − 2 m i , i + 1 − ∑ l + 1 k = i + 2 ( m ik − m i + 1, k )] v m , i = 1, . . . , l Acting by F k , k + 1 on v m F k , k + 1 v m = q − ∑ k − 1 i = 1 ( m ik − m i , k + 1 ) v m + ǫ k , k + 1 k − 1 j − 1 i = 1 ( m ik − m i , k + 1 ) [ m jk ] q v m − ǫ jk + ǫ j , k + 1 , q − ∑ ∑ + k = 1, . . . , l j = 1 Acting by F 1, l + 1 on v m F 1, l + 1 v m = q ∑ l i = 2 m 1 i v m + ǫ 1, l + 1

U q ( gl l + 1 ) -module � V λ Basis E i v λ = 0, q X v λ = q λ ( X ) v λ , λ ∈ k ∗ i = 1, . . . , l , X ∈ k l + 1 , l + 1 � � V λ ∼ 23 · · · F m 1, l + 1 1, l + 1 · · · F m l , l + 1 � q ν K i v m , F i v m , E i v m | v m = F m 12 12 F m 13 13 F m 23 l , l + 1 v 0 � ∈ Z ⊗ ( l + 1 ) l /2 � m = m 12 , m 13 , m 23 , . . . , m 1, l + 1 , . . . , m l , l + 1 + Acting by E k , k + 1 on v m E k , k + 1 v m = q λ k − λ k + 1 − 2 m k , k + 1 − ∑ l + 1 s = k + 2 ( m ks − m k + 1, s ) k − 1 q ∑ k − 1 i = j + 1 ( m ik − m i , k + 1 ) [ m j , k + 1 ] q v m + ǫ jk − ǫ j , k + 1 ∑ × j = 1 l + 1 + [ λ k − λ k + 1 − ∑ ( m ks − m k + 1, s ) − m k , k + 1 + 1 ] q [ m k , k + 1 ] q v m − ǫ k , k + 1 s = k + 2 l + 1 q − λ k + λ k + 1 − 2 + ∑ l + 1 i = j ( m ki − m k + 1, i ) [ m kj ] q v m − ǫ kj + ǫ k + 1, j , ∑ − k = 1, . . . , l j = k + 2