Yangian Flags via Quasideterminants

Transcription

1 Yangian Flags via Quasideterminants Aaron Lauve LaCIM Université du Québec à Montréal Advanced Course on Quasideterminants and Universal Localization CRM, Barcelona, February 2007 Key Idea Proceed with Caution Watch Out!

2 Yangian flags via quasideterminants Abstract The Yangian Y n, by construction, is a deformation of the universal enveloping algebra of the general linear Lie algebra. Analogous to its classical counterpart, Y n has a nice (i.e., Ore) localization. As there is a (nonzero)(yangian) determinant in Y n, one may also view Y n as a deformation of the (coordinate ring of the) general linear group. From this point of view, existence of the localization allows us to use the quasideterminant to construct the coordinate ring Fl(Y n ) for a flag variety for Y n. Analogous to its classical counterpart, Fl(Y n ) is simultaneously a comodule algebra for Y n (viewed it as a general linear group) and a model for irreducible representations for Y n (viewed as a general linear algebra). 1

3 Overview I. Two actions of GL n (C) on polynomials C[X n n ] A. one, a model for the action of GL n on flags B. the other, a model for irreducible representations of gl n II. Flag varieties for GL n (D) over division rings D A. noncommutative flags B. coordinatization via the quasideterminant C. recaptures the classical story upon specialization III. Yangians Y n for gl n A. defined as a deformation of U(gl n ) B. view Y n as a deformation of C[GL n ] and use (II.) to mimic (I.A.) C. treating Y n as usual,... (I.B.) works too! 2

4 Notation SETS & SEQUENCES [n] = {1, 2,...,n} ( [n] ) d = sets of size d [n] d = sequences of size d MATRIX OPERATIONS A ij delete row i and column j A I,J keep rows I and columns J A J = A [d],j for J [n] d SET/SEQUENCE OPERATIONS I J = (i 1,..., i r, j 1,...,j s ) SEQUENCES & PERMUTATIONS l(σ) length of permutation σ l(i J) length of permutation I J 3

5 I.A. Flag Varieties & Coordinate Rings Let X = (x ij ) 1 i,j n be a matrix of indeterminants. Write C[X] for the polynomials in the x ij. Define an action GL n C[X] C[X] by g F(X) = F(g 1 X). Example. Fix n = 3, and consider g 1 = Then, 5 F(X) = x 23 [g F](X) = 7x x 23. Moreover, detx {1,3} detx {1,2,3} g (2x11 )(7x x 23 ) (7x x 21 )(2x 13 ) = 8 det X {1,3} g 24 detx{1,2,3}. 4

6 I.A. Flag Varieties & Coordinate Rings Observation. The minors projective invariants for the subgroup { (I) := detx [d],i I ( [3]) } d ; d = 2, 3 [ ] 0 GL 3. in C[X] are This is not surprising... 5

7 I.A. Flag Varieties & Coordinate Rings Definition. A flag Φ of shape γ is an increasing chain of subspaces of V : Φ : 0 = W 0 W 1 W r = V (dimw i /W i 1 = γ i ). Definition. Flags are right cosets: Fl(γ) = P γ \GL n (C), where 0. P γ =.. Focus on γ = (1, 1,...,1) for clarity. Write Fl(n) in this case. Fl(n) is made into a variety by the Plücker embedding: η : A { deta I : I [n] d, 1 d < n }, a map into P γ := PC n PC n2 PC nn 1. 6

8 I.A. Flag Varieties & Coordinate Rings Theorem (Schur 01; Hodge 43). Let C[P γ ] = C[f I : I ( [n]) d ; 1 d < n] be the homogeneous coordinate ring for P γ. The image of η ( Fl(n) ) in P γ is cut out by the homogeneous relations below: (A I ) Alternating: For all I [n] d, all permutations σ S d : f I = ( 1) l(σ) f σi (= 0 when I has repetitions) (Y I,J ) Young Symmetry: For all 1 r d e, I ( [n] ) ( d r, and J [n] ) e+r ( 1) l(λ J\Λ) f I Λ f J\Λ = 0 Λ J, Λ =r e.g. (Y,{124} ) : 0 = f 1 f 24 f 2 f 14 + f 4 f 12. Call the quotient (of C[P γ ] modulo these relations) the flag algebra Fl(n). Theorem. F l(n) is isomorphic to the subalgebra of C[X] generated by the minors { ( (I) : I [n] ) } d ; 1 d < n of X = (xij ). 7

9 Taft and Towber s Work (Briefly) It would be very important to define flag spaces for quantum groups. [Manin 88] Their Goal. Build a quantum flag algebra Fl q (n) with the following properties: 1. Fl q (n) reduces to Fl(n) when q e.g., Fl q (n) and Fl(n) share the same basis (semi-standard Young tableaux). 2. Fl q (n) is a comodule algebra for the quantum group C[GL q (n)]. 3. Fl q (n) is isomorphic to the subalgebra of C[GL q (n)] generated by the minors {det q X I : I ( [n]) } d, 1 d < n of its matrix of generators X = (xij ). Theorem (T-T 91). Success! Put Fl q (n) := C f I : I [n] d, 1 d < n modulo (A I ) q : f I = ( q) l(σ) f σi (Y I,J ) q : Λ ( q) l(λ J\Λ) f I Λ f J\Λ = 0 (C I,J ) q : For all 1 d < e, I ( [n]) ( d, and J [n] e f J f I = Λ J J\Λ =d ) : ( q) l(λ J\Λ) f J\Λ f Λ I 8

10 I.B. Polynomial Representations of gl n Keep X and C[X] as above. Define an action GL n C[X] C[X] by g F(X) = F(X g). This action also looks nice on the minors (I). Example. Fix n = 3, and consider g = Then X = (a b c) maps to X g = (2a + 7b 4b 3c). So g acts on minors as follows: (1, 3) (1, 2, 3) g (2a + 7b, 3c) = 6 (1, 3) + 21 (2, 3) g 24 (1, 2, 3) + 84 (2, 2, 3) = 24 (1, 2, 3) 9

11 I.B. Polynomial Representations of gl n Take the derivative of this action, and get a familiar module for gl n Map semi-standard Young tableaux T onto polynomials T C[X] as follows: (2, 4, 6) (4) (1, 2, 3) (1, 4, 6) (4, 6) (6) Fact. For λ a partition with at most n parts, the irreducible gl n module V λ is exactly the C-span of the T, the set running over semi-standard tableaux T of shape λ with fillings from [n]. That is, Fl(n) λ V λ as gl n -modules. 10

12 II.A. Noncommutative Flags Follow the work of Taft and Towber Start from noncommutative flags, not from the algebra of functions F l(γ). Hopefully arrive at correct generalizations of F l(γ). Preliminary steps are identical to GL n (C) case Fix a skew-field D and a free (left) D-module V = D n The two competing definitions for flags are again equivalent. Definition. The noncommutative flags F l(γ) of shape γ over D are the cosets P γ \GL n (D). Questions 1. Can we find good coordinates for Fl(γ)? 2. Can we characterize F l(γ) via relations on these coordinates? 3. Do specializations yield appropriate versions of (A I ), (Y I,J ), and (C I,J )? 11

13 II.B. Quasideterminantal Coordinatization A main organizing tool in noncommutative algebra. [G-G-R-W 05] Definition (Gelfand-Retakh 91). Given an n n matrix A over some ring R, the (ij)-quasideterminant A ij is defined whenever A ij is invertible, and in that case, 1 A ij = = Example (n = 2). Properties. ij A 11 = a 11 a 12 a 1 22 a 21 A 12 = a 12 a 11 a 1 21 a 22 In the commutative case, it looks like ±det A/detA ij. Has a Cramer s rule. Is zero (or undefined) when A is not full rank

14 II.B. Quasideterminantal Coordinatization Definition. The (left) quasi-plücker coordinate of A associated to (i, j, K), for i, j [n] and K [n] \ {i, j}, is the ratio p K ij (A) := A i K 1 si A j K sj (s [n]). Properties (G-R 97; L. 04). The quasi-plücker coordinates p K ij (A) satisfy p K ij (A) is independent of s appearing in definition. p K ij (g A) = pk ij (A) for all g GL n(d). (Question 1: Yes) If F(A) is some rational function in the a ij which is P γ (D)-invariant, then F is a rational function in the p K ij (A). (Question 2: Maybe) For I J 1, pairs of coordinates satisfy (A i,j,k I ) Alternating: pij k I (YI,J i ) Young Symmetry: (C???? ) Any More? pjk i I = pj I ik. l J pi il (A) pj\l li (A) = 1. 13

15 II.C. Specialization to T -generic Flags Fix a K-algebra A(n) on n 2 generators T = (t ij ). Definition. A ring map A(n) D into a division ring D is called T -inverting if T and its submatrices are invertible over D. In this case, we say that D has T -generic flags, and that A(n) is the ring of coordinate functions for the T -generic matrices. Definition. Let Det be a map from square submatrices of T to A(n). Call Det an amenable determinant if: (i) Det T i,j = t ij ; (ii) ( I, J) some version of row- or column-expansion of DetT I,J holds; (iii) ( I I, J J) DetT I,J and DetT I,J almost commute. (Question 3: To some extent) Theorem (L. 05). If A(n), D, and Det are as above, then the quasi-plücker relations give (A I ), and (Y I,J ), and (some) (C I,J ) relations in A(n) for minors { Det T I }. 14

16 II.C. Specialization to T -generic Flags Example. Let T be a matrix of commuting indeterminants. Then the quasi-plücker coordinates become ratios of determinants. the quasi-plücker relations reduce to those which cut out Fl(n) in P γ. Example. Let T be the q-generic matrix from quantum group theory. Then A(n) = C q [GL n ] has an (Ore) field of fractions, so quasi-plücker minors make sense when specializing a matrix of formal noncommuting variables to T. Moreover the quasi-plücker coordinates become ratios of quantum determinants. the quasi-plücker relations reduce to those used to define Fl q (γ) in case γ = (d, n d). for arbitrary γ, some of the commuting relations (C I,J ) q are missing, but... 15

17 III.A. The Yangian Y n Let t (r) ij be a collection of noncommuting variables (1 i, j n; r = 1, 2,...). For convenience, write t (0) ij = δ ij. Definition. The Yangian Y n for gl n is the algebra C t (r) ij i, j [n]; r N modulo the relations ( r, s 0) [ t (r+1) ij, t (s) ] [ (r) kl t ij, ] t(s+1) (r) kl = t kj t(s) il t (s) kj t(r) il. In particular, [ (1) t ij, ] [ t(1) kl δij, t (2) ] kl = δkj t (1) il δ il t (1) kj. Theorem. The mapping E ij t (1) ij is an embedding U(gl n ) Y n. 16

18 III.B. The Yangian as deformation of C[GL n ] Collect the t (r) ij as power series: t ij (u) := δ ij + r 1 t(r) ij u r. Collect as a matrix of generators: T(u) := (t ij (u)). Definition. The Yangian Y n is the C-algebra generated by T(u) modulo the relations [ tij (u), t kl (v) ] = 1 u v (t kj(u)t il (v) t kj (v)t il (u)). (i.e., equate the coefficients of all u r v s occuring on each side, letting u, v commute). a determinant function t - -(u) is defined for T(u) which is invertible in Y n [[u 1 ]]. Y n more precisely Y n [[u 1 ]] starts to look like our A(n). Even better, t - -(u) has the properties required of our Det... 17

19 III.B. The Yangian as deformation of C[GL n ] For any a C, the shift t ij (u + a) of t ij (u) is defined by r t(r) ij (u+a) r, i.e., the power series got by expanding (u+a) r = ) p 0 a p u p r for each r. Definition. For any submatrix T I,J (u) of T(u) of order d, the quantum determinant t I J (u) is given by t I J(u) = ( r p σ S d ( 1) l(σ) t iσ1 j 1 (u)t iσ2 j 2 (u 1) t iσd j d (u d + 1). Define t I J (u + a) analogous to t ij(u + a). Properties. Det is amenable: t I J (u) is alternating in I, J; t I J (u) = i I ( 1)i 1 t ij (u)t I\i J\j (u 1); [ t I J (u), t I J (v)] = 0. 18

20 III.B. Toward Fl(Y n ) for the Yangians Simplify notation: If I = d, write t I (u) for t [d] I (u). Corollary. Fix I ( [n]) ( d and J [n] ) e for any 1 d e. The quantum column minors of T(u) satisfy the following relations in Y n [[u 1 ]]: (A I ) u : (Y I,J ) u : t I (u) = ( 1) l(σ) t σi (u) ( q) l(λ J\Λ) t I Λ (u+d)t J\Λ (u+e+r) = 0 Λ J, Λ =r (C I,J ) u : t J (u+e)t I (u+d) = Λ J, J\Λ =d ( 1) l(λ J\Λ) t J\Λ (u+e)t Λ I (u+d) Can we reproduce Taft and Towber s goals? 19

21 Taft and Towber s Work (Briefly) It would be very important to define flag spaces for quantum groups. [Manin 88] Their Goal. Build a quantum flag algebra Fl q (n) with the following properties: 1. Fl q (n) reduces to Fl(n) when q e.g., Fl q (n) and Fl(n) share the same basis (semi-standard Young tableaux). 2. Fl q (n) is a comodule algebra for the quantum group C[GL q (n)]. 3. Fl q (n) is isomorphic to the subalgebra of C[GL q (n)] generated by the minors {det q X I : I ( [n]) } d, 1 d < n of its matrix of generators X = (xij ). Theorem (Taft-Towber 91). Success! Put Fl q (n) :=... 20

22 III.B. Toward Fl(Y n ) for the Yangians Partial results Collect the formal noncommuting variables f (r) I (coordinate functions for the t (r) I ) as a power series f I (u) := r f(r) Theorem. Put F(Y n ) := C f (r) I modulo (AI ) u, (Y I,J ) u, and (C I,J ) u. We have: I u r. 2. F(Y n ) is a right comodule algebra for Y n. 3. There is a ring map F(Y n ) Y n onto the subalgebra of Y n generated by the (coefficients of powers of u 1 appearing in the power series) quantum minors t I (u). Limits to the analogy (missing relations) 3. F(Y n ) Y n is not injective. For example [ (t I ) (r), (t I ) (s)] = 0 ( r, s). 1. F(Y n ) does not reduce to Fl(n) in any sense. One at least needs relations (Y I,J ) v,w and (C I,J ) v,w for any v, w, not just v = u + a, w = u + b. 21

23 III.C. Highest Weight Representations of Y n Highest weight modules/vectors Definition. A module M λ(u) for Y n is called a highest weight module, with weight(s) λ(u) = (λ 1 (u),...,λ n (u)), if: there is a vector ξ M λ(u) such that Y n ξ = M λ(u) ; t aa ξ = λ a (u) ξ for all 1 a n and moreover, λ a (u) 1 + u 1 C[[u 1 ]]; t ab ξ = 0 for all 1 a < b n. Criteria for irreducibility and finite-dimensionality of the M λ(u) are known (Drinfeld 88, Billig-Futorny-Molev 05). There are highest weight vectors all over the place inside F(Y n )... 22

24 III.C. Highest Weight Representations of Y n For f (r) J F(Y n ), define a T(u)-action by: t ab (u) f (r) J = δ ab f (r) J + δ b J u 1 f (r) j 1 a j d, where j 1 a j d indicates replacement of b by a in J. Extend this to an action on monomials f (r 1) J 1 f (r k) J k derivations. by letting t ab (u) act by Theorem. This action respects the multiplicative relations in Y n and F(Y n ), making F(Y n ) a module algebra, decomposing as highest-weight modules. 23

25 Compare with Parabolic Subalgebras, I Fix a composition γ = (γ 1,...,γ r ) of n. Denote its partial sums by γ = {γ 1, γ 1 + γ 2,...,γ γ r 1 }. Factor T(u) over D as L D U with D block-diagonal of shape γ. Then, L = (E ij (u)) and U = (F ij (u)) are matrices with matrix entries. (e.g., F ij (u) has shape γ i γ j ) Theorem (Brundan-Kleshchev 05). Let Yγ (resp. Y γ + ) denote the parabolic subalgebra of D generated by the (coefficients of the powers of u 1 occurring in the power-series in the) entries of the E s (resp. F s). 1. Y γ and Y + γ are actually subalgebras of Y n. 2. They generate Y n, together with the coefficients of t [i] (u) (i γ ). 24

26 Compare with Parabolic Subalgebras, II Fix γ, γ, and Y + γ as above. Instead of studying F q (Y n ), one can study the pre flag algebra: The subalgebra of D generated by the quasi-plücker coordinates p K ij (T(u)), for all choices (i, j, K) with K + 1 γ. (Or rather, generated by the coefficients of the powers of u 1 appearing therein.) Corollary. The algebra Y + γ the left quasi-plücker coordinates p K ab is the subalgebra of D generated by (the coefficients of... ) for K = [k] \ a, a k < b, and k γ. Conclude: the relations l [j] p[i 1] il the generators of Y + γ. p [j]\l lk = 1 give some new relations among 25

27 Questions Are there any remaining quasi-plücker identites to discover? Find enough relations to make F q (Y n ) Y n injective (and a comodule map). Can modifications to the definition of t ab (u) f (r) J (highest weight) Y n -modules inside F(Y n )? be made to recover all irreducible References [1] A. Lauve, lauve or lauve@lacim.uqam.ca [2] J. Brundan and A. Kleshchev, Parabolic presentations of the Yangian Y (gl n ). [3] I. Gelfand, S. Gelfand, V. Retakh, and R. L. Wilson, Quasideterminants. [4] E. J. Taft and J. Towber, Quantum deformation of flag schemes and Grassmann schemes, I. [5] Y. Billig, V. Futorny, A. Molev, Verma modules for Yangians. 26