(Finite) W-algebras associated to truncated current Lie algebras

Transcription

1 1/ 15 (Finite) W-algebras associated to truncated current Lie algebras Xiao He (Université Laval) June 4, 2018 Interactions of quantum affine algebras with cluster algebras, current algebras and categorification A conference celebrating the 60 th birthday of Prof. Vyjayanthi Chari The Catholic University of America

2 2/ 15 Contents A brief review of W-algebras in the literature Truncated current Lie algebras Finite W-algebras associated to truncated current Lie algebras

3 3/ 15 Outline A brief review of W-algebras in the literature Truncated current Lie algebras Finite W-algebras associated to truncated current Lie algebras

4 4/ 15 Notation Z: integers, C: complex numbers. over C unless explicit mention. g: fin.dim. s.s. Lie algebra over C. e g nonzero nilpotent element. {e, f, h} g an sl 2 -triple: [e, f ] = h, [h, e] = 2e, [h, f ] = 2f. We fix a non-degenerate invariant bilinear form ( ) on g. Invariance means ([x, y] z) = (x [y, z]) x, y, z g.

5 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g.

6 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g. Let g = i Z g(i), where g(i) = {x g [h, x] = ix}. (1)

7 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g. Let g = i Z g(i), where g(i) = {x g [h, x] = ix}. (1) (1) is called a Dynkin grading of g.

8 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g. Let g = i Z g(i), where g(i) = {x g [h, x] = ix}. (1) (1) is called a Dynkin grading of g. We have e g(2), and

9 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g. Let g = i Z g(i), where g(i) = {x g [h, x] = ix}. (1) (1) is called a Dynkin grading of g. We have e g(2), and ( ) [e, ] : g(i) g(i + 2) is inj. for i 1 and surj. for i 1.

10 5/ 15 Nilpotent element and good Z-grading We can always embed e into an sl 2 -triple {e, f, h} in g. Let g = i Z g(i), where g(i) = {x g [h, x] = ix}. (1) (1) is called a Dynkin grading of g. We have e g(2), and ( ) [e, ] : g(i) g(i + 2) is inj. for i 1 and surj. for i 1. Definition A Z-grading Γ : g = i Z g(i) is called good if e g(2), s.t. the above condition ( ) is satisfied, and e is called a good element w.r.t. Γ. We call Γ even if g(i) = 0 odd i.

11 6/ 15 Various W-algebras Given a good Z-grading of g with good element e, one can construct the following four W-algebras.

12 6/ 15 Various W-algebras Given a good Z-grading of g with good element e, one can construct the following four W-algebras. C[JS] classical affine W-algebra gr W k (g, e) (quantum) affine W-algebra Zhu Zhu s C 2 -algebra Zhu C[S] classical finite W-algebra gr W fin (g, e) (quantum) finite W-algebra S is a Poisson variety sitting in g, and JS the arc space of S.

13 7/ 15 Outline A brief review of W-algebras in the literature Truncated current Lie algebras Finite W-algebras associated to truncated current Lie algebras

14 8/ 15 Truncated current Lie algebras (TCLA) The level p truncated current Lie algebra associated to g is ( ) g p := g C[t]/t p+1 C[t], with Lie bracket: [a t i, b t j ] = [a, b] t i+j, where t i+j = 0 when i + j > p.

15 8/ 15 Truncated current Lie algebras (TCLA) The level p truncated current Lie algebra associated to g is ( ) g p := g C[t]/t p+1 C[t], with Lie bracket: [a t i, b t j ] = [a, b] t i+j, where t i+j = 0 when i + j > p. An element x g can be considered as the element x 1 g p.

16 8/ 15 Truncated current Lie algebras (TCLA) The level p truncated current Lie algebra associated to g is ( ) g p := g C[t]/t p+1 C[t], with Lie bracket: [a t i, b t j ] = [a, b] t i+j, where t i+j = 0 when i + j > p. An element x g can be considered as the element x 1 g p. Let x = p i=0 x i t i and y = p i=0 y i t i, with x i, y i g.

17 8/ 15 Truncated current Lie algebras (TCLA) The level p truncated current Lie algebra associated to g is ( ) g p := g C[t]/t p+1 C[t], with Lie bracket: [a t i, b t j ] = [a, b] t i+j, where t i+j = 0 when i + j > p. An element x g can be considered as the element x 1 g p. Let x = p i=0 x i t i and y = p i=0 y i t i, with x i, y i g. Define (x y) p := (x i y j ). i+j=p

18 8/ 15 Truncated current Lie algebras (TCLA) The level p truncated current Lie algebra associated to g is ( ) g p := g C[t]/t p+1 C[t], with Lie bracket: [a t i, b t j ] = [a, b] t i+j, where t i+j = 0 when i + j > p. An element x g can be considered as the element x 1 g p. Let x = p i=0 x i t i and y = p i=0 y i t i, with x i, y i g. Define (x y) p := (x i y j ). i+j=p Lemma ( ) p is a non-degenerate invariant sym. bilinear form on g p.

19 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i).

20 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i).

21 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i). Lemma The Z-grading Γ p of g p is good with good element e.

22 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i). Lemma The Z-grading Γ p of g p is good with good element e. Let χ p = (e ) p g p.

23 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i). Lemma The Z-grading Γ p of g p is good with good element e. Let χ p = (e ) p g p. The following skew-symmetric bilinear form on g p ( 1) is non-degenerate. x y := χ p ([x, y]) = (e [x, y]) p.

24 9/ 15 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i). Lemma The Z-grading Γ p of g p is good with good element e. Let χ p = (e ) p g p. The following skew-symmetric bilinear form on g p ( 1) is non-degenerate. x y := χ p ([x, y]) = (e [x, y]) p. Choose a Lagrangian subspace l p of g p ( 1).

25 Preparation for finite W-algebras Given a good Z-grading Γ : g = i Z g(i) with good element e, there exists h Γ g, s.t. [h Γ, y] = iy, y g(i). Let g p (i) := {x g p [h Γ, x] = ix } and Γ p : g p = i Z g p (i). Lemma The Z-grading Γ p of g p is good with good element e. Let χ p = (e ) p g p. The following skew-symmetric bilinear form on g p ( 1) is non-degenerate. x y := χ p ([x, y]) = (e [x, y]) p. Choose a Lagrangian subspace l p of g p ( 1). Let ( ) m l,p = i 2 g p (i) l p. 9/ 15

26 10/ 15 Outline A brief review of W-algebras in the literature Truncated current Lie algebras Finite W-algebras associated to truncated current Lie algebras

27 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p.

28 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p. Let Q χp := U(g p )/I χp.

29 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p. Let Q χp := U(g p )/I χp. The adjoint action of m l,p on U(g p ) preserves I χp, hence it induces an adjoint action on Q χp.

30 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p. Let Q χp := U(g p )/I χp. The adjoint action of m l,p on U(g p ) preserves I χp, hence it induces an adjoint action on Q χp. Let H χp := Q ad m l,p χ p = {u + I χp Q χp [x, u] I χp, x m l,p }.

31 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p. Let Q χp := U(g p )/I χp. The adjoint action of m l,p on U(g p ) preserves I χp, hence it induces an adjoint action on Q χp. Let H χp := Q ad m l,p χ p = {u + I χp Q χp [x, u] I χp, x m l,p }. Definition (via Whittaker model) The finite W-algebra associated to (g p, e) is defined to be the associative algebra H χp with the multiplication (u + I χp ) (v + I χp ) := uv + I χp.

32 11/ 15 Finite W-algebras associated to TCLAs Let I χp be the left ideal of U(g p ) generated by the elements x χ p (x) for all x m l,p. Let Q χp := U(g p )/I χp. The adjoint action of m l,p on U(g p ) preserves I χp, hence it induces an adjoint action on Q χp. Let H χp := Q ad m l,p χ p = {u + I χp Q χp [x, u] I χp, x m l,p }. Definition (via Whittaker model) The finite W-algebra associated to (g p, e) is defined to be the associative algebra H χp with the multiplication (u + I χp ) (v + I χp ) := uv + I χp. Remark There are other equivalent definitions.

33 12/ 15 Quantization of Slodowy slice and Kostant s theorem One can find an sl 2 -triple {e, f, h}, s.t. f g( 2) and h g(0).

34 12/ 15 Quantization of Slodowy slice and Kostant s theorem One can find an sl 2 -triple {e, f, h}, s.t. f g( 2) and h g(0). Definition The Slodowy slice in g p through χ p is S χp := χ p + ker(ad f ), here ad = coadjoint action on g p. S χp has a Poisson structure.

35 12/ 15 Quantization of Slodowy slice and Kostant s theorem One can find an sl 2 -triple {e, f, h}, s.t. f g( 2) and h g(0). Definition The Slodowy slice in g p through χ p is S χp := χ p + ker(ad f ), here ad = coadjoint action on g p. S χp has a Poisson structure. Theorem (Quantization of jets of Slodowy slice) There is a filtration {K n H χp } n 0 on H χp s.t. the associated graded gr K H χp is isomorphic to C[S χp ] as Poisson algebras.

36 12/ 15 Quantization of Slodowy slice and Kostant s theorem One can find an sl 2 -triple {e, f, h}, s.t. f g( 2) and h g(0). Definition The Slodowy slice in g p through χ p is S χp := χ p + ker(ad f ), here ad = coadjoint action on g p. S χp has a Poisson structure. Theorem (Quantization of jets of Slodowy slice) There is a filtration {K n H χp } n 0 on H χp s.t. the associated graded gr K H χp is isomorphic to C[S χp ] as Poisson algebras. A regular nilpotent element e g is called principal, where regular means its adjoint orbit is dense in the nilpotent cone.

37 12/ 15 Quantization of Slodowy slice and Kostant s theorem One can find an sl 2 -triple {e, f, h}, s.t. f g( 2) and h g(0). Definition The Slodowy slice in g p through χ p is S χp := χ p + ker(ad f ), here ad = coadjoint action on g p. S χp has a Poisson structure. Theorem (Quantization of jets of Slodowy slice) There is a filtration {K n H χp } n 0 on H χp s.t. the associated graded gr K H χp is isomorphic to C[S χp ] as Poisson algebras. A regular nilpotent element e g is called principal, where regular means its adjoint orbit is dense in the nilpotent cone. Theorem (Truncated current version of Kostant s theorem) Let e be principal. Then the finite W-algebra H χp associated to (g p, e) is isomorphic to the center of U(g p ).

38 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p.

39 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p.

40 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M.

41 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M. Remark Q χp is a Whittaker g p -module and H χp = Wh(Q χp ).

42 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M. Remark Q χp is a Whittaker g p -module and H χp = Wh(Q χp ). Let g p -Wmod χ p be the category of f.g. Whittaker g p -modules and H χp -Mod the category of f.g. left H χp -modules.

43 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M. Remark Q χp is a Whittaker g p -module and H χp = Wh(Q χp ). Let g p -Wmod χ p be the category of f.g. Whittaker g p -modules and H χp -Mod the category of f.g. left H χp -modules. Theorem (Skryabin equivalence for TCLA) (1) Let M g p -Wmod χ p. Then Wh(M) H χp -Mod.

44 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M. Remark Q χp is a Whittaker g p -module and H χp = Wh(Q χp ). Let g p -Wmod χ p be the category of f.g. Whittaker g p -modules and H χp -Mod the category of f.g. left H χp -modules. Theorem (Skryabin equivalence for TCLA) (1) Let M g p -Wmod χ p. Then Wh(M) H χp -Mod. (2) Let N H χp -Mod. Then Q χp Hχ p N g p-wmod χ p.

45 13/ 15 Whittaker modules and Skryabin equivalence Definition (Whitt.=Whittaker) A g p -module M is called a Whitt. module if a χ p (a) acts loc. nil. on M, a m l,p. An element x in a Whitt. module is M is called a Whitt. vector if (a χ p (a)) x = 0, a m l,p. Let Wh(M) be the collection of Whit. vectors of a Whitt. module M. Remark Q χp is a Whittaker g p -module and H χp = Wh(Q χp ). Let g p -Wmod χ p be the category of f.g. Whittaker g p -modules and H χp -Mod the category of f.g. left H χp -modules. Theorem (Skryabin equivalence for TCLA) (1) Let M g p -Wmod χ p. Then Wh(M) H χp -Mod. (2) Let N H χp -Mod. Then Q χp Hχ p N g p-wmod χ p. (3) Wh( ) and Q χp Hχ give an equivalence of categories p between g p -Wmod χ p and H χp -Mod.

46 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1.

47 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1. Non-degenerate bilinear forms and good Z-gradings exist on them,

48 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1. Non-degenerate bilinear forms and good Z-gradings exist on them,one can define W-algebras associated to truncated multicurrent algebras, and the above theorems still hold.

49 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1. Non-degenerate bilinear forms and good Z-gradings exist on them,one can define W-algebras associated to truncated multicurrent algebras, and the above theorems still hold. Affine W-algebras associated to TCLAs can be defined as in the semisimple case,

50 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1. Non-degenerate bilinear forms and good Z-gradings exist on them,one can define W-algebras associated to truncated multicurrent algebras, and the above theorems still hold. Affine W-algebras associated to TCLAs can be defined as in the semisimple case, they are vertex algebras and their Zhu algebras are finite W-algebras associated to TCLAs.

51 14/ 15 Two remarks Consider more generally truncated multicurrent algebras, i.e., Lie algebras of the following form, g C[t 1 ]/ t n C[t 1 ]/ t n r +1 r, where n 1,, n r 1. Non-degenerate bilinear forms and good Z-gradings exist on them,one can define W-algebras associated to truncated multicurrent algebras, and the above theorems still hold. Affine W-algebras associated to TCLAs can be defined as in the semisimple case, they are vertex algebras and their Zhu algebras are finite W-algebras associated to TCLAs. More properties of these affine W-algebras are still under study.

52 Thanks! 15/ 15