Quantizations of Kac-Moody algebras

Transcription

1 FES-C UNAM MEXICO 1 4 of July 2013 Noncommutatve Rngs and ther Applcatons LENS, France

2 Quantzatons Quantzed envelopng algebras: Drnfel d (1985), Jmbo (1985).

3 Quantzatons Quantzed envelopng algebras: Drnfel d (1985), Jmbo (1985). Multparameter verson: N. Reshetkhn (1990); P. Cotta-Ramusno, M. Rnald (1991); C. De Concn, V.G. Kac, C. Proces (1992); M. Constantn, M. Varagnolo (1994).

4 Quantzatons Quantzed envelopng algebras: Drnfel d (1985), Jmbo (1985). Multparameter verson: N. Reshetkhn (1990); P. Cotta-Ramusno, M. Rnald (1991); C. De Concn, V.G. Kac, C. Proces (1992); M. Constantn, M. Varagnolo (1994). Deformatons of Hall algebras: C. M. Rngel (1996).

5 Quantzatons Quantzed envelopng algebras: Drnfel d (1985), Jmbo (1985). Multparameter verson: N. Reshetkhn (1990); P. Cotta-Ramusno, M. Rnald (1991); C. De Concn, V.G. Kac, C. Proces (1992); M. Constantn, M. Varagnolo (1994). Deformatons of Hall algebras: C. M. Rngel (1996). Two-parameter quantum groups: G. Benkart, S. Wtherspoon (2004); N. Bergeron, Y. Gao, N. Hu (2005).

6 Hopf algebras Algebras and coalgebras: m : H H H : H H H product; coproduct;

7 Hopf algebras Algebras and coalgebras: m : H H H product; Examples: : H H H coproduct; k[g], (g) = g g, g G; U(L), (l) = l l, l L.

8 Hopf algebras Algebras and coalgebras: m : H H H product; : H H H coproduct; Examples: k[g], (g) = g g, g G; U(L), (l) = l l, l L. Both are cocommutatve: (a) = a (1) a (2) = a (2) a (1).

9 Hopf algebras Algebras and coalgebras: m : H H H product; Examples: : H H H Both are cocommutatve: coproduct; k[g], (g) = g g, g G; U(L), (l) = l l, l L. (a) = a (1) a (2) = a (2) a (1). Thm. (Carter Kostant). Every cocommutatve Hopf algebra (over C) has the form C[G]#U(L).

10 Hopf algebras Algebras and coalgebras: m : H H H product; Examples: : H H H Both are cocommutatve: coproduct; k[g], (g) = g g, g G; U(L), (l) = l l, l L. (a) = a (1) a (2) = a (2) a (1). Thm. (Carter Kostant). Every cocommutatve Hopf algebra (over C) has the form C[G]#U(L). A quantum group = a Hopf algebra whch s nether commutatve nor cocommutatve, but t s close to.

11 Quantzatons In what extend a quantum unversal envelopng algebra of a Kac-Moody algebra g s defned by mult-degrees of ts defnng relatons?

12 Quantzatons In what extend a quantum unversal envelopng algebra of a Kac-Moody algebra g s defned by mult-degrees of ts defnng relatons? R g s a class of character Hopf algebras defned by the same number of defnng relatons of the same degrees as g s.

13 Gabber-Kac representaton Cartan matrx A = a j : an ntegral n n matrx a = 2, a j 0 for j, a j = 0 mples a j = 0.

14 Gabber-Kac representaton Cartan matrx A = a j : an ntegral n n matrx a = 2, a j 0 for j, a j = 0 mples a j = 0. Generators: e, f, h, 1 n.

15 Gabber-Kac representaton Cartan matrx A = a j : an ntegral n n matrx a = 2, a j 0 for j, a j = 0 mples a j = 0. Generators: e, f, h, 1 n. Relatons: [h, h j ] = 0, [h, e j ] = a j e j, [h, f j ] = a j f j ; [e, f j ] = 0 f j, [e, f ] = h ; (ad e ) 1 a j e j = 0, (ad f ) 1 a j f j = 0 f j, (ad a) m b = [... [[b, a], a],..., a]. }{{} m

16 Quantfcaton of Cartan subalgebra h exp(h) = 1 + h + h2 2 + h3 3! + h4 4! + (h) = 1 h + h 1; (exp(h)) = exp(h) exp(h).

17 Quantfcaton of Cartan subalgebra h exp(h) = 1 + h + h2 2 + h3 3! + h4 4! + (h) = 1 h + h 1; (exp(h)) = exp(h) exp(h). [h, h j ] = 0 G = exp(h) s a commutatve group; [h, e j ] = a j e j, G acts on x j = q(e j ) by a character χ j ; [h, f j ] = a j f j, G acts on x j = q(f j ) by a character χ j.

18 Quantfcaton of Cartan subalgebra h exp(h) = 1 + h + h2 2 + h3 3! + h4 4! + (h) = 1 h + h 1; (exp(h)) = exp(h) exp(h). [h, h j ] = 0 G = exp(h) s a commutatve group; [h, e j ] = a j e j, G acts on x j = q(e j ) by a character χ j ; [h, f j ] = a j f j, G acts on x j = q(f j ) by a character χ j. g 1 y j g = χ j (g)y j, χ j = (χj ) 1. g 1 y j g = χ j (g)y j,

19 Hopf algebra structure Group-lke element: (g) = g g, ε(g) = 1. The grouplkes span a Hopf subalgebra =k [G].

20 Hopf algebra structure Group-lke element: (g) = g g, ε(g) = 1. The grouplkes span a Hopf subalgebra =k [G]. A skew-prmtve element: (a) = a h + f a, wth f, h G.

21 Hopf algebra structure Group-lke element: (g) = g g, ε(g) = 1. The grouplkes span a Hopf subalgebra =k [G]. A skew-prmtve element: (a) = a h + f a, wth f, h G. Coproduct: (y ± ) = y ± h ± + f ± y ±, h ±, f ± G.

22 Hopf algebra structure Group-lke element: (g) = g g, ε(g) = 1. The grouplkes span a Hopf subalgebra =k [G]. A skew-prmtve element: (a) = a h + f a, wth f, h G. Coproduct: (y ± ) = y ± h ± + f ± y ±, h ±, f ± G. g 1 y ± j g = χ j ± (g)y j, χ j = (χj ) 1, g G.

23 Second and thrd groups of relatons [e, f j ] = 0 f j, [e, f ] = h ; df R j = α j y y j + β j y j y = 0, R (y, y ) = df α y y + β y y k µ k g k = 0, g k G.

24 Second and thrd groups of relatons [e, f j ] = 0 f j, [e, f ] = h ; df R j = α j y y j + β j y j y = 0, R (y, y ) = df α y y + β y y k µ k g k = 0, g k G. (ad e ) 1 a j e j = 0, (ad f ) 1 a j f j = 0 f j : S j (y 1 a j S j (y, y j ) = df 1 a j, y j ) = df k=0 k=0 γ jk y k y j y 1 a j k = 0, j; δ jk (y ) k y j (y ) 1 a j k = 0, j.

25 Second and thrd groups of relatons H = G y, y R j = 0, S ± j = 0, 2n 3 + 3n 2 parameters α j, β j, µ k, γ jk, δ jk.

26 Second and thrd groups of relatons H = G y, y R j = 0, S ± j = 0, 2n 3 + 3n 2 parameters α j, β j, µ k, γ jk, δ jk. Theorem 1. The only way to keep the Hopf algebra structure on H s to replace the Le operaton n Gabber-Kac relatons wth a skew commutator: [u, v] = χ v (h u )uv χ u (f v )vu.

27 Second and thrd groups of relatons H = G y, y R j = 0, S ± j = 0, 2n 3 + 3n 2 parameters α j, β j, µ k, γ jk, δ jk. Theorem 1. The only way to keep the Hopf algebra structure on H s to replace the Le operaton n Gabber-Kac relatons wth a skew commutator: [u, v] = χ v (h u )uv χ u (f v )vu. Remans just n 2 parameters, p j = χ (h j )χ (f j ), of 2n 3 + 3n 2.

28 Indecomposable Cartan matrces A = a j. Defnton. A quantfcaton H s regular f p j p j = p a j.

29 Indecomposable Cartan matrces A = a j. Defnton. A quantfcaton H s regular f p j p j = p a j. All known quantzatons are regular.

30 Indecomposable Cartan matrces A = a j. Defnton. A quantfcaton H s regular f p j p j = p a j. All known quantzatons are regular. Theorem 2. There exsts just a fnte number of algebra structures for exceptonal quantzatons of a gven Kac-Moody algebra.

31 Indecomposable Cartan matrces A = a j. Defnton. A quantfcaton H s regular f p j p j = p a j. All known quantzatons are regular. Theorem 2. There exsts just a fnte number of algebra structures for exceptonal quantzatons of a gven Kac-Moody algebra. Theorem 3. If one of the parameters p j p j, 1, j n s not a root of 1 then the quantfcaton H s regular.

32 Indecomposable Cartan matrces A = a j. Defnton. A matrx A s symmetrzable f there exsts natural d 1, d 2,..., d n such that, d a j = d j a j, 1, j n. An n-tuple (m 1, m 2,..., m n ) s a modular symmetrzaton f m a j m j a j (mod N), 0 m, m j < N, 1, j n.

33 Indecomposable Cartan matrces A = a j. Defnton. A matrx A s symmetrzable f there exsts natural d 1, d 2,..., d n such that, d a j = d j a j, 1, j n. An n-tuple (m 1, m 2,..., m n ) s a modular symmetrzaton f m a j m j a j (mod N), 0 m, m j < N, 1, j n. Theorem 4. The algebrac structure of a regular quantfcaton s defned by two ndependent parameters q k and m M so that p = q d ξ m, m = (m 1, m 2,..., m n ), where ξ s a fxed Nth prmtve root of 1.

34 Indecomposable Cartan matrces A = a j. Defnton. A matrx A s symmetrzable f there exsts natural d 1, d 2,..., d n such that, d a j = d j a j, 1, j n. An n-tuple (m 1, m 2,..., m n ) s a modular symmetrzaton f m a j m j a j (mod N), 0 m, m j < N, 1, j n. Theorem 4. The algebrac structure of a regular quantfcaton s defned by two ndependent parameters q k and m M so that p = q d ξ m, m = (m 1, m 2,..., m n ), where ξ s a fxed Nth prmtve root of 1. In all known quantzatons, m = (0, 0,..., 0).

35 The number of exceptonal quantzatons Here ϕ, 0 s the Fbonacc sequence 1, 1, 2, 3, 5, 8, 13, 21,.... A n ϕ 2n 2 G 2 12 E (1) B n 2ϕ 2n 2 A (1) 1 6 E (1) C n 2ϕ 2n A (1) l ϕ 2n + ϕ 2n 2 2 A (2) 2 24 E C (1) l 4ϕ 2n A (2) 2l 4ϕ 2n E G (1) 2 38 D (2) l+1 4ϕ 2n + 2 E F (1) 4 91 E (2) 6 80 F 4 40 E (1) D (3) 4 19 D n ϕ 2n + ϕ 2n 7 2 D (1) l 15 ϕ 2n ϕ 2n 8 2 A (2) 2l 1 2ϕ 2n 2 + 2ϕ 2n B (1) l ϕ 2n+1 + 5ϕ 2n 5 2

36 THE END THANK YOU