An extended Freudenthal Magic Square in characteristic 3 Alberto Elduque Universidad de Zaragoza From Lie Algebras to Quantum Groups Coimbra, 2006 Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 1 / 29 Exceptional Lie algebras G2, F4, E6, E7, E8 G2 = der O (Cartan 1914) F = der H ( ) 4 3 O (Chevalley-Schafer 1950) E6 = str0 H3(O) Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 2 / 29 Exceptional Lie algebras G2, F4, E6, E7, E8 G2 = der O (Cartan 1914) F = der H ( ) 4 3 O (Chevalley-Schafer 1950) E6 = str0 H3(O) Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 2 / 29 Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, Then T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J is a Lie algebra (char 6= 3) under a suitable Lie bracket. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 3 / 29 Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, Then T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J is a Lie algebra (char 6= 3) under a suitable Lie bracket. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 3 / 29 Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, Then T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J is a Lie algebra (char 6= 3) under a suitable Lie bracket. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 3 / 29 Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, Then T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J is a Lie algebra (char 6= 3) under a suitable Lie bracket. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 3 / 29 Freudenthal Magic Square T (C, J) H3(k) H3(k × k) H3(Mat2(k)) H3(C(k)) k A1 A2 C3 F4 k × k A2 A2 ⊕ A2 A5 E6 Mat2(k) C3 A5 D6 E7 C(k) F4 E6 E7 E8 Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 4 / 29 Tits construction rearranged 0 3 2 0 J = H3(C ) ' k ⊕ ⊕i=0ιi (C ) , 2 2 0 J0 ' k ⊕ ⊕i=0ιi (C ) , 0 2 0 der J ' tri(C ) ⊕ ⊕i=0ιi (C ) , T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J 2 2 0 0 2 0 ' der C ⊕ (C0 ⊗ k ) ⊕ ⊕i=0C0 ⊗ ιi (C ) ⊕ tri(C ) ⊕ (⊕i=0ιi (C )) 0 2 0 ' tri(C) ⊕ tri(C ) ⊕ ⊕i=0ιi (C ⊗ C ) (Barton-Sudbery, Landsberg-Manivel, Allison-Faulkner) Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 5 / 29 Tits construction rearranged 0 3 2 0 J = H3(C ) ' k ⊕ ⊕i=0ιi (C ) , 2 2 0 J0 ' k ⊕ ⊕i=0ιi (C ) , 0 2 0 der J ' tri(C ) ⊕ ⊕i=0ιi (C ) , T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J 2 2 0 0 2 0 ' der C ⊕ (C0 ⊗ k ) ⊕ ⊕i=0C0 ⊗ ιi (C ) ⊕ tri(C ) ⊕ (⊕i=0ιi (C )) 0 2 0 ' tri(C) ⊕ tri(C ) ⊕ ⊕i=0ιi (C ⊗ C ) (Barton-Sudbery, Landsberg-Manivel, Allison-Faulkner) Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 5 / 29 Tits construction rearranged 0 3 2 0 J = H3(C ) ' k ⊕ ⊕i=0ιi (C ) , 2 2 0 J0 ' k ⊕ ⊕i=0ιi (C ) , 0 2 0 der J ' tri(C ) ⊕ ⊕i=0ιi (C ) , T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J 2 2 0 0 2 0 ' der C ⊕ (C0 ⊗ k ) ⊕ ⊕i=0C0 ⊗ ιi (C ) ⊕ tri(C ) ⊕ (⊕i=0ιi (C )) 0 2 0 ' tri(C) ⊕ tri(C ) ⊕ ⊕i=0ιi (C ⊗ C ) (Barton-Sudbery, Landsberg-Manivel, Allison-Faulkner) Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 5 / 29 Symmetric composition algebras Nicer formulas are obtained if symmetric composition algebras are used, instead of the more classical Hurwitz algebras. (S, ∗, q) ( q(x ∗ y) = q(x)q(y), q(x ∗ y, z) = q(x, y ∗ z). Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 6 / 29 Symmetric composition algebras Nicer formulas are obtained if symmetric composition algebras are used, instead of the more classical Hurwitz algebras. (S, ∗, q) ( q(x ∗ y) = q(x)q(y), q(x ∗ y, z) = q(x, y ∗ z). Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 6 / 29 Symmetric composition algebras: examples Para-Hurwitz algebras: C Hurwitz algebra with norm q and standard involution , but with new multiplication x ∗ y =x ¯y¯. Okubo algebras: In characteristic 6= 3 these are the forms of sl3, ∗, q) with ω − ω2 x ∗ y = ωxy − ω2yx − tr(xy)1, 3 1 q(x) = tr(x2), q(x, y) = tr(xy). 2 (ω a cubic root of 1.) A different definition is needed in characteristic 3. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 7 / 29 Symmetric composition algebras: examples Para-Hurwitz algebras: C Hurwitz algebra with norm q and standard involution , but with new multiplication x ∗ y =x ¯y¯. Okubo algebras: In characteristic 6= 3 these are the forms of sl3, ∗, q) with ω − ω2 x ∗ y = ωxy − ω2yx − tr(xy)1, 3 1 q(x) = tr(x2), q(x, y) = tr(xy). 2 (ω a cubic root of 1.) A different definition is needed in characteristic 3. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 7 / 29 Symmetric composition algebras: classification Theorem (Okubo, Osborn, Myung, P´erez-Izquierdo, E.) With some exceptions in dimension 2, any symmetric composition algebras is either a para-Hurwitz algebra (dimension 1, 2, 4 or 8), or an Okubo algebra (dimension 8). Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 8 / 29 Symmetric composition algebras: classification Theorem (Okubo, Osborn, Myung, P´erez-Izquierdo, E.) With some exceptions in dimension 2, any symmetric composition algebras is either a para-Hurwitz algebra (dimension 1, 2, 4 or 8), or an Okubo algebra (dimension 8). Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 8 / 29 Symmetric composition algebras: classification Theorem (Okubo, Osborn, Myung, P´erez-Izquierdo, E.) With some exceptions in dimension 2, any symmetric composition algebras is either a para-Hurwitz algebra (dimension 1, 2, 4 or 8), or an Okubo algebra (dimension 8). Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 8 / 29 Triality algebra (S, ∗, q) a symmetric composition algebra 3 tri(S) = {(d0, d1, d2) ∈ so(S, q) : d0(x ∗ y) = d1(x) ∗ y + x ∗ d2(y) ∀x, y ∈ S} is the triality Lie algebra of S. 0 if dim S = 1, 2-dim’l abelian if dim S = 2, tri(S) = 3 so(S0, q) if dim S = 4, so(S, q) if dim S = 8. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 9 / 29 Triality algebra (S, ∗, q) a symmetric composition algebra 3 tri(S) = {(d0, d1, d2) ∈ so(S, q) : d0(x ∗ y) = d1(x) ∗ y + x ∗ d2(y) ∀x, y ∈ S} is the triality Lie algebra of S. 0 if dim S = 1, 2-dim’l abelian if dim S = 2, tri(S) = 3 so(S0, q) if dim S = 4, so(S, q) if dim S = 8. Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 9 / 29 The Lie algebra g(S, S 0) Let S and S0 be two symmetric composition algebras. Consider 0 0 2 0 g(S, S ) = tri(S) ⊕ tri(S ) ⊕ ⊕i=0ιi (S ⊗ S ) , 0 0 where ιi (S ⊗ S ) is just a copy of S ⊗ S , with bracket given by: tri(S) ⊕ tri(S0) is a Lie subalgebra of g(S, S0), 0 0 [(d0, d1, d2), ιi (x ⊗ x )] = ιi di (x) ⊗ x , 0 0 0 0 0 0 [(d0, d1, d2), ιi (x ⊗ x )] = ιi x ⊗ di (x ) , 0 0 0 0 [ιi (x ⊗ x ), ιi+1(y ⊗ y )] = ιi+2 (x ∗ y) ⊗ (x ∗ y ) (indices modulo 3), 0 0 0 0 0 i 0i 0 [ιi (x ⊗ x ), ιi (y ⊗ y )] = q (x , y )θ (tx,y ) + q(x, y)θ (tx0,y 0 ), 1 1 where tx,y = q(x,.)y − q(y,.)x, 2 q(x, y)1 − rx ly , 2 q(x, y)1 − lx ry Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 10 / 29 The Lie algebra g(S, S 0) Let S and S0 be two symmetric composition algebras. Consider 0 0 2 0 g(S, S ) = tri(S) ⊕ tri(S ) ⊕ ⊕i=0ιi (S ⊗ S ) , 0 0 where ιi (S ⊗ S ) is just a copy of S ⊗ S , with bracket given by: tri(S) ⊕ tri(S0) is a Lie subalgebra of g(S, S0), 0 0 [(d0, d1, d2), ιi (x ⊗ x )] = ιi di (x) ⊗ x , 0 0 0 0 0 0 [(d0, d1, d2), ιi (x ⊗ x )] = ιi x ⊗ di (x ) , 0 0 0 0 [ιi (x ⊗ x ), ιi+1(y ⊗ y )] = ιi+2 (x ∗ y) ⊗ (x ∗ y ) (indices modulo 3), 0 0 0 0 0 i 0i 0 [ιi (x ⊗ x ), ιi (y ⊗ y )] = q (x , y )θ (tx,y ) + q(x, y)θ (tx0,y 0 ), 1 1 where tx,y = q(x,.)y − q(y,.)x, 2 q(x, y)1 − rx ly , 2 q(x, y)1 − lx ry Alberto Elduque (Universidad de Zaragoza) Freudenthal Magic Supersquare June 30, 2006 10 / 29 The Lie algebra g(S, S 0) Let S and S0 be two symmetric composition algebras.