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Tits Construction, Freudenthal Magic Square, and Modular Simple Lie Superalgebras

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Tits Construction, Freudenthal Magic Square, and Modular Simple Lie Superalgebras Tits construction, Freudenthal Magic Square, and modular simple Lie superalgebras Alberto Elduque Universidad de Zaragoza Lie Algebras and their Applications in honor of Professor Hyo Chul Myung on his 70th birthday 1 Freudenthal Magic Square 2 Symmetric composition algebras 3 Composition superalgebras 4 Freudenthal Supermagic Square 5 Back to Tits construction 6 Some conclusions Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 2 / 38 1 Freudenthal Magic Square 2 Symmetric composition algebras 3 Composition superalgebras 4 Freudenthal Supermagic Square 5 Back to Tits construction 6 Some conclusions Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 3 / 38 G2 = der O (Cartan 1914) F = der H ( ) 4 3 O (Chevalley-Schafer 1950) E6 = str0 H3(O) Exceptional Lie algebras G2, F4, E6, E7, E8 Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 4 / 38 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) Tits construction . July 30, 2007 4 / 38 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= 2, 3) under a suitable Lie bracket: 1 1 [a ⊗ x, b ⊗ y] = tr(xy)D + [a, b] ⊗ xy − tr(xy)1 + 2t(ab)d . 3 a,b 3 x,y Tits construction (1966) Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 5 / 38 then T (C, J) = der C ⊕ (C0 ⊗ J0) ⊕ der J is a Lie algebra (char 6= 2, 3) under a suitable Lie bracket: 1 1 [a ⊗ x, b ⊗ y] = tr(xy)D + [a, b] ⊗ xy − tr(xy)1 + 2t(ab)d . 3 a,b 3 x,y Tits construction (1966) C a Hurwitz algebra (unital composition algebra), J a central simple Jordan algebra of degree 3, Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 5 / 38 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= 2, 3) under a suitable Lie bracket: 1 1 [a ⊗ x, b ⊗ y] = tr(xy)D + [a, b] ⊗ xy − tr(xy)1 + 2t(ab)d . 3 a,b 3 x,y Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 5 / 38 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) Tits construction . July 30, 2007 6 / 38 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 ) (Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel) 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 ) , Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 7 / 38 (Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel) 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 ) Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 7 / 38 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 ) (Vinberg, Allison-Faulkner, Barton-Sudbery, Landsberg-Manivel) Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 7 / 38 1 Freudenthal Magic Square 2 Symmetric composition algebras 3 Composition superalgebras 4 Freudenthal Supermagic Square 5 Back to Tits construction 6 Some conclusions Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 8 / 38 (S, ∗, q) ( q(x ∗ y) = q(x)q(y), q(x ∗ y, z) = q(x, y ∗ z). Symmetric composition algebras Nicer formulas are obtained if symmetric composition algebras are used, instead of the more classical Hurwitz algebras. Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 9 / 38 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) Tits construction . July 30, 2007 9 / 38 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. Symmetric composition algebras: examples Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 10 / 38 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. Symmetric composition algebras: examples Para-Hurwitz algebras: C Hurwitz algebra with norm q and standard involution , but with new multiplication x ∗ y =x ¯y¯. Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 10 / 38 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) Tits construction . July 30, 2007 10 / 38 Symmetric composition algebras: classification Theorem (Okubo, Osborn, Myung, P´erez-Izquierdo, E.) With some exceptions in dimension 2, any symmetric composition algebra is either a para-Hurwitz algebra (dimension 1, 2, 4 or 8), or an Okubo algebra (dimension 8). Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 11 / 38 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. 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. Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 12 / 38 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) Tits construction . July 30, 2007 12 / 38 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 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: Alberto Elduque (Universidad de Zaragoza) Tits construction . July 30, 2007 13 / 38 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) Tits construction .
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