Neuigkeiten über Lie-Algebren

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1 1 PSLn+1(q), Ln+1(q) q n(n+1)/ n (q i+1 1) (n+1,q 1) i=1 The Tits group F4 () is not a group of Lie type, but is the (index ) commutator subgroup of F4 (). It is usually given honorary Lie type status. The groups starting on the second row are the classical groups. The sporadic suzuki group is unrelated to the families of Suzuki groups. Copyright c 01 Ivan Andrus q 36 (q 1 1)(q 9 1)(q 8 1) (q 6 1)(q 5 1)(q 1) (3, q 1) q 63 (, q 1) 9 (q i 1) i=1 i =, q 10 (q 30 1)(q 4 1) (q 0 1)(q 18 1)(q 14 1) (q 1 1)(q 8 1)(q 1) For sporadic groups and families, alternate names in the upper left are other names by which they may be known. For specific non-sporadic groups these are used to indicate isomorphims. All such isomorphisms appear on the table except the family Bn( m ) = Cn( m ). Finite simple groups are determined by their order with the following exceptions: Bn(q) and Cn(q) for q odd, n > ; A8 = A3() and A(4) of order q 4 (q 1 1)(q 8 1) (q 6 1)(q 1) 1 q 6 (q 6 1)(q 1) q 1 (q 8 + q 4 + 1) (q 6 1)(q 1) q 36 (q 1 1)(q 9 + 1)(q 8 1) (q 6 1)(q 5 + 1)(q 1) (3, q + 1) q (q + 1)(q 1) q 1 (q 6 + 1)(q 4 1) (q 3 + 1)(q 1) q 3 (q 3 + 1)(q 1) On+1(q), Ωn+1(q) q n n (q i 1) (, q 1) i=1 q n n (q i 1) (, q 1) i=1 q n(n 1) (q n n 1 1) (q i 1) (4,q n 1) q n(n 1) (q n n 1 +1) (q i 1) (4,q n +1) i=1 q n(n+1)/ n+1 (q i ( 1) i ) (n+1,q+1) i= F7, HHM, HT H The Periodic Table Of Finite Simple Groups 0, C1, Z1 Dynkin Diagrams of Simple Lie Algebras 1 An 1 3 n F4 C A1(4), A1(5) A5 60 A1(9), B() A3() A6 360 A7 50 A A An n! A() A1(7) 168 G(3) A1(8) 504 A1(11) 660 A1(13) 1 09 A1(17) 448 An(q) Bn Cn E6() E6(3) E6(4) E6(q) 1 3 n 1 3 n E7() E7(3) E7(4) E7(q) E8() E8(3) E8(4) E8(q) F4() F4(3) F4(4) F4(q) Dn E6,7,8 G(3) G(4) G(5) G(q) 3 4 n D 4 ( 3 ) D 4 (3 3 ) D 4 (4 3 ) D 4 (q 3 ) E 6 ( ) E 6 (3 ) E 6 (4 ) E 6 (q ) G B ( 3 ) 9 10 B ( 5 ) B ( 7 ) B( n+1 ) Tits F 4 () F 4 ( 3 ) F 4 ( 5 ) F 4( n+1 ) G (3 3 ) G (3 5 ) G (3 7 ) G(3 n+1 ) A3(4) B(3) 5 90 B(4) B3() B(5) B(7) Bn(q) C3(3) C3(5) C4(3) C3(7) D4() D4(3) D5() D4(5) D 4 ( ) D 4 (3 ) D 5 ( ) D 4 (4 ) G() A (9) A (16) A (5) A 3 (9) C3(9) D5(3) D 4 (5 ) A (64) Meinolf PSpn(q) O + n (q) O PSUn+1(q) n Cn(q) Dn(q) D n(q ) A n(q ) Geck (q) i=1 C3 3 C5 5 C7 7 C11 11 C13 13 Zp Cp p Neuigkeiten über Lie-Algebren Kolloquium Tübingen 17. Juli 017 Alternating Groups Classical Chevalley Groups Chevalley Groups Classical Steinberg Groups Steinberg Groups Suzuki Groups Ree Groups and Tits Group Sporadic Groups Cyclic Groups Alternates Symbol Order M11 M1 M M3 M Sz O NS, O S 3 1 Suz O N Co3 Co Co J(1), J(11) J F5, D HN H J J LyS Ly H JM J F3, E Th J4 HS M() M(3) Fi Fi McL He Ru F3+, M(4) F F1, M1 Fi 4 B M

2 The greatest mathematical paper of all time A. J. Coleman, The Mathematical Intelligencer 11 (1989), 938 Euclid s Elements, Newton s Principia,... and in the past 00 years: Wilhelm Killing ( ), Die Zusammensetzung der stetigen, endlichen Transformationsgruppen II, Mathematische Annalen 33 (1888), MathSciNet Review MR by Jean Dieudonné: Many mathematicians will disagree, [...]. One may however observe that: (1) nobody had tackled the problem before Killing; () he solved it by methods he invented; (3) nobody collaborated with him for that solution; (4) the result became a most important milestone in modern mathematics. I think it is not so easy to find papers exhibiting all those features

3 Lie algebra = infinitesimal version of a transformation group A vector space L equipped with a product x y such that. x x = 0,. x (y z) + y (z x) + z (x y) = 0 (Jacobi identity). Usually write x y as [x, y] (Lie bracket). Example. Vector product in L = R 3 : Example. Matrices: L = M n (C) with bracket [A, B] = A B B A; denote gl n (C). Subalgebra sl n (C) = {A M n (C) Trace(A) = 0}.

4 An infinite-dimensional example Start with R = C[X, X 1 ] (Laurent polynomials). A linear map D : R R is called a derivation if D(f g) = f D(g) + D(f ) g for all f, g R. Example: D = usual (formal) derivative with respect to X. D(X n ) = nx n 1, D(X 1 ) = X, etc. Let L = vector space of all derivations of R. Exercise: If D, D L, then [D, D ] = D D D D L. So L is a Lie algebra. For m Z, we set L m (f ) = X m+1 D(f ) for f R. Then:. {L m m Z} is a vector space basis of L.. [L m, L n ] = (m n)l m+n for all m, n Z. L is called Witt algebra important in mathematical physics.

5 A subspace U L is called an ideal if [u, x] U and [x, u] U for all x L, u U. In this case, L = L/U also is a Lie algebra. So L built up from U and L. Definition. L is called simple if L {0}, the bracket is not identically zero and there is no proper ideal in L. Cartan Killing ( 1890): The finite-dimensional simple Lie algebras over C are classified by Dynkin diagrams. A n D n G > F 4 > B n < C n > E 6 E 7 E 8

6 Infinite families: Lie algebras of matrices A n sl n+1 (C), B n so n+1 (C), C n sp n (C), D n so n (C). Exceptional algebras: dim g = 14, dim f 4 = 5, dim e 6 = 78, dim e 7 = 133, dim e 8 = 48. S. GARIBALDI, E 8, the most exceptional group. Bull. AMS 53 (016), The Lie algebra e 8 or Lie group E 8 was first sighted by a human being sometime in summer or early fall of 1887, by Wilhelm Killing as part of his program to classify the semisimple finite-dimensional Lie algebras over the complex numbers. [...] Since then, it has been a source of fascination for mathematicians and others in its role as the largest of the exceptional Lie algebras. [...] If we know some statement for all groups except E 8, then we do not really know it.

7 How does a Dynkin diagram determine a simple Lie algebra? If diagram has nodes 1,..., n, then form Cartan matrix A = (a ij ) 1 i,j n where:. a ii = for all i; a ij = 0 if i j are not joined by an edge;. if i j are joined by a single edge, set a ij = a ji = 1;. if i j are joined by m edges (arrow towards j), set a ij = 1, a ji = m. B 3 1 < 3 A = Chevalley generators ( épinglage ) of corresponding Lie algebra L:. L = e i, f i, h i 1 i n Lie,. [e i, f i ] = h i, [e i, f j ] = 0 (if i j), [h i, e j ] = a ij e j, [h i, f j ] = a ji f j.. H := h 1,..., h n C Cartan subalgebra of L. This determines L up to isomorphism.

8 Vector space basis of L Consider R n with standard basis {α 1,..., α n }. Define s i : R n R n, α j α j a ij α i. Then s i = id n and W = s 1,..., s n GL n (R) is a finite reflection group. Φ := {w(α i ) w W, 1 i n} is the abstract root system associated with A. Corresponding Lie algebra L has basis {h 1,..., h n } {e α α Φ}. Root strings (Killing): Let α, β Φ, α ±β. Let p = p α,β 0 and q = q α,β 0 be maximal such that β qα,..., β α, β, β + α,..., β + pα Φ. C. CHEVALLEY (1955): e α can be chosen such that e αi = ±e i, e αi = ±f i and [e α, e β ] = ±(q α,β + 1)e α+β whenever α, β, α + β Φ (and with this normalization, the e α are unique up to sign).

9 An aside: root systems and lattices Let Γ := all Z-linear combinations of roots α Φ. Then Γ is a lattice in R n. Conversely, let Γ R n be a lattice.. Γ is integral if (x, y) Z for x, y Γ; then Γ is even if (x, x) Z for x Γ.. Dual lattice Γ := {x R n (x, y) Z for all y Γ}; if Γ integral, then Γ Γ.. Γ is unimodular if Γ is integral and Γ = Γ. Up to isomorphism, there is only one even unimodular lattice in R 8. (History goes back to around 1870, before Lie and Killing.) This is given by Γ 8 := { (x i ) R 8 x i Z, x i x j Z, i x i Z }. The set R := {x Γ 8 (x, x) = } contains precisely 40 vectors; these form the root system associated with the Dynkin diagram E 8. (A further famous example in dimension 4: the Leech lattice Conway s finite simple groups. See, e.g., J-P. Serre, A Course in Arithmetic, or W. Ebeling, Lattices and Codes, Springer-Verlag.)

10 C. CHEVALLEY ( 1955/1960): Algebraic analogues of Lie groups. Simple Lie algebra L + field k group G L (k). (Mimic exponential exp(e α) over arbitrary field k; then G L (k) = exp(te α) α Φ, t k.). k algebraically closed: G L (k) simple algebraic group over k, SL n (k), Sp n (k), SO n (k), E 8 (k),.... k finite new families (at the time) of finite simple groups, e.g., E 8 (F q ) = q 48 + lower powers of q (where q is a prime power). Textbook references:. BOURBAKI, Groupes et algèbres de Lie, 1968/1975;. STEINBERG, Lectures on Chevalley groups, 1967/68 (now available from AMS);. HUMPHREYS, Introduction to Lie algebras and representation theory, 197;. CARTER, Simple groups of Lie type, 197;. ERDMANN WILDON, Introduction to Lie algebras, 006.

11 So what are the news about Lie algebras? GEORGE LUSZTIG, arxiv: (3 pages): The Lie group of type E 8 can be obtained from the graph E 8 by a method of Chevalley (1955), simplified using theory of canonical basis (1990). 3 more papers: one by Lusztig, two by myself.

12 Why look for a simplification?. Construction of L is an issue, especially for exceptional types.. Chevalley s construction of G L (k) relies on choice of e α s. J. TITS 1966: Start with vector space M of correct dimension, fix basis {h 1,..., h n } {e α α Φ} and try to define Lie bracket for basis elements. Relies on systematic study of sign choices in Chevalley s normalisaztion of e α. J-P. SERRE 1966: Start with free Lie algebra generators {e i, f i 1 i n} and add defining relations (nowadays called Serre relations ) to obtain a presentation. Very elegant, does not resolve issue of making choices for the e α s. C.-M. RINGEL 1990: Fix orientation of Dynkin diagram and use the representation theory of quivers and Hall polynomials. Then e α s correspond to well-defined indecomposable objects in certain category of modules. Pre-cursor of canonical bases, still relies on n 1 choices of orientations.

13 The simplified construction Recall root strings : Let α, β Φ, α ±β. Then p α,β := max{i 0 β + iα Φ} and q α,β := max{j 0 β jα Φ}. Let M be a vector space over C of the correct dimension, with a given basis {u 1,..., u n } {v α α Φ}. Define linear maps e i, f i : M M by e i (u j ) := a ji v αi, e i (v α ) := f i (u j ) := a ji v αi, f i (v α ) := (Note: Matrices of e i, f i have entries in Z 0 in fact, in {0, 1,, 3}.) Theorem. LUSZTIG (1990) + G. (Proc. Amer. Soc., 017) (q αi,α + 1)v α+αi if α + α i Φ, u i if α = α i, 0 otherwise; (p αi,α + 1)v α αi if α α i Φ, u i if α = α i, 0 otherwise. Consider the Lie algebra gl(m) and let L := e i, f i 1 i n Lie gl(m). Then L is a simple Lie algebra corresponding to the given Dynkin diagram.

14 Example: Type G with Cartan matrix A = ( ) R with standard basis {α 1, α }. Matrices s 1, s GL (R) given by ( ) ( ) s 1 =, s 0 1 = Applying s 1, s repeatedly to α 1 = (1, 0), α = (0, 1), we obtain Φ = {±(1, 0), ±(0, 1), ±(1, 1), ±(1, ), ±(1, 3), ±(, 3)}.. Simple Lie algebra L = e 1, e, f 1, f Lie gl 14 (C). For example: e 1 = e 1, e f 1, f upper triangular lower triangular (A dot stands for 0)

15 Resolving sign issue in Chevalley s basis We can transport the basis {u 1,..., u n } {v α α Φ} from M to L and obtain: Corollary.. There is a basis { h 1,..., h n } {e α α Φ} of L such that: [e i, h j ] = a ji e αi, [e i, e α ] = [f i, h j ] = a ji e αi, [f i, e α ] =. This basis of L is unique up to a global constant. (q αi,α + 1)e α+αi if α + α i Φ, h i if α = α i, 0 otherwise; (p αi,α + 1)e α αi if α α i Φ, h i if α = α i, 0 otherwise.. In particular, the structure constants in the equations [e α, e β ] = N αβ e α+β (for α, β, α + β Φ) are uniquely determined up to a global constant.

16 Where do the formulae for the action of e i, f i come from?. 1985: Introduction of quantised enveloping algebras of Lie algebras (Drinfeld, Jimbo), as Hopf algebra deformations depending on a parameter v. Origins in mathematical physics, quantum integrable systems,.... Early 1990 s: Discovery of canonical bases (Lusztig) and crystal bases (Kashiwara). Specialisation v 1 gives rise to canonical bases in irreducible representations of ordinary simple Lie algebras. LUSZTIG ( J. Comb. Algebra 017). Consider L gl(m), as above. This is an irreducible representation and {u 1,..., u n } {v α α Φ} is the canonical basis of M in the above sense. ( Explains why formulae are natural.) Explicit/simple construction of L and, hence, also of G L (k), useful for:. algorithmic problems: nilpotent orbits, matrix group recognition project,...;. teaching courses on Lie algebras ( existence theorem ).

17 My motivation for studying Lie algebras and Chevalley groups Group theory = study of symmetries Continuous Lie groups Discrete finite groups. Atoms of symmetry: finite simple groups.. A highlight of 0th century mathematics: Classification. (first announced 1981 completed 004: Aschbacher, Smith 1000 pages of proof). nd generation proof: D. Gorenstein, R. Lyons, R. Solomon, Math. Survey and Monographs, Amer. Math. Soc., 1994?? (currently 6 volumes).

18 q n(n+1)/ n (q i+1 1) (n+1,q 1) i= q 36 (q 1 1)(q 9 1)(q 8 1) (q 6 1)(q 5 1)(q 1) (3, q 1) q 63 9 (q i 1) (, q 1) i =, q 10 (q 30 1)(q 4 1) (q 0 1)(q 18 1)(q 14 1) (q 1 1)(q 8 1)(q 1) q 4 (q 1 1)(q 8 1) (q 6 1)(q 1) q 6 (q 6 1)(q 1) q 1 (q 8 + q 4 + 1) (q 6 1)(q 1) q 36 (q 1 1)(q 9 + 1)(q 8 1) (q 6 1)(q 5 + 1)(q 1) (3, q + 1) q (q + 1)(q 1) q 1 (q 6 + 1)(q 4 1) (q 3 + 1)(q 1) q 3 (q 3 + 1)(q 1) q n n (q i 1) (, q 1) i=1 q n n (q i 1) (, q 1) i=1 q n(n 1) (q n n 1 1) (q i 1) (4,q n 1) q n(n 1) (q n n 1 +1) (q i 1) (4,q n +1) q n(n+1)/ n+1 (q i ( 1) i ) (n+1,q+1) The Periodic Table Of Finite Simple Groups 0, C1, Z1 1 Dynkin Diagrams of Simple Lie Algebras 1 An 1 3 n 1 F C A1(4), A1(5) A5 A() A1(7) Bn 1 3 n Dn 3 4 n 4 G 1 A3(4) B(3) C3(3) D4() D 4 ( ) G() A (9) C3 60 A1(9), B() A6 168 G(3) A1(8) Cn 1 3 n E6,7, B(4) C3(5) D4(3) D 4 (3 ) A (16) 3 C Tits A7 A1(11) E6() E7() E8() F4() G(3) 3D 4 ( 3 ) E 6 ( ) B ( 3 ) F 4 () G (3 3 ) B3() C4(3) D5() D 5 ( ) A (5) C A3() A8 A1(13) E6(3) E7(3) E8(3) F4(3) G(4) 3D 4 (3 3 ) E 6 (3 ) B ( 5 ) F 4 ( 3 ) G (3 5 ) B(5) C3(7) D4(5) D 4 (4 ) A 3 (9) C A9 A1(17) E6(4) E7(4) E8(4) F4(4) G(5) 3D 4 (4 3 ) E 6 (4 ) B ( 7 ) F 4 ( 5 ) G (3 7 ) B(7) C3(9) D5(3) D 4 (5 ) A (64) C PSLn+1(q), Ln+1(q) On+1(q), Ωn+1(q) PSpn(q) O + n (q) O n (q) PSUn+1(q) Zp An An(q) E6(q) E7(q) E8(q) F4(q) G(q) 3D 4 (q 3 ) E 6 (q ) B( n+1 ) F 4( n+1 ) G(3 n+1 ) Bn(q) Cn(q) Dn(q) D n(q ) A n(q ) Cp n! i=1 i=1 i=1 i= p Alternating Groups Classical Chevalley Groups Alternates Chevalley Groups Classical Steinberg Groups Symbol Steinberg Groups Suzuki Groups Order Ree Groups and Tits Group Sporadic Groups Cyclic Groups For sporadic groups and families, alternate names in the upper left are other names by which they The Tits group may be known. For specific non-sporadic groups F4 () is not a group of Lie type, but is the (index ) commutator subgroup of F4 (). these are used to indicate isomorphims. All such isomorphisms appear on the table except the family Bn( m ) = Cn( m ). It is usually given honorary Lie type status. The groups starting on the second row are the classical groups. The sporadic suzuki group is unrelated with the following exceptions: Finite simple groups are determined by their order to the families of Suzuki groups. Bn(q) and Cn(q) for q odd, n > ; A8 = A3() and A(4) of order Copyright c 01 Ivan Andrus. M Sz Suz M O NS, O S O N M M M Co3 Co Co J(1), J(11) J F5, D HN H J J LyS Ly H JM J F3, E Th J4 HS M() M(3) Fi Fi McL F3+, M(4) Fi F7, HHM, HT H He F B Ru F1, M1 M