Nonassociative Lie Theory

Transcription

1 Ivan P. Shestakov The International Conference on Group Theory in Honor of the 70th Birthday of Professor Victor D. Mazurov Novosibirsk, July 16-20, 2013 Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences

2 Lie Algebras Lie Groups Associative Algebras Hopf Algebras Formal Groups Sabinin Algebras Analytic Loops Nonassociative Algebras Nonassociative Hopf Algebras Formal Loops

3 Local Loops and Tangent Algebras. Definition Quasigroup (nonassociative group) is a set with a binary operation Q, where equations a x = b, y a = b have unique solutions in Q for any a, b Q. Loop is a quasigroup with the unit element e. The solutions of the equations above define the operations of left and right division a \ b = x and b/a = y. In terms of these operations, we can give Equivalent definition: Loop is an algebraic system M,, \, /, e such that a \ (a b) = b = a (a \ b) (a b)/b = a = (a/b) b e a = a = a e

4 Local loops. Let M be smooth finite-dimensional manifold. A local multiplication on open U M is a smooth map F : U U M. If there exists e U with the property that F e U = Id(U) = F U e, the local multiplication F is called unital, or a local loop. The point e is referred to as the unit. Notation: F(x, y), x y, xy.

5 Left and right divisions For any local loop there exist two local multiplications V V M with V U, denoted by x/y and y \ x. As above, they are defined by (x/y) y = x and y (y \ x) = x. and are called the right and the left divison, respectively. The existence of both divisions follows from the fact that the right and left multiplication maps R y = F U y : U M, L y = F y U : U M are close to the inclusion map U M when y is close to e. In particular, if y is sufficiently close to e, both maps R y and L y are one-to-one and their images contain a neighbourhood of e. We take V to be the largest neighbourhood on which both divisions are defined.

6 Example 1: Invertible elements in algebras. Call an element a of a unital algebra invertible if both equations ax = 1 and xa = 1 have a unique solution. Let A be a finite-dimensional unital algebra over R. Then the invertible elements of A form a local loop. This local loop is not necessarily a loop. Consider, for instance, the generalized Cayley-Dickson algebras C n on R 2n. When n > 3 there exist pairs of invertible elements in C n whose product is zero.

7 Example 2: Homogeneous spaces. Let M be a homogeneous space for a Lie group G and U M a neighbourhood of a point e M. Consider the mapping p : G M, g g(e). Assume that we are given a section of p over U, that is, a smooth map i : U G such that i(e) is the unit in G and p i = Id U. Then M is a local loop, with the multiplication U U M defined as (x, y) p(i(x)i(y)). When p is actually a homomorphism of Lie groups, that is, when the stabilizer G e of the element e is a normal subgroup in G, this local loop structure is the same thing as the product on M restricted to U U. There are many important examples of homogeneous spaces, among them spheres, hyperbolic spaces and Grassmannians.

8 Example 3: Analytic local loops. Consider an n-tuple of power series F(x, y) = (F 1 (x, y),..., F n (x, y)) where x, y R n, and assume that all of them converge in some neighbourhood of the origin in R 2n. Then the map (x, y) F(x, y) defines a local loop on R n, with the origin as the unit, if and only if F(0, y) = y and F (x, 0) = x for all x, y R n. A local loop on an analytic manifold whose multiplication can be written in this form in some coordinate chart is called analytic.

9 Tangent algebras A.I.Malcev (1955): Analytic loop L the tangent algebra T (L), [x, y] = [y, x] Moufang loop L, a(b(ac)) = ((ab)a)c Alternative algebra A, a(bb) = (ab)b, a(ab) = (aa)b Alternative algebra A Malcev algebra T (L), [a, a] = 0, [J(a, b, c), a] = J(a, b, [a, c]) Malcev algebra A( ), [a, b] = ab ba Moufang loop U(A)

10 Malcev algebras, alternative algebras, and Moufang loops E.N.Kuzmin (1971): Malcev algebras Moufang loops I.P.Shestakov (2004): Moufang loops Alternative algebras Malcev Problem: Malcev algebras??? Alternative algebras

11 Akivis algebras M.Akivis (1976): Analytic loop L Akivis algebra Ak(L) Akivis algebra (A, +, [, ],,, ): [x, x] = 0 ( 1) σ x σ(1), x σ(2), x σ(3) = J(x 1, x 2, x 3 ) σ S 3 Algebra B Ak B = B, [x, y], (x, y, z), where [x, y] = xy yx, (x, y, z) = xy)z x(yz). I.Shestakov (1999): Every Akivis algebra A can be embedded into the algebra Ak B for a suitable algebra B.

12 Local loops and Sabinin algebras L 1, L 2 local analytic loops. L 1 Ak(L 1 ) = Ak(L 2 ) L 2 L 1 = L2 L.Sabinin, P.Mikheev (1987): Local loops analytic Hyperalgebras (Sabinin algebras) L Sab(L) L

13 Primitive elements of bialgebras Bialgebra B = B, +, m, : B, +, m an algebra: m : B B B B, +, a coalgebra: : B B B is a homomorphism of algebras. Prim (B, ) = {w B (w) = w w}. 1. B = F X, free associative algebra (char =0). (x i ) = x i x i. Prim (F X, ) = Lie X.

14 Primitive elements of bialgebras 1. B = F{X}, free nonassociative algebra (char =0). (x i ) = x i x i. K.Strambach: Prim (F{X}, ) = Ak X? I.Sh.+ U.Umirbaev (2001): p = (x 2, x, x) x(x, x, x) (x, x, x)x Prim (F{X}, ), p Ak X Problem: To describe Prim (F {X}, ).

15 Primitive elements of bialgebras I.Sh.+ U.Umirbaev: Prim (F{X}, ) is generated (starting with X) by [x, y], (x, y, z) and p(x 1,..., x n ; y 1,..., y m ; z). Let u = x 1 x 2 x n, v = y 1 y 2 y m ; denote p(x 1,..., x n ; y 1,..., y m ; z) as p(u, v, z). Then the equality (u, v, z) = (u),(v) u (1) v (1) p(u (2), v (2), z) defines the primitive elements p(u, v, z) inductively. p(x 1, y 1, z) = (x, y, z) p(x 1 x 2, y, z) = (x 1 x 2, y, z) x 1 (x 2, y, z) x 2 (x 1, y, z).

16 Primitive elements of bialgebras Theorem I.Sh.+ U.U.: Let C,, δ be a unital bialgebra over a field F of characteristic 0. Then the space Prim (C, δ) is closed relatively the operations p(u, v, z). If C is generated as an algebra by Prim (C, δ) then C has a PBW-base over Prim (C, δ).

17 Sabinin algebras V a vector space, T (V ) the tensor algebra over V, : T (V ) T (V ) T (V ), v 1 v + v 1, v V. ;, : T (V ) V V V, w y z w; y, z. w; y, y = 0, w u v w ; y, z w v u w ; y, z + (1) w (2) ; u, v w (w) w ; y, z = 0, x; y, z + x,y,z( w w (1) ; w (2) ; y, z, x ) = 0. (w) x, y, z, u, v V ; w, w T (V ).

18 Sabinin algebras Examples: - Lie algebras: 1; a, b = [a, b], x; a, b = 0, x VT (V ). - Lie triple systems: 1; a, b = 0, u; a, b = [a, b, u], u V ; x; a, b = 0, x V i, i > 1. - Malcev algebras: 1; a, b = [a, b],...

19 Shestakov-Umirbaev functor I.Sh.+ U.U.: Let A be an arbitrary algebra. Define 1; a, b = [a, b] a; b, c = (a, b, c) (a, c, b) a 1,..., a n ; b, c = p(a 1 a n ; b, c) p(a 1 a n ; c, b), where a, b, c, a 1,..., a n A. Then A ( ) = A, is a Sabinin algebra. If A is a bialgebra then Prim A is a subalgebra of the Sabinin algebra A ( ).

20 Universal enveloping algebra Let : Alg Sab, A A ( ), be the functor from the category Alg of algebras to the category Sab of Sabinin algebras, then admits a left adjoined functor U : Sab Alg such that for every V Sab and A Alg there is a bijection Hom Sab (V, A ( ) ) = Hom Alg (U(V ), A). Moreover, there exists a canonical Sabinin algebra homomorphism i : V U(V ) ( ) = U(V ) such that for any algebra A and a Sabinin algebra homomorphism ϕ : V A ( ) there exists a unique homomorphism ϕ : U(V ) A satisfying the equality ϕ i = ϕ. The algebra U(V ) is called the universal enveloping algebra of a Sabinin algebra V.

21 Universal enveloping algebra The algebra U(V ) has a natural structure of a bialgebra, the comultiplication : U(V ) U(V ) U(V ) is defined by the condition τ(v) 1 τ(v) + τ(v) 1, v V. Moreover, τ(v ) = Prim (U(V ), ). Problem: Is τ : V U(V ) a monomorphism? J.M.Pérez Izquierdo +I.Sh.: - Yes, for Malcev algebras. J.M.Pérez Izquierdo: - Yes, for Bol algebras. J.M.Pérez Izquierdo: - Yes, in general case.

22 Nonassociative Hopf algebras H,,, ɛ, /, \ is an H-bialgebra if 1) H,,, ɛ is a bialgebra with a counit ɛ, 2) \ : H H H, / : H H H satisfy (a) a (1) \ (a (2) b) = ɛ(a)b = (a) a (1) (a (2) \ b) (b) (a b (1))/b (2) = ɛ(b)a = (b) (a/b (1)) b (2)

23 Nonassociative Milnor-Moore Theorem Examples: 1) Loop algebra FL, (l) = l l. 2) U(V ), V a Sabinin algebra. J.M.Pérez Izquierdo: Nonassociative Milnor-Moore Theorem. If H is an H- bialgebra over a field of characteristic 0 generated by Prim H then H = U(Prim H).

24 Free algebras as H-algebras A Hopf algebra is an H-bialgebra: a \ b = S(a)b, a/b = as(b). H = F{X}, a free nonassociative algebra: (x) = 1 x + x 1, x X ɛ(1) = 1, ɛ(x) = 0, x X Let u F{X}, x, y X. 1 \ u = u/1 = u 1 \ x u + x \ 1 u = ɛ(x)u = 0 x \ u = xu u x/1 + u 1/x = ɛ(x)u = 0 u/x = ux

25 Free algebras as H-algebras (xy) = xy 1 + x y + y x + 1 xy 0 = uɛ(xy) = u (xy)/1 + u x/y + u y/x + u 1/(xy) u/(xy) = u(xy) + (ux)y + (uy)x F{X},,, ɛ, \, / is a cocommutative and coassociative H-bialgebra. Moreover, F{X} = U(Sab X ) Sab X = Prim (F{X}, )

26 Formal loops A formal map from a vector space V over a field k to a vector space W is a linear map k[v ] W. A formal multiplication on a vector space V is a formal map from V V to V. Given that k[v V ] is canonically isomorphic to k[v ] k[v ], this is the same a linear map k[v ] k[v ] V. A formal multiplication F is a formal loop if F 1 k[v ] = π V = F k[v ] 1, where π V : k[v ] V be the projection of a polynomial onto its linear part.

27 Formal loops, bialgebras of distributions, and Sabinin algebras G local analytic loop Sabinin algebra g G formal loop Sabinin algebra g. J.Mostovoy + J.M.Pérez Izquierdo (2009): k[g], the bialgebra Formal loop G G : k[v ] k[v ] V of distributions on G supported at 1 Sabinin algebra g g = V, ;, U(g), the universal enveloping algebra k[g] = U(g) Prim (k[g]) = Prim (U(g)) = g.

28 Linear formal loops Example: A a finite-dimensional algebra over k, x y = x + y + xy gives a formal loop structure G(A). A formal loop G is called linear if there existe a finite-dimensional algebra A such that G G(A). G is linear g satisfies Ado s theorem Formal Moufang loops are linear. There exist non-linear formal loops.

29 Moufang-Hopf algebras M, a Malcev algebra; U(M) is not alternative. L, a Moufang loop; k[l] is not alternative. U(M) and k[l] are bialgebras. : U(M) U(M) U(M), (m) = 1 m + m 1 : k[l] k[l] k[l], The bialgebra k[l] evidently satisfies (g) = g g [(zx (1) )y]x (2) = z(x (1) y x (2) ) ( ) (x) (x) A linearization of the Moufang identity. The bialgebra U(M) satisfies (*) as well!!!

30 Moufang-Hopf algebras A cocommutative and coassociative H-bialgebra satisfying ( ) is called a Moufang-Hopf algebra. Every Moufang-Hopf algebra has an antipode mapping S with its standard properties. Then a/b = as(b), b \ a = S(b)a.

31 Groups with triality G, a group; ρ, σ Aut G, σ 2 = ρ 3 = 1, σρσ = ρ 2. G is a group with triality (relative to σ, ρ) if it satisfies the identity (g 1 g σ )(g 1 g σ ) ρ (g 1 g σ ) ρ2 = 1. Doro, Glauberman, Grishkov, Zavarnitsin: M(G) = {g 1 g σ g G} G with triality is a Moufang loop with m n = m ρ nm ρ2 Groups witht triality Moufang loops

32 Lie algebras with triality A Lie algebra L is called a Lie algebra with triality if Aut L S 3 and every x L satisfies the identity σ S 3 ( 1) sgn (σ) σ(x) = 0. P.Mikheev, A.Grishkov: Lie algebras witht triality Malcev algebras

33 Hopf algebras with triality G.Benkart, S.Madariaga, J.Pérez Izquierdo: Cocommutative Hopf algebras witht triality Moufang-Hopf algebras H, a cocommutative Hopf algebra; ρ, σ Aut H, σ 2 = ρ 3 = 1, σρσ = ρ 2. H is with triality if it satisfies the identity P(u (1) )(P(u (2) )) ρ (P(u (3) )) ρ2 = ɛ(u)1. (u) where P(u) = (u) σ(u (1))S(u (2) ).

34 Hopf algebras with triality If H is a cocommutative Hopf algebra witht triality then the set MH(H) = {P(u) u H} is a Moufang-Hopf algebra with comultiplication inheritid from H and the multiplication u v = (u) ρ 2 (S(u (1) ))vρ(s(u (2) ))