Nonassociative Lie Theory Ivan P . Shestakov The International - - PowerPoint PPT Presentation
Nonassociative Lie Theory 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
Nonassociative Lie Theory
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
Ivan Shestakov Nonassociative Lie Theory
Analytic Loops Nonassociative Algebras Nonassociative Hopf Algebras Formal Loops
Local Loops and Tangent Algebras.
Definition Quasigroup (nonassociative group) is a set with a binary
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
Ivan Shestakov Nonassociative Lie Theory
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.
Ivan Shestakov Nonassociative Lie Theory
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 Ry = F|U×y : U → M, Ly = 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 Ry and Ly 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.
Ivan Shestakov Nonassociative Lie Theory
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 Cn on R2n. When n > 3 there exist pairs of invertible elements in Cn whose product is zero.
Ivan Shestakov Nonassociative Lie Theory
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
unit in G and p ◦ i = IdU. 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 Ge 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.
Ivan Shestakov Nonassociative Lie Theory
Example 3: Analytic local loops.
Consider an n-tuple of power series F(x, y) = (F1(x, y), . . . , Fn(x, y)) where x, y ∈ Rn, and assume that all of them converge in some neighbourhood of the origin in R2n. Then the map (x, y) → F(x, y) defines a local loop on Rn, with the origin as the unit, if and only if F(0, y) = y and F(x, 0) = x for all x, y ∈ Rn. A local loop on an analytic manifold whose multiplication can be written in this form in some coordinate chart is called analytic.
Ivan Shestakov Nonassociative Lie Theory
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 ⇒ Malcev algebra T(L), [a, a] = 0, [J(a, b, c), a] = J(a, b, [a, c]) Alternative algebra A, a(bb) = (ab)b, a(ab) = (aa)b ⇒ Malcev algebra A(−), [a, b] = ab − ba Alternative algebra A ⇒ Moufang loop U(A)
Ivan Shestakov Nonassociative Lie Theory
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
Ivan Shestakov Nonassociative Lie Theory
Akivis algebras
M.Akivis (1976): Analytic loop L ⇒ Akivis algebra Ak(L) Akivis algebra (A, +, [·, ·], ·, ·, ·): [x, x] =
(−1)σxσ(1), xσ(2), xσ(3) = J(x1, x2, x3) 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.
Ivan Shestakov Nonassociative Lie Theory
Local loops and Sabinin algebras
L1, L2 local analytic loops. L1 → Ak(L1) ∼ = Ak(L2) ← L2 L1 ∼ = L2 L.Sabinin, P.Mikheev (1987): Local analytic loops ⇔ Hyperalgebras (Sabinin algebras) L → Sab(L) → L
Ivan Shestakov Nonassociative Lie Theory
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 ⊗ 1 + 1 ⊗ w}.
∆(xi) = xi ⊗ 1 + 1 ⊗ xi. Prim (FX, ∆) = LieX.
Ivan Shestakov Nonassociative Lie Theory
Primitive elements of bialgebras
∆(xi) = xi ⊗ 1 + 1 ⊗ xi. K.Strambach: Prim (F{X}, ∆) = AkX? I.Sh.+ U.Umirbaev (2001): p = (x2, x, x) − x(x, x, x) − (x, x, x)x ∈ Prim (F{X}, ∆), p ∈ AkX Problem: To describe Prim (F{X}, ∆).
Ivan Shestakov Nonassociative Lie Theory
Primitive elements of bialgebras
I.Sh.+ U.Umirbaev: Prim (F{X}, ∆) is generated (starting with X) by [x, y], (x, y, z) and p(x1, . . . , xn; y1, . . . , ym; z). Let u = x1x2 · · · xn, v = y1y2 · · · ym; denote p(x1, . . . , xn; y1, . . . , ym; z) as p(u, v, z). Then the equality (u, v, z) =
u(1)v(1)p(u(2), v(2), z) defines the primitive elements p(u, v, z) inductively. p(x1, y1, z) = (x, y, z) p(x1x2, y, z) = (x1x2, y, z) − x1(x2, y, z) − x2(x1, y, z).
Ivan Shestakov Nonassociative Lie Theory
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).
Ivan Shestakov Nonassociative Lie Theory
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 +
w(1) ⊗ w(2); u, v ⊗ w′; y, z = 0,
(w ⊗ x; y, z +
w(1); w(2); y, z, x) = 0. x, y, z, u, v ∈ V; w, w′ ∈ T(V).
Ivan Shestakov Nonassociative Lie Theory
Sabinin algebras
Examples:
V; x; a, b = 0, x ∈ V ⊗i, i > 1.
Ivan Shestakov Nonassociative Lie Theory
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) a1, . . . , an; b, c = p(a1 · · · an; b, c) − p(a1 · · · an; c, b), where a, b, c, a1, . . . , an ∈ A. Then A(∼) = A, · · · is a Sabinin algebra. If A is a bialgebra then Prim A is a subalgebra of the Sabinin algebra A(∼).
Ivan Shestakov Nonassociative Lie Theory
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 HomSab(V, A(∼)) ∼ = HomAlg(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.
Ivan Shestakov Nonassociative Lie Theory
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.
Ivan Shestakov Nonassociative Lie Theory
Nonassociative Hopf algebras
H, ·, ∆, ǫ, /, \ is an H-bialgebra if 1) H, ·, ∆, ǫ is a bialgebra with a counit ǫ, 2) \ : H ⊗ H → H, / : H ⊗ H → H satisfy
ǫ(a)b =
(a) a(1) · (a(2) \ b)
ǫ(b)a =
(b)(a/b(1)) · b(2)
Ivan Shestakov Nonassociative Lie Theory
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).
Ivan Shestakov Nonassociative Lie Theory
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 ǫ(1) = 1, ǫ(x) = 0, x ∈ X Let u ∈ F{X}, x, y ∈ X. 1 \ u = u/1 = u 1 \ x · u + x \ 1 · u = ǫ(x)u = 0 ⇒ x \ u = −xu u · x/1 + u · 1/x = ǫ(x)u = 0 ⇒ u/x = −ux
Ivan Shestakov Nonassociative Lie Theory
Free algebras as H-algebras
∆(xy) = xy ⊗ 1 + x ⊗ y + y ⊗ x + 1 ⊗ xy 0 = uǫ(xy) = u · (xy)/1 + u · x/y + u · y/x + u · 1/(xy) u/(xy) = −u(xy) + (ux)y + (uy)x F{X}, ·, ∆, ǫ, \, / is a cocommutative and coassociative H-bialgebra. Moreover, F{X} ∼ = U(SabX) SabX = Prim (F{X}, ∆)
Ivan Shestakov Nonassociative Lie Theory
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 πV : k[V] → V be the projection of a polynomial onto its linear part.
Ivan Shestakov Nonassociative Lie Theory
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): Formal loop G G : k[V] ⊗ k[V] → V ⇒ k[G], the bialgebra
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.
Ivan Shestakov Nonassociative Lie Theory
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.
Ivan Shestakov Nonassociative Lie Theory
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], ∆(g) = g ⊗ g The bialgebra k[L] evidently satisfies
[(zx(1))y]x(2) =
z(x(1)y · x(2)) (∗) “A linearization” of the Moufang identity. The bialgebra U(M) satisfies (*) as well !!!
Ivan Shestakov Nonassociative Lie Theory
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.
Ivan Shestakov Nonassociative Lie Theory
Groups with triality
G, a group; ρ, σ ∈ Aut G, σ2 = ρ3 = 1, σρσ = ρ2. G is a group with triality (relative to σ, ρ) if it satisfies the identity (g−1gσ)(g−1gσ)ρ(g−1gσ)ρ2 = 1. Doro, Glauberman, Grishkov, Zavarnitsin: G with triality ⇒ M(G) = {g−1gσ | g ∈ G} is a Moufang loop with m · n = m−ρnm−ρ2 Groups witht triality ⇔ Moufang loops
Ivan Shestakov Nonassociative Lie Theory
Lie algebras with triality
A Lie algebra L is called a Lie algebra with triality if Aut L ⊃ S3 and every x ∈ L satisfies the identity
(−1)sgn (σ)σ(x) = 0. P .Mikheev, A.Grishkov: Lie algebras witht triality ⇔ Malcev algebras
Ivan Shestakov Nonassociative Lie Theory
Hopf algebras with triality
G.Benkart, S.Madariaga, J.Pérez Izquierdo: Cocommutative Hopf alge- bras witht triality ⇔ Moufang-Hopf alge- bras H, a cocommutative Hopf algebra; ρ, σ ∈ Aut H, σ2 = ρ3 = 1, σρσ = ρ2. H is with triality if it satisfies the identity
P(u(1))(P(u(2)))ρ(P(u(3)))ρ2 = ǫ(u)1. where P(u) =
(u) σ(u(1))S(u(2)).
Ivan Shestakov Nonassociative Lie Theory
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 =
ρ2(S(u(1)))vρ(S(u(2)))
Ivan Shestakov Nonassociative Lie Theory