Lie Objects Matthieu Deneufchˆ atel Laboratoire d’Informatique de Paris Nord, Universit´ e Paris 13 S´ eminaire CALIN, 28 Juin 2011
Outline Lie and Enveloping Algebras 1 Example 2 Two Theorems 3 CQMM Theorem Poincar´ e-Birkhoff-Witt Theorem Duality 4 Lie exponential 5 Group of characters of an algebra 6 M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 2 / 24
Lie and Enveloping Algebras Outline Lie and Enveloping Algebras 1 Example 2 Two Theorems 3 CQMM Theorem Poincar´ e-Birkhoff-Witt Theorem Duality 4 Lie exponential 5 Group of characters of an algebra 6 M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 3 / 24
Lie and Enveloping Algebras Lie Algebra k a field of characteristic zero. Definition A Lie algebra G is a vector space endowed with a bilinear operation [ · , · ] : G × G → G satisfying the following relations, ∀ a , b , c ∈ G : [ a , a ] = 0; [ a , [ b , c ]] + [ b , [ c , a ]] + [ c , [ a , b ]] = 0 . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 4 / 24
Lie and Enveloping Algebras Lie Algebra k a field of characteristic zero. Definition A Lie algebra G is a vector space endowed with a bilinear operation [ · , · ] : G × G → G satisfying the following relations, ∀ a , b , c ∈ G : [ a , a ] = 0; [ a , [ b , c ]] + [ b , [ c , a ]] + [ c , [ a , b ]] = 0 . Each associative algebra A has a natural Lie algebra structure A L with the bracket defined by : [ a , b ] = ab − ba . In terms of categories : f : k − UAA − → k − Lie algebra . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 4 / 24
Lie and Enveloping Algebras Enveloping algebra Let G be a Lie algebra. It is possible to associate to G an associative algebra called enveloping algebra of G denoted by U ( G ). Universal problem. There exists a unital associative algebra U ( G ) and a Lie algebra homomorphism φ 0 : G → U ( G ) L such that, for any associative algebra A , any Lie algebra homomorphism φ : G → A L , there is a unique algebra homomorphism f : U ( G ) → A making the following diagram commute: φ 0 U ( G ) G φ f A L M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 5 / 24
Lie and Enveloping Algebras Enveloping algebra g : k − Lie algebra − → k − UAA . g is the left adjoint of f . Universal problem. There exists a unital associative algebra U ( G ) and a Lie algebra homomorphism φ 0 : G → U ( G ) L such that, for any associative algebra A , any Lie algebra homomorphism φ : G → A L , there is a unique algebra homomorphism f : U ( G ) → A making the following diagram commute: φ 0 G U ( G ) φ f A L M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 5 / 24
Example Outline Lie and Enveloping Algebras 1 Example 2 Two Theorems 3 CQMM Theorem Poincar´ e-Birkhoff-Witt Theorem Duality 4 Lie exponential 5 Group of characters of an algebra 6 M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 6 / 24
Example Example : Free Lie Algebra X an alphabet. Theorem There exists a Lie algebra Lie k � X � over k unique up to isomorphism and freely generated by X . It is called Free Lie Algebra. Construction : Lie Monomials : ∀ x ∈ X , x is a Lie monomial ; if u and v are Lie monomials, then so is [ u , v ] = uv − vu (concatenation product). M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 7 / 24
Example Example : Free Lie Algebra X an alphabet. Theorem There exists a Lie algebra Lie k � X � over k unique up to isomorphism and freely generated by X . It is called Free Lie Algebra. Construction : Lie Monomials : ∀ x ∈ X , x is a Lie monomial ; if u and v are Lie monomials, then so is [ u , v ] = uv − vu (concatenation product). Lie Polynomials and Series : respectively finite and infinite k -linear combinations of Lie monomials. → Lie k � X � . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 7 / 24
Example Link with the free associative algebra k � X � For P , Q ∈ k � X � , their Lie bracket is [ P , Q ] = PQ − QP . The smallest submodule of k � X � closed under this bracket and containing X is the free Lie algebra Lie k � X � . k � X � is the enveloping algebra of Lie k � X � : k � X � = U ( Lie k � X � ) . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 8 / 24
Example Link with the free associative algebra k � X � For P , Q ∈ k � X � , their Lie bracket is [ P , Q ] = PQ − QP . The smallest submodule of k � X � closed under this bracket and containing X is the free Lie algebra Lie k � X � . k � X � is the enveloping algebra of Lie k � X � : k � X � = U ( Lie k � X � ) . Coproduct ∆ on k � X � (homomorphism of k -algebra defined on the letters): ∆( x ) = x ⊗ 1 + 1 ⊗ x . Lie polynomials (Friedrich) The following conditions are equivalent : P ∈ k � X � is a Lie polynomial ; ∆( P ) = P ⊗ 1 + 1 ⊗ P ( P is primitive). M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 8 / 24
Two Theorems Outline Lie and Enveloping Algebras 1 Example 2 Two Theorems 3 CQMM Theorem Poincar´ e-Birkhoff-Witt Theorem Duality 4 Lie exponential 5 Group of characters of an algebra 6 M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 9 / 24
Two Theorems CQMM Theorem Example of k � X � ( k � X � , conc , 1 X ∗ , ∆ , ǫ ) is a cocommutative graded bialgebra : � k � X � = k = n � X � , n ≥ 0 � where P ∈ k = n � X � means that P = � P | w � w . | w | = n M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 10 / 24
Two Theorems CQMM Theorem Example of k � X � ( k � X � , conc , 1 X ∗ , ∆ , ǫ ) is a cocommutative graded bialgebra : � k � X � = k = n � X � , n ≥ 0 � where P ∈ k = n � X � means that P = � P | w � w . | w | = n k � X � = U ( Lie k � X � ) . Lie polynomials are primitive elements : ∀ P ∈ Lie k ( X ) , ∆( P ) = P ⊗ 1 + 1 ⊗ P . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 10 / 24
Two Theorems CQMM Theorem CQMM Theorem Let B be a bialgebra. It is graded if : B = � n ≥ 0 B n ; µ ( B p , B q ) ⊂ B p + q , ∀ p , q ∈ N ; � ∆( B n ) ⊂ B p ⊗ B q , ∀ n ∈ N ; p + q = n connected if B 0 = k 1 B . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 11 / 24
Two Theorems CQMM Theorem CQMM Theorem Let B be a bialgebra. It is graded if : B = � n ≥ 0 B n ; µ ( B p , B q ) ⊂ B p + q , ∀ p , q ∈ N ; � ∆( B n ) ⊂ B p ⊗ B q , ∀ n ∈ N ; p + q = n connected if B 0 = k 1 B . Let B be a cocommutative graded connected bialgebra. Cartier-Quillen-Milnor-Moore Theorem B is the enveloping algebra of its primitive elements. M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 11 / 24
Two Theorems Poincar´ e-Birkhoff-Witt Theorem Theorem ( X , < ), Lyn ( X ). Lyn ( X ) is a (totally ordered) basis of Lie k � X � . Poincar´ e-Birkhoff-Witt Let ( g i ) i ∈ I be a totally ordered basis of a Lie algebra G . Then the “decreasing” products g α = g α 1 i 1 . . . g α p i p , i 1 > · · · > i p , α i ∈ N , form a basis of U ( G ). Thus, Lyn ( X ) induces a basis of k � X � = U ( Lie k ( X )) in the following way: M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 12 / 24
Two Theorems Poincar´ e-Birkhoff-Witt Theorem PBW Basis For l ∈ Lyn ( X ), let us define ( P l ) l ∈ Lyn ( X ) by : � l if | l | = 1; P l = [ l 1 , l 2 ] otherwise, with l = l 1 l 2 the standard factorization of l . If w = l α 1 i 1 . . . l α k with l i 1 > · · · > l i k , i k l i 1 . . . P α k P w = P α 1 l ik . P w is homogeneous for the multidegree (finely homogeneous). � P w = w + ∗ v where the star denotes coefficients in Z . v > w ∈ X ∗ ( P l ) l ∈ Lyn ( X ) is a basis of Lie k ( X ) and ( P w ) w ∈ X ∗ is a basis of k � X � . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 13 / 24
Duality Outline Lie and Enveloping Algebras 1 Example 2 Two Theorems 3 CQMM Theorem Poincar´ e-Birkhoff-Witt Theorem Duality 4 Lie exponential 5 Group of characters of an algebra 6 M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 14 / 24
Duality Dual basis Duality bracket : � u | v � = δ u , v ⇒ k �� X �� ∼ ( k � X � ) ∗ : � � S | P � = � S | w �� P | w � . w ∈ X ∗ M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 15 / 24
Duality Dual basis Duality bracket : � u | v � = δ u , v w if | w | = 1; xS u if w = xu and w is a Lyndon word; S w = α 1 α k S . . . S l i 1 l ik i 1 . . . l α k otherwise, if w = l α 1 i k α 1 ! . . . α k ! k = S S k − 1 for k > 0 and S 0 = 1. with S M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 15 / 24
Duality Dual basis Duality bracket : � u | v � = δ u , v w if | w | = 1; xS u if w = xu and w is a Lyndon word; S w = α 1 α k S . . . S l i 1 l ik i 1 . . . l α k otherwise, if w = l α 1 i k α 1 ! . . . α k ! k = S S k − 1 for k > 0 and S 0 = 1. with S Theorem � S u | P v � = δ u , v . M. Deneufchˆ atel (LIPN - P13) Lie Objects 28/06/2011 15 / 24
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