Some possible exponentiations over the enveloping algebras - PowerPoint PPT Presentation

Possible Exponentia- tions over Some possible exponentiations over the enveloping algebras universal enveloping algebra of sl 2 ( C ) Sonia L’Innocente Sonia L’Innocente Department of Mathematics Institute of Mathematics University of Camerino University of Mons-Hainaut Italy Belgium MODNET Conference in Barcelona Final Conference of the Research Training Network in Model Theory 3-7 November 2008, Barcelona, Spain Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 1 / 28

Possible Exponentia- tions over enveloping algebras Sonia L’Innocente Seminar’s aim We want to illustrate the main results of the work: Some possible exponentiations over the universal enveloping algebra of sl 2 ( C ) (S.L ’I., A. Macintyre, F . Point). where some methods from model theory of modules and some techniques of ultraproducts are applied. Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 2 / 28

Possible Exponentia- Outline tions over enveloping algebras Sonia L’Innocente Our setting Some results in this framework 1 Our Setting Exponential map over Some results in this framework U = U C Exponential maps and ultraproducts 2 Exponentiation over U = U C Exponential maps and ultraproducts Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 3 / 28

Possible Exponentia- Outline tions over enveloping algebras Sonia L’Innocente Our setting Some results in this framework 1 Our Setting Exponential map over Some results in this framework U = U C Exponential maps and ultraproducts 2 Exponentiation over U = U C Exponential maps and ultraproducts Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 4 / 28

Possible Exponentia- tions over enveloping Our setting algebras Let k be an algebraically closed field of characteristic 0. Sonia L’Innocente Consider the simple Lie algebra sl 2 ( k ) of Our setting all 2 × 2 traceless matrices over k Some results in this framework Exponential with the bracket operation [ x , y ] = xy − yx . map over U = U C Recall that a basis of sl 2 ( k ) is Exponential maps and ultraproducts � 0 � 0 � 1 � � � 1 0 0 x = y = h = . 0 0 1 0 0 − 1 So, [ x , y ] = h , [ h , x ] = 2 x , [ h , y ] = − 2 y . We focus on the universal enveloping algebra of sl 2 ( k ) , denoted by U k . Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 5 / 28

Possible Exponentia- tions over enveloping Our setting algebras Let k be an algebraically closed field of characteristic 0. Sonia L’Innocente Consider the simple Lie algebra sl 2 ( k ) of Our setting all 2 × 2 traceless matrices over k Some results in this framework Exponential with the bracket operation [ x , y ] = xy − yx . map over U = U C Recall that a basis of sl 2 ( k ) is Exponential maps and ultraproducts � 0 � 0 � 1 � � � 1 0 0 x = y = h = . 0 0 1 0 0 − 1 So, [ x , y ] = h , [ h , x ] = 2 x , [ h , y ] = − 2 y . We focus on the universal enveloping algebra of sl 2 ( k ) , denoted by U k . Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 5 / 28

Possible Definition Exponentia- tions over A universal enveloping algebra of sl 2 ( k ) over k is enveloping algebras an associative algebra (with a unit) U k with Sonia L’Innocente a (Lie algebra) homomorphism i : sl 2 ( k ) → U k such that Our setting if A is any associative k -algebra with the homomorphism Some results in this framework f : sl 2 ( k ) → A , Exponential map over then there exists a unique homomorphism: U = U C Exponential maps and ultraproducts Θ : U k → A such that the diagram sl 2 ( k ) → U k ↓ ւ A commutes. Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 6 / 28

Possible Definition Exponentia- tions over A universal enveloping algebra of sl 2 ( k ) over k is enveloping algebras an associative algebra (with a unit) U k with Sonia L’Innocente a (Lie algebra) homomorphism i : sl 2 ( k ) → U k such that Our setting if A is any associative k -algebra with the homomorphism Some results in this framework f : sl 2 ( k ) → A , Exponential map over then there exists a unique homomorphism: U = U C Exponential maps and ultraproducts Θ : U k → A such that the diagram sl 2 ( k ) → U k ↓ ւ A commutes. Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 6 / 28

Possible Definition Exponentia- tions over A universal enveloping algebra of sl 2 ( k ) over k is enveloping algebras an associative algebra (with a unit) U k with Sonia L’Innocente a (Lie algebra) homomorphism i : sl 2 ( k ) → U k such that Our setting if A is any associative k -algebra with the homomorphism Some results in this framework f : sl 2 ( k ) → A , Exponential map over then there exists a unique homomorphism: U = U C Exponential maps and ultraproducts Θ : U k → A such that the diagram sl 2 ( k ) → U k ↓ ւ A commutes. Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 6 / 28

Possible Definition Exponentia- tions over A universal enveloping algebra of sl 2 ( k ) over k is enveloping algebras an associative algebra (with a unit) U k with Sonia L’Innocente a (Lie algebra) homomorphism i : sl 2 ( k ) → U k such that Our setting if A is any associative k -algebra with the homomorphism Some results in this framework f : sl 2 ( k ) → A , Exponential map over then there exists a unique homomorphism: U = U C Exponential maps and ultraproducts Θ : U k → A such that the diagram sl 2 ( k ) → U k ↓ ւ A commutes. Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 6 / 28

Possible Exponentia- tions over enveloping algebras Sonia L’Innocente The Poincar´ e-Birkhoff-Witt Theorem Our setting The k -algebra U k has as basis (over k ) Some results in this framework Exponential map over U = U C { x n y l h s : n , l , s ≥ 0 } Exponential maps and ultraproducts where { x , y , h } is the basis of sl 2 ( k ) . Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 7 / 28

We will use these algebraic properties of U k : Possible Exponentia- • U k has a Z -graded k -algebra. Let U κ, m be the tions over enveloping subalgebra of elements of grade m . We have algebras Sonia � L’Innocente U k = U k , m ; m ∈ Z Our setting x m U k , 0 = U k , 0 x m ; Some results in for m > 0 , U k , m = this framework Exponential y | m | U k , 0 = U k , 0 y | m | . for m < 0 , U k , m = map over U = U C Exponential maps and ultraproducts • A key role is played by the Casimir operator of U k : c = 2 xy + 2 yx + h 2 which generates the center of U k • By PBW basis of U k , we can see that the 0-component of U k U k 0 = k [ c , h ] Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 8 / 28

We will use these algebraic properties of U k : Possible Exponentia- • U k has a Z -graded k -algebra. Let U κ, m be the tions over enveloping subalgebra of elements of grade m . We have algebras Sonia � L’Innocente U k = U k , m ; m ∈ Z Our setting x m U k , 0 = U k , 0 x m ; Some results in for m > 0 , U k , m = this framework Exponential y | m | U k , 0 = U k , 0 y | m | . for m < 0 , U k , m = map over U = U C Exponential maps and ultraproducts • A key role is played by the Casimir operator of U k : c = 2 xy + 2 yx + h 2 which generates the center of U k • By PBW basis of U k , we can see that the 0-component of U k U k 0 = k [ c , h ] Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 8 / 28

Possible Exponentia- tions over Simple finite dim. representations enveloping algebras Let λ be a positive integer. Sonia L’Innocente Consider the vector space k [ X , Y ] . Our setting Some results in Any simple ( λ + 1 ) -dim. sl 2 ( k ) -module V λ can be described this framework as the subspace of k [ X , Y ] Exponential map over of all homogenous polynomials in X and Y of degree λ . U = U C Exponential maps and ultraproducts According to the following basis of monomials X λ , X λ − 1 Y , . . . , XY λ − 1 , Y λ , we have λ kX λ − j Y j . � V λ = j = 0 Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 9 / 28

Possible Exponentia- tions over Simple finite dim. representations enveloping algebras Let λ be a positive integer. Sonia L’Innocente Consider the vector space k [ X , Y ] . Our setting Some results in Any simple ( λ + 1 ) -dim. sl 2 ( k ) -module V λ can be described this framework as the subspace of k [ X , Y ] Exponential map over of all homogenous polynomials in X and Y of degree λ . U = U C Exponential maps and ultraproducts According to the following basis of monomials X λ , X λ − 1 Y , . . . , XY λ − 1 , Y λ , we have λ kX λ − j Y j . � V λ = j = 0 Sonia L’Innocente (Camerino ∼ Mons) Possible Exponentiations over enveloping algebras 9 / 28