Classification of complex semisimple Lie algebras by root systems - PowerPoint PPT Presentation

Classification of complex semisimple Lie algebras by root systems Ian Xiao Supervised by: Dr. Jeroen Schillewaert Department of Mathematics, University of Auckland February 27, 2019

Lie algebras Vector space g with a bilinear map [ , ] : g × g → g such that ◮ [ x , x ] = 0 ◮ [ x , [ y , z ]] + [ y , [ z , x ]] + [ z , [ x , y ]] = 0

Lie algebras Vector space g with a bilinear map [ , ] : g × g → g such that ◮ [ x , x ] = 0 ◮ [ x , [ y , z ]] + [ y , [ z , x ]] + [ z , [ x , y ]] = 0 Examples ◮ R 3 with cross product. ◮ The space of n × n matrices over any field k with Lie bracket [ X , Y ] = XY − YX ◮ Any associative algebra A , with Lie bracket given by [ x , y ] = xy − yx

Lie algebras More definitions ◮ Lie subalgebras: subspaces [ h , h ] ⊆ h ◮ Ideals: subalgebras [ I , g ] ⊆ I ◮ Quotient Lie algebras: g / I ◮ Lie homomorphisms: [ φ ( x ) , φ ( y )] = φ ([ x , y ]) ◮ Extensions and semidirect products ◮ Derivations: Der ( g ) = { D ∈ End ( g ) | D ( ab ) = D ( a ) b + aD ( b ) }

Lie algebras More definitions ◮ Lie subalgebras: subspaces [ h , h ] ⊆ h ◮ Ideals: subalgebras [ I , g ] ⊆ I ◮ Quotient Lie algebras: g / I ◮ Lie homomorphisms: [ φ ( x ) , φ ( y )] = φ ([ x , y ]) ◮ Extensions and semidirect products ◮ Derivations: Der ( g ) = { D ∈ End ( g ) | D ( ab ) = D ( a ) b + aD ( b ) } Basic results ◮ Der ( g ) is a Lie algebra ◮ Isomorphism theorems ◮ Correspondence theorem

Lie algebra representations and modules Homomorphism ρ : g → gl V , where gl V = End ( V ) with [ x , y ] = xy − yx .

Lie algebra representations and modules Homomorphism ρ : g → gl V , where gl V = End ( V ) with [ x , y ] = xy − yx . Example Adjoint representation ad : g → ad ( g ) given by x �→ ( y �→ [ x , y ]) . Note that ad ( g ) ⊆ gl g .

Lie algebra representations and modules Homomorphism ρ : g → gl V , where gl V = End ( V ) with [ x , y ] = xy − yx . Example Adjoint representation ad : g → ad ( g ) given by x �→ ( y �→ [ x , y ]) . Note that ad ( g ) ⊆ gl g . A g -module is a vector space V together with a linear operator · : g × V → V which preserves the Lie bracket for gl g

Lie algebra representations and modules Homomorphism ρ : g → gl V , where gl V = End ( V ) with [ x , y ] = xy − yx . Example Adjoint representation ad : g → ad ( g ) given by x �→ ( y �→ [ x , y ]) . Note that ad ( g ) ⊆ gl g . A g -module is a vector space V together with a linear operator · : g × V → V which preserves the Lie bracket for gl g Theorem There is a one-to-one correspondence between Lie representations and Lie modules.

Lie algebra representation More definitions Submodules, irreducible modules, indecomposable modules, quotient modules. Let V and W be g -modules. A g -module homomorphism from V to W is a linear map φ : V → W such that φ ( x · v ) = x · φ ( v ) for all x ∈ g and v ∈ V .

Lie algebra representation More definitions Submodules, irreducible modules, indecomposable modules, quotient modules. Let V and W be g -modules. A g -module homomorphism from V to W is a linear map φ : V → W such that φ ( x · v ) = x · φ ( v ) for all x ∈ g and v ∈ V . Theorem Schur’s Lemma . Let g be a Lie algebra over C , and let V be a finite dimensional irreducible g -module. Then φ : V → V is a g -module homomorphism if and only if φ ∈ span { I V } .

Nilpotent Lie algebras Lower central series : g = g 0 ⊃ g 1 ⊃ g 2 ⊃ ... with g 1 := [ g , g ] and g i + 1 := [ g , g i ] for each i . A Lie algebra is nilpotent if its lower central series terminates.

Nilpotent Lie algebras Lower central series : g = g 0 ⊃ g 1 ⊃ g 2 ⊃ ... with g 1 := [ g , g ] and g i + 1 := [ g , g i ] for each i . A Lie algebra is nilpotent if its lower central series terminates. Example The space of strictly upper triangular matrices.

Nilpotent Lie algebras Lower central series : g = g 0 ⊃ g 1 ⊃ g 2 ⊃ ... with g 1 := [ g , g ] and g i + 1 := [ g , g i ] for each i . A Lie algebra is nilpotent if its lower central series terminates. Example The space of strictly upper triangular matrices. Ad-nilpotency A linear map α ∈ End ( V ) is nilpotent if α n = 0 for some n ∈ N . A Lie algebra g is called ad-nilpotent if ad ( x ) is nilpotent for each x ∈ g .

Nilpotent Lie algebras Lower central series : g = g 0 ⊃ g 1 ⊃ g 2 ⊃ ... with g 1 := [ g , g ] and g i + 1 := [ g , g i ] for each i . A Lie algebra is nilpotent if its lower central series terminates. Example The space of strictly upper triangular matrices. Ad-nilpotency A linear map α ∈ End ( V ) is nilpotent if α n = 0 for some n ∈ N . A Lie algebra g is called ad-nilpotent if ad ( x ) is nilpotent for each x ∈ g . Engel’s Theorem A Lie algebra is nilpotent if and only if it is ad-nilpotent.

Solvable Lie algebras A Lie algebra g is solvable if it has a terminating derived series g = g ( 0 ) ⊃ g ( 1 ) ⊃ ... ⊃ g ( r ) = 0, where g ( i + 1 ) = [ g ( i ) , g ( i ) ] .

Solvable Lie algebras A Lie algebra g is solvable if it has a terminating derived series g = g ( 0 ) ⊃ g ( 1 ) ⊃ ... ⊃ g ( r ) = 0, where g ( i + 1 ) = [ g ( i ) , g ( i ) ] . Example The space of upper triangular matrices.

Solvable Lie algebras A Lie algebra g is solvable if it has a terminating derived series g = g ( 0 ) ⊃ g ( 1 ) ⊃ ... ⊃ g ( r ) = 0, where g ( i + 1 ) = [ g ( i ) , g ( i ) ] . Example The space of upper triangular matrices. Lie’s Theorem Let g ⊂ gl V be a solvable Lie subalgebra where V is n -dimensional over C . There exists a basis { v 1 , ..., v n } of V such that every x ∈ g is represented by an upper triangular matrix.

Solvable Lie algebras A Lie algebra g is solvable if it has a terminating derived series g = g ( 0 ) ⊃ g ( 1 ) ⊃ ... ⊃ g ( r ) = 0, where g ( i + 1 ) = [ g ( i ) , g ( i ) ] . Example The space of upper triangular matrices. Lie’s Theorem Let g ⊂ gl V be a solvable Lie subalgebra where V is n -dimensional over C . There exists a basis { v 1 , ..., v n } of V such that every x ∈ g is represented by an upper triangular matrix. Corollary If g is a finite dimensonal solvable Lie algebra over C , then all irreducible g -modules are one-dimensional.

Solvable Lie algebras Corollary Let ρ : g → gl V be a Lie representation where V is a n -dimensional vector space over C , and g is solvable. Then there exists a basis of V such that each ρ ( x ) ∈ ρ ( g ) is represented by an upper triangular matrix.

Solvable Lie algebras Corollary Let ρ : g → gl V be a Lie representation where V is a n -dimensional vector space over C , and g is solvable. Then there exists a basis of V such that each ρ ( x ) ∈ ρ ( g ) is represented by an upper triangular matrix. Proposition Let g be a finite dimensional Lie algebra over C , then g is solvable if and only if g ( 1 ) is nilpotent.

Solvable Lie algebras Corollary Let ρ : g → gl V be a Lie representation where V is a n -dimensional vector space over C , and g is solvable. Then there exists a basis of V such that each ρ ( x ) ∈ ρ ( g ) is represented by an upper triangular matrix. Proposition Let g be a finite dimensional Lie algebra over C , then g is solvable if and only if g ( 1 ) is nilpotent. Cartan’s criterion for solvability Let g be a finite dimensional Lie algebra over C . Then g is solvable if and only if tr ( ad ( x ) ◦ ad ( y )) = 0 for all x ∈ g and y ∈ g ( 1 ) .

Killing form and semisimple Lie algebras The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g : → k given by κ ( x , y ) = tr ( ad ( x ) ◦ ad ( y )) . This allows us to restate Cartan’s criterion for solvability:

Killing form and semisimple Lie algebras The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g : → k given by κ ( x , y ) = tr ( ad ( x ) ◦ ad ( y )) . This allows us to restate Cartan’s criterion for solvability: Cartan’s criterion for solvability Let g be a finite dimensional Lie algebra over C . Then g is solvable if and only if κ ( x , y ) = 0 for all x ∈ g and y ∈ g ( 1 ) .

Killing form and semisimple Lie algebras The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g : → k given by κ ( x , y ) = tr ( ad ( x ) ◦ ad ( y )) . This allows us to restate Cartan’s criterion for solvability: Cartan’s criterion for solvability Let g be a finite dimensional Lie algebra over C . Then g is solvable if and only if κ ( x , y ) = 0 for all x ∈ g and y ∈ g ( 1 ) . Example The special linear Lie algebra of order n is given by sl n = { x ∈ gl V | tr ( x ) = 0 } . E.g. sl 2 ( C ) , with basis { e , f , h } � 0 � � 0 � � 1 � 1 0 0 e = , f = , and h = . 0 0 1 0 0 − 1