Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras On the curvatures of subalgebras of nilpotent Lie algebras Ana Hini´ c Gali´ c La Trobe University, Australia coauthors: Grant Cairns, Yury Nikolayevsky (La Trobe University, Australia) Marcel Nicolau (Universitat Aut` onoma de Barcelona, Spain) PADGE2012, KULeuven, Belgium August 29, 2012 Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Table of contents Nilpotent Lie algebras 1 Curvatures of a nilpotent Lie algebras 2 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras 3 Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Intrinsic and extrinsic sectional curvatures Intrinsic and extrinsic Ricci curvatures Intrinsic and extrinsic scalar curvatures Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Nilpotent Lie algebras Let g be an n -dimensional Lie algebra over R . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Nilpotent Lie algebras Let g be an n -dimensional Lie algebra over R . • Defined the following ideals: C 0 ( g ) = g , C 1 ( g ) = [ g , g ] , C k +1 ( g ) = [ C k ( g ) , g ] , for all k ≥ 0. Then we have the descending central series of g : g = C 0 ( g ) ⊃ C 1 ( g ) ⊃ · · · ⊃ C k ( g ) ⊃ . . . . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Nilpotent Lie algebras Let g be an n -dimensional Lie algebra over R . • Defined the following ideals: C 0 ( g ) = g , C 1 ( g ) = [ g , g ] , C k +1 ( g ) = [ C k ( g ) , g ] , for all k ≥ 0. Then we have the descending central series of g : g = C 0 ( g ) ⊃ C 1 ( g ) ⊃ · · · ⊃ C k ( g ) ⊃ . . . . Definition A Lie algebra g is called nilpotent if there is an integer k such that C k ( g ) = { 0 } . The smallest integer k such that C k ( g ) = { 0 } is called the nilindex of g . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Examples of nilpotent Lie algebras Every abelian Lie algebra is nilpotent with the nilindex equal to 1. 1 The Heisenberg algebra h 2 k +1 defined in the basis { X 1 , X 2 , . . . , X 2 k +1 } by 2 [ X 2 i − 1 , X 2 i ] = X 2 k +1 , i = 1 , . . . , k . The nilindex is equal to 2. The n -dimensional algebra m 0 ( n ) defined in a basis { X 1 , . . . , X n } by the 3 brackets [ X 1 , X i ] = X i +1 for all 2 ≤ i ≤ n − 1 . The nilindex is equal to n − 1. Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Metric Lie algebras • Let ( G , g ) be a simply-connected Lie group with left-invariant Riemannian metric g . • Then ( g , �· , ·� ) is the corresponding Lie algebra of G equipped with an inner product (a metric Lie algebra ). Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Metric Lie algebras • Let ( G , g ) be a simply-connected Lie group with left-invariant Riemannian metric g . • Then ( g , �· , ·� ) is the corresponding Lie algebra of G equipped with an inner product (a metric Lie algebra ). • If g is a metric Lie algebra, with inner product �· , ·� , the Levi-Civita connection on g is given by: 2 �∇ X Y , Z � = � [ X , Y ] , Z � + � [ Z , X ] , Y � + � [ Z , Y ] , X � , ∀ X , Y , Z ∈ g . (1) • Decomposition: ∇ X Y = 1 2[ X , Y ] + U ( X , Y ) , where � U ( X , Y ) , Z � = 1 2 ( � [ Z , X ] , Y � + � [ Z , Y ] , X � ) . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Sectional curvature • For a metric Lie algebra g , the sectional curvature for X , Y ∈ g : R ( X , Y , X , Y ) K ( X , Y ) = − � X , X �� Y , Y � − � X , Y � 2 . (2) • The numerator k of the curvature function K for X , Y ∈ g is equal to k ( X , Y ) = − R ( X , Y , X , Y ) = � U ( X , Y ) � 2 − � U ( X , X ) , U ( Y , Y ) � − 3 4 � [ X , Y ] � 2 (3) − 1 2 � [ X , [ X , Y ]] , Y � − 1 2 � [ Y , [ Y , X ]] , X � . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Sectional curvature • For a metric Lie algebra g , the sectional curvature for X , Y ∈ g : R ( X , Y , X , Y ) K ( X , Y ) = − � X , X �� Y , Y � − � X , Y � 2 . (2) • The numerator k of the curvature function K for X , Y ∈ g is equal to k ( X , Y ) = − R ( X , Y , X , Y ) = � U ( X , Y ) � 2 − � U ( X , X ) , U ( Y , Y ) � − 3 4 � [ X , Y ] � 2 (3) − 1 2 � [ X , [ X , Y ]] , Y � − 1 2 � [ Y , [ Y , X ]] , X � . Theorem (Wolf, 1964) Let ( G , g ) be a connected nonabelian nilpotent Lie group and let g be the corresponding Lie algebra. Then there exist two-dimensional subspaces π 1 , π 2 , π 3 ⊂ g such that the sectional curvatures satisfy K ( π 1 ) < 0 < K ( π 3 ) and K ( π 2 ) = 0 . Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Ricci curvature • Ricci curvature tensor: Ric( X , Y ) = � n i =1 R ( E i , X , Y , E i ) where { E 1 , E 2 , . . . , E n } is an orthonormal basis for g . • Ricci curvature in the direction of X ∈ g ( X � = 0) is R ic ( X ) = Ric( X , X ) . (4) � X � 2 Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Ricci curvature • Ricci curvature tensor: Ric( X , Y ) = � n i =1 R ( E i , X , Y , E i ) where { E 1 , E 2 , . . . , E n } is an orthonormal basis for g . • Ricci curvature in the direction of X ∈ g ( X � = 0) is R ic ( X ) = Ric( X , X ) . (4) � X � 2 • I. Dotti, 1982: For a metric Lie algebra g , the Ricci curvature function in a direction X ∈ g is given by n � ric ( X ) = Ric( X , X ) = R ( E i , X , X , E i ) i =1 n n n = − 1 � [ X , E i ] � 2 − 1 2 B ( X , X ) + 1 � X , [ E i , E j ] � 2 − � � � � [ U ( E i , E i ) , X ] , X � , 2 4 i =1 i , j =1 i =1 (5) where B ( X , Y ) = tr(ad( X ) ◦ ad( Y )) is the Killing form. Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebras Nilpotent Lie algebras Sectional curvature Curvatures of a nilpotent Lie algebras Ricci curvature Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Scalar curvature Lemma Let g be a nilpotent metric Lie algebra and { E 1 , E 2 , . . . , E n } an orthonormal basis. Then for all X , Y ∈ g i , j =1 � X , [ E i , E j ] � 2 − 1 (a) ric ( X ) = 1 � n � n i =1 � [ X , E i ] � 2 , 4 2 (b) Ric( X , Y ) = 1 � n i , j =1 � [ E i , E j ] , X �� [ E i , E j ] , Y � − 1 � n i =1 � [ X , E i ] , [ Y , E i ] � . 4 2 Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
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