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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`

• noma 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

1

Nilpotent Lie algebras

2

Curvatures of a nilpotent Lie algebras Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

3

Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras 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:

   C0(g) = g , C1(g) = [g, g], Ck+1(g) = [Ck(g), g], for all k ≥ 0. Then we have the descending central series of g: g = C0(g) ⊃ C1(g) ⊃ · · · ⊃ Ck(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:

   C0(g) = g , C1(g) = [g, g], Ck+1(g) = [Ck(g), g], for all k ≥ 0. Then we have the descending central series of g: g = C0(g) ⊃ C1(g) ⊃ · · · ⊃ Ck(g) ⊃ . . . . Definition A Lie algebra g is called nilpotent if there is an integer k such that Ck(g) = {0}. The smallest integer k such that Ck(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

1

Every abelian Lie algebra is nilpotent with the nilindex equal to 1.

2

The Heisenberg algebra h2k+1 defined in the basis {X1, X2, . . . , X2k+1} by [X2i−1, X2i] = X2k+1 , i = 1, . . . , k. The nilindex is equal to 2.

3

The n-dimensional algebra m0(n) defined in a basis {X1, . . . , Xn} by the brackets [X1, Xi] = Xi+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

Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Metric Lie algebras Sectional curvature Ricci curvature 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

Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Metric Lie algebras Sectional curvature Ricci curvature 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∇XY , Z = [X, Y ], Z + [Z, X], Y + [Z, Y ], X, ∀X, Y , Z ∈ g. (1)

• Decomposition:

∇XY = 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

Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Sectional curvature

• For a metric Lie algebra g, the sectional curvature for X, Y ∈ g:

K(X, Y ) = − R(X, Y , X, Y ) X, XY , 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 − 1 2[X, [X, Y ]], Y − 1 2[Y , [Y , X]], X. (3)

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Sectional curvature

• For a metric Lie algebra g, the sectional curvature for X, Y ∈ g:

K(X, Y ) = − R(X, Y , X, Y ) X, XY , 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 − 1 2[X, [X, Y ]], Y − 1 2[Y , [Y , X]], X. (3) 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 π1, π2, π3 ⊂ g such that the sectional curvatures satisfy K(π1) < 0 < K(π3) and K(π2) = 0.

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Ricci curvature

• Ricci curvature tensor: Ric(X, Y ) = n

i=1 R(Ei, X, Y , Ei) where

{E1, E2, . . . , En} is an orthonormal basis for g.

• Ricci curvature in the direction of X ∈ g (X = 0) is

Ric(X) = Ric(X, X) X2 . (4)

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Ricci curvature

• Ricci curvature tensor: Ric(X, Y ) = n

i=1 R(Ei, X, Y , Ei) where

{E1, E2, . . . , En} is an orthonormal basis for g.

• Ricci curvature in the direction of X ∈ g (X = 0) is

Ric(X) = Ric(X, X) X2 . (4)

• I. Dotti, 1982: For a metric Lie algebra g, the Ricci curvature function in a

direction X ∈ g is given by ric(X) = Ric(X, X) =

n

• i=1

R(Ei, X, X, Ei) = −1 2

n

• i=1

[X, Ei]2 − 1 2B(X, X) + 1 4

n

• i,j=1

X, [Ei, Ej]2 −

n

• i=1

[U(Ei, Ei), X], X, (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

Nilpotent Lie algebras Curvatures of a nilpotent Lie algebras Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Lemma Let g be a nilpotent metric Lie algebra and {E1, E2, . . . , En} an orthonormal

• basis. Then for all X, Y ∈ g

(a) ric(X) = 1

4

n

i,j=1X, [Ei, Ej]2 − 1 2

n

i=1 [X, Ei]2,

(b) Ric(X, Y ) = 1

4

n

i,j=1[Ei, Ej], X[Ei, Ej], Y − 1 2

n

i=1[X, Ei], [Y , Ei].

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Lemma Let g be a nilpotent metric Lie algebra and {E1, E2, . . . , En} an orthonormal

• basis. Then for all X, Y ∈ g

(a) ric(X) = 1

4

n

i,j=1X, [Ei, Ej]2 − 1 2

n

i=1 [X, Ei]2,

(b) Ric(X, Y ) = 1

4

n

i,j=1[Ei, Ej], X[Ei, Ej], Y − 1 2

n

i=1[X, Ei], [Y , Ei].

Theorem (Milnor, 1976) For any left-invariant metric on a nonabelian nilpotent Lie group there exists a direction of strictly negative Ricci curvature and a direction of strictly positive Ricci curvature.

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Scalar curvature

The scalar curvature of a metric Lie algebra g is given by s =

n

• i=1

Ric(Ei) = −1 4

n

• i,j=1

[Ei, Ej]2 − 1 2

n

• i=1

B(Ei, Ei) −

n

• i=1

U(Ei, Ei)2. (6)

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 Metric Lie algebras Sectional curvature Ricci curvature Scalar curvature

Scalar curvature

The scalar curvature of a metric Lie algebra g is given by s =

n

• i=1

Ric(Ei) = −1 4

n

• i,j=1

[Ei, Ej]2 − 1 2

n

• i=1

B(Ei, Ei) −

n

• i=1

U(Ei, Ei)2. (6) Lemma Suppose that g is an n-dimensional metric Lie algebra with an orthonormal basis {E1, . . . , En} with respect to which the commutator coefficients are c k

i,j ;

that is, [Ei, Ej] = n

k=1 c k i,j Ek.

If g is nilpotent, then its scalar curvature is s = −1 4

n

• i,j,k=1
• c k

i,j

2 .

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 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

Consider a Lie subgroup H of a simply-connected Lie group G and the corresponding Lie algebras h and g respectively.

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 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

Consider a Lie subgroup H of a simply-connected Lie group G and the corresponding Lie algebras h and g respectively.

• ∇h: the Levi-Civita connection on H defined by the restriction of ·, ·g to h.
• The second fundamental form of g, defined by Gauss’ formula as

α(X, Y ) = ∇X Y − ∇h

XY ,

has the explicit form α(X, Y ) = 1 2

r

• j=1

([fj, X], Y + [fj, Y ], X) fj (7) for an orthonormal basis {f1, . . . , fr} of the orthogonal complement h⊥ of h.

• Let {h1, . . . , hm} be an orthonormal basis for a subalgebra h 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 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

Let h be an ideal of a nilpotent Lie algebra g and f ∈ h⊥.

• Then [f , h] ⊂ h and ad(f ) is nilpotent, so the restriction of ad(f ) to h is

nilpotent and hence tr(ad(f ) |h) = 0.

• Then (7) implies

m

• i=1

α(hi, hi) = 0 so the mean curvature m

i=1 α(hi, hi) of h is equal to zero.

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 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

Let h be an ideal of a nilpotent Lie algebra g and f ∈ h⊥.

• Then [f , h] ⊂ h and ad(f ) is nilpotent, so the restriction of ad(f ) to h is

nilpotent and hence tr(ad(f ) |h) = 0.

• Then (7) implies

m

• i=1

α(hi, hi) = 0 so the mean curvature m

i=1 α(hi, hi) of h is equal to zero.

Lemma Let H be a connected Lie subgroup of a simply-connected nilpotent Lie group G endowed with a left-invariant Riemannian metric and let the corresponding Lie algebras be h and g, respectively. If h is an ideal of g, then H is a minimal submanifold 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 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

Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras

• The intrinsic curvatures of h:

Rh, K h, Rich, s(h) the curvature operator, sectional curvature, Ricci curvature and scalar curvature, respectively, defined by ∇h.

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 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

Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras

• The intrinsic curvatures of h:

Rh, K h, Rich, s(h) the curvature operator, sectional curvature, Ricci curvature and scalar curvature, respectively, defined by ∇h.

• The extrinsic curvatures of h:

Rh

e , K h e , Rich e , se(h) the curvature operator, sectional curvature, Ricci curvature

and scalar curvature by using the Levi-Civita connection ∇ 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 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

Intrinsic and extrinsic sectional curvatures

• For X, Y ∈ h, the extrinsic sectional curvature is just K(X, Y ) and the

corresponding formula is given by K(X, Y ) = K h(X, Y ) − α(X, X), α(Y , Y ) − α(X, Y )2 X2Y 2 − X, Y 2 , for linearly independent (not necessary orthonormal) vector fields X, Y ∈ h.

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 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

• For X, Y ∈ h, the extrinsic Ricci curvature tensor is

Rich

e (X, Y ) = m

• i=1

R(hi, X, Y , hi).

• The extrinsic Ricci curvature in a direction of a unit vector field X ∈ h is

Rich

e (X) = Rich e (X, X).

(8) Theorem Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis {h1, . . . , hm} for h. If X ∈ h, then the Ricci curvature function satisfies rich

e (X) = rich(X) + m

• i=1

α(X, hi)2, while for X = 0, the extrinsic Ricci curvature satisfies Rich

e (X) = Rich(X) + m

• i=1

α(X, hi)2 X2 . (9) In particular, Rich

e (X) ≥ Rich(X).

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 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

Corollary If a is an abelian ideal of a nilpotent Lie algebra g, then the extrinsic Ricci curvature Rica

e (X) is nonnegative for all X ∈ a.

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 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

Corollary If a is an abelian ideal of a nilpotent Lie algebra g, then the extrinsic Ricci curvature Rica

e (X) is nonnegative for all X ∈ a.

You cannot just replace the Ricci curvature by the sectional curvature. It is not true that for an abelian ideal a, one has K(X, Y ) ≥ 0 for all X, Y ∈ a. Indeed, consider the five-dimensional algebra generated by the orthonormal basis {X1, . . . , X5} with relations [X1, X2] = X3, [X1, X4] = X5. X2, . . . , X5 generate a four-dimensional abelian ideal. However, for X = (X2 + X3)/2 and Y = (X4 + X5)/2 we have K(X, Y ) = −1/4.

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 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

Theorem Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis {h1, . . . , hm} for h. Then se(h) = s(h) +

m

• i,j=1

α(hi, hj)2. (10) In particular, se(h) ≥ s(h).

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 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

Theorem Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis {h1, . . . , hm} for h. Then se(h) = s(h) +

m

• i,j=1

α(hi, hj)2. (10) In particular, se(h) ≥ s(h). Corollary If a is an abelian ideal of a nilpotent Lie algebra g, then the extrinsic scalar curvature se(a) of a is nonnegative.

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 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

Corollary If a1, a2 are abelian ideals of a nilpotent Lie algebra g, with a1 ⊂ a2, then se(a1) ≤ se(a2). An ideal with the maximal extrinsic scalar curvature may not be abelian. Consider the five-dimensional Lie algebra g generated by the orthonormal basis {X1, . . . , X5} with the relations [X1, X2] = 2X3, [X1, X3] = 1 2X4, [X1, X4] = X5, [X2, X3] = 1 2X5. Let hk be the ideal generated by Xk, . . . , X5. So one has the series g = h1 ⊃ · · · ⊃ h5. k 1 2 3 4 5 se(hk)

−11 4 5 2 3 4 1 2

The maximum occurs at h2, which is not abelian. The algebra g in this example has the property that it has a unique maximal abelian ideal, h3.

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 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

Corollary If X, Y generate a two-dimensional ideal a of a nilpotent Lie algebra g, then K(X, Y ) ≥ 0.

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 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

Corollary If X, Y generate a two-dimensional ideal a of a nilpotent Lie algebra g, then K(X, Y ) ≥ 0. You cannot just replace ideal by subalgebra Consider the five-dimensional algebra generated by the orthonormal basis {X1, . . . , X5} with relations [X1, X2] = X3, [X1, X4] = X5. Then X = (X2 + X3)/2 and Y = (X4 + X5)/2 generate a two-dimensional abelian subalgebra, while K(X, Y ) = −1/4.

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 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

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Ana Hini´ c Gali´ c La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras