Lie Theory and Algebraic Geometry
Lie Theory and Algebraic Geometry
D. Calaque
ETH Z¨urich
November 28, 2011 Lie Theory and Algebraic Geometry Symmetries
Symmetries (Lie Theory) C
A B
informally: a group is a set of transformations that are invertible that one can compose that preserves something
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry C
A B
that are invertible that one can compose that preserves something
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations C
A B
that one can compose that preserves something
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible C
A B
that preserves something
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose C
A B
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something C
A B
Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something
C
A B Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something
B
C A Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something
A
B C Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something
A
B C Lie Theory and Algebraic Geometry Symmetries What is... a group?
group: mathematical object behind the concept of symmetry informally: a group is a set of transformations that are invertible that one can compose that preserves something
A
C B invented by Ern˜oRubik in 1974
world record: 5,66 sec (Feliks Zemdegs)
43’252’003’274’489’856’000 elements
God’s number: 20 Rokicki-Kociemba-Davidson-Dethridge (2010)
Generators = elementary moves God’s number = diameter of the group in terms generators
Lie Theory and Algebraic Geometry Symmetries Digression... symmetry group of the Rubik’s cube 43’252’003’274’489’856’000 elements
God’s number: 20 Rokicki-Kociemba-Davidson-Dethridge (2010)
Generators = elementary moves God’s number = diameter of the group in terms generators
Lie Theory and Algebraic Geometry Symmetries Digression... symmetry group of the Rubik’s cube
invented by Ern˜oRubik in 1974
world record: 5,66 sec (Feliks Zemdegs) Lie Theory and Algebraic Geometry Symmetries Digression... symmetry group of the Rubik’s cube
invented by Ern˜oRubik in 1974
world record: 5,66 sec (Feliks Zemdegs)
43’252’003’274’489’856’000 elements
God’s number: 20 Rokicki-Kociemba-Davidson-Dethridge (2010)
Generators = elementary moves God’s number = diameter of the group in terms generators √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degrees 3 and 4, yes in degree > 4, no - counter-example: x 5 − x + 1 = 0 Given P(x), is there a criterium for the answer to be yes?
solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832)
Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? in degrees 3 and 4, yes in degree > 4, no - counter-example: x 5 − x + 1 = 0 Given P(x), is there a criterium for the answer to be yes?
solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832)
Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degree > 4, no - counter-example: x 5 − x + 1 = 0 Given P(x), is there a criterium for the answer to be yes?
solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832)
Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degrees 3 and 4, yes Given P(x), is there a criterium for the answer to be yes?
solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832)
Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degrees 3 and 4, yes in degree > 4, no - counter-example: x 5 − x + 1 = 0 solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832)
Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degrees 3 and 4, yes in degree > 4, no - counter-example: x 5 − x + 1 = 0 Given P(x), is there a criterium for the answer to be yes? Lie Theory and Algebraic Geometry Symmetries What is... a Galois group?
Can one solve a polynomial equation P(x) = 0 by radicals? √ 2 −b± b2−4ac in degree 2, ax + bx + c = 0, yes- x = 2a in degrees 3 and 4, yes in degree > 4, no - counter-example: x 5 − x + 1 = 0 Given P(x), is there a criterium for the answer to be yes?
solutions: S = {x1,..., xk } Galois group: permutations of S such that any alg. eq. satisfied by x1,..., xk is still satisfied after they have been permuted. solvability by radicals ⇔ Evariste Galois Galois group is “solvable” (1811 - 1832) ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c
symmetries = translations
not discrete!
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
solo: y(x) = R x g(t)dt +c
symmetries = translations
not discrete!
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown) ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
+c
symmetries = translations
not discrete!
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
symmetries = translations
not discrete!
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
not discrete!
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c
symmetries = translations ⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c
symmetries = translations
not discrete! − Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable
Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c
symmetries = translations
not discrete!
⇒ Lie group Lie Theory and Algebraic Geometry Symmetries What is... a Lie group?
Sophus Lie (1842 - 1899): algebraic → differential equations
y 0(x) = g(x)(y unknown)
solo: y(x) = R x g(t)dt +c
symmetries = translations
not discrete!
⇒ Lie group
− Picard-Vessiot theory (a-k-a differential Galois theory) : solvability by quadrature ⇔ Lie group connected and solvable affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) infinitesimally: [ d , y d ] = d studied at ETH dy dy dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
b d (infinitesimal) translations: f (y + b) = e dy f (y)
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) infinitesimally: [ d , y d ] = d studied at ETH dy dy dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) d d d infinitesimally: [ dy , y dy ] = dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) studied at ETH
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) d d d infinitesimally: [ dy , y dy ] = dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) studied at ETH
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 d d d infinitesimally: [ dy , y dy ] = dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
becomes non-abelian: 2(y + 1) 6= 2y + 1
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup
Armand Borel (1923 - 2003) studied at ETH d d d infinitesimally: [ dy , y dy ] = dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
becomes non-abelian: 2(y + 1) 6= 2y + 1
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y) Armand Borel (1923 - 2003) studied at ETH d d d infinitesimally: [ dy , y dy ] = dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y)
Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) studied at ETH Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v]
Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y) Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) infinitesimally:[ d , y d ] = d studied at ETH dy dy dy Lie Theory and Algebraic Geometry Symmetries What is... a Lie algebra?
Definition: Lie algebra def= infinitesimal Lie group b d (infinitesimal) translations: f (y + b) = e dy f (y) affine transformations: y 7→ ay + b, a > 0 symmetry group of y 00 − y 0 = 0 first example of a Borel subgroup added (infinitesimal) dilations log(a)y d f (ay) = e dy f (y) Armand Borel becomes non-abelian: 2(y + 1) 6= 2y + 1 (1923 - 2003) infinitesimally:[ d , y d ] = d studied at ETH dy dy dy Lie bracket [u, v] is the infinitesimal default of commutativity: exp (u) exp (v) exp (v) exp (u)−1 = exp [u, v] SO(2) −→
: G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group)
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. SO(2) −→
: G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group)
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. SO(2) −→
: G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Lie groups as symmetries of spaces. Exam- ple:
SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group)
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 SO(2) −→
: G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Exam- ple:
SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group)
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. SO(2) −→
: G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group) homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(2) −→ SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group): G/H non axiomatic approach to non-euclidan geometries
Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(2) −→ SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group): G/H homogeneous space = Klein geometry
Felix Klein (1849 - 1925) Lie Theory and Algebraic Geometry Symmetries What is... a Klein geometry?
A non-solvable example: group SO(n) of rotational symmetries. Matrices A such that AAt = At A = id and det(A) = 1. Infinitesimal rotations are then skew-symmetric matrices:
A = exp (a) ⇒ a + at = 0 Lie groups as symmetries of spaces. Exam- ple:
SO(2) −→ SO(3) −→ S2
homogeneous spaces (spaces with a transi- tive action of a Lie group): G/H homogeneous space = Klein geometry
non axiomatic approach to non-euclidan geometries ( 1 ⊕ · · · ⊕ lk )
i
Under reasonable assumption r = n o a n nilpotent and a abelian.
Need to classify nilpotent and simple Lie algebras. Nilpotent ones are known only in dimension ≤ 6.
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o l with r solvable and l semi-simple.
Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967)
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries ( 1 ⊕ · · · ⊕ lk )
i
Under reasonable assumption r = n o a n nilpotent and a abelian.
Need to classify nilpotent and simple Lie algebras. Nilpotent ones are known only in dimension ≤ 6.
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o l with r solvable and l semi-simple.
Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967) semi-
Under reasonable assumption r = n o a n nilpotent and a abelian.
Need to classify nilpotent and simple Lie algebras. Nilpotent ones are known only in dimension ≤ 6.
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o (l1 ⊕ · · · ⊕ lk )
with r solvable and li simple.
Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967) semi-
Need to classify nilpotent and simple Lie algebras. Nilpotent ones are known only in dimension ≤ 6.
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o (l1 ⊕ · · · ⊕ lk )
with r solvable and li simple.
Under reasonable assumption r = n o a n nilpotent and a abelian. Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967) semi-
Nilpotent ones are known only in dimension ≤ 6.
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o (l1 ⊕ · · · ⊕ lk )
with r solvable and li simple.
Under reasonable assumption r = n o a n nilpotent and a abelian. Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967) Need to classify nilpotent and simple Lie algebras. semi-
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries
Levi–Mal’tsev decomposition Any finite dimensional Lie algebra g decomposes as
g = r o (l1 ⊕ · · · ⊕ lk )
with r solvable and li simple.
Under reasonable assumption r = n o a n nilpotent and a abelian. Eugenio Levi Anatoly Malcev (1883 - 1917) (1909 - 1967) Need to classify nilpotent and simple Lie algebras. Nilpotent ones are known only in dimension ≤ 6. Simple Lie algebras over R classified by Cartan (Riem. sym. spaces) Ichiro Satake Dynkin diagrams −→ Satake diagrams
Wilhelm Killing (1847-1923) conjectured a classification Elie´ Cartan (1869-1951) completed and proved it Eugene Dynkin modern approach Dynkin diagrams
Complete list of Dynkin diagrams
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Simple Lie algebras over R classified by Cartan (Riem. sym. spaces) Ichiro Satake Dynkin diagrams −→ Satake diagrams
Elie´ Cartan (1869-1951) completed and proved it Eugene Dynkin modern approach Dynkin diagrams
Complete list of Dynkin diagrams
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Wilhelm Killing (1847-1923) conjectured a classification Simple Lie algebras over R classified by Cartan (Riem. sym. spaces) Ichiro Satake Dynkin diagrams −→ Satake diagrams
Eugene Dynkin modern approach Dynkin diagrams
Complete list of Dynkin diagrams
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Wilhelm Killing (1847-1923) conjectured a classification Elie´ Cartan (1869-1951) completed and proved it Simple Lie algebras over R classified by Cartan (Riem. sym. spaces) Ichiro Satake Dynkin diagrams −→ Satake diagrams
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Wilhelm Killing (1847-1923) conjectured a classification Elie´ Cartan (1869-1951) completed and proved it Eugene Dynkin modern approach Dynkin diagrams
Complete list of Dynkin diagrams Ichiro Satake Dynkin diagrams −→ Satake diagrams
Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Wilhelm Killing (1847-1923) conjectured a classification Elie´ Cartan (1869-1951) completed and proved it Eugene Dynkin modern approach Dynkin diagrams
Simple Lie algebras over R classified by Cartan (Riem. sym. spaces)
Complete list of Dynkin diagrams Lie Theory and Algebraic Geometry Symmetries Classifying infinitesimal symmetries - continued
Simple Lie algebras over C Wilhelm Killing (1847-1923) conjectured a classification Elie´ Cartan (1869-1951) completed and proved it Eugene Dynkin modern approach Dynkin diagrams
Simple Lie algebras over R classified by Cartan (Riem. sym. spaces) Ichiro Satake Dynkin diagrams −→ Satake diagrams A few Satake diagrams Lie Theory and Algebraic Geometry Loci
Loci (Algebraic Geometry) n locus in C of solutions of polynomial equations in n variables 2 (plane) algebraic curve: {(x, y) ∈ C |P(x, y) = 0}
x2 + xy + y 2 = 1 x3 = y 2 examples of algebraic surfaces:
Kummer surface Togliatti surface
Lie Theory and Algebraic Geometry Loci What is... an affine algebraic variety? 2 (plane) algebraic curve: {(x, y) ∈ C |P(x, y) = 0}
x2 + xy + y 2 = 1 x3 = y 2 examples of algebraic surfaces:
Kummer surface Togliatti surface
Lie Theory and Algebraic Geometry Loci What is... an affine algebraic variety?
n locus in C of solutions of polynomial equations in n variables x2 + xy + y 2 = 1 x3 = y 2 examples of algebraic surfaces:
Kummer surface Togliatti surface
Lie Theory and Algebraic Geometry Loci What is... an affine algebraic variety?
n locus in C of solutions of polynomial equations in n variables 2 (plane) algebraic curve: {(x, y) ∈ C |P(x, y) = 0} examples of algebraic surfaces:
Kummer surface Togliatti surface
Lie Theory and Algebraic Geometry Loci What is... an affine algebraic variety?
n locus in C of solutions of polynomial equations in n variables 2 (plane) algebraic curve: {(x, y) ∈ C |P(x, y) = 0}
x2 + xy + y 2 = 1 x3 = y 2 Lie Theory and Algebraic Geometry Loci What is... an affine algebraic variety?
n locus in C of solutions of polynomial equations in n variables 2 (plane) algebraic curve: {(x, y) ∈ C |P(x, y) = 0}
x2 + xy + y 2 = 1 x3 = y 2 examples of algebraic surfaces:
Kummer surface Togliatti surface Three problems with this statement: parallel lines missing point tangential points not enough points self-intersections Etienne´ Bezout´ (1730 - 1783) too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point tangential points not enough points self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2.
Etienne´ Bezout´ (1730 - 1783) parallel lines missing point tangential points not enough points self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement:
Etienne´ Bezout´ (1730 - 1783) missing point tangential points not enough points self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines
Etienne´ Bezout´ (1730 - 1783) tangential points not enough points self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point
Etienne´ Bezout´ (1730 - 1783) not enough points self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point tangential points
Etienne´ Bezout´ (1730 - 1783) self-intersections too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point tangential points not enough points
Etienne´ Bezout´ (1730 - 1783) too many points
Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point tangential points not enough points self-intersections Etienne´ Bezout´ (1730 - 1783) Lie Theory and Algebraic Geometry Loci What is... B´ezout’stheorem?
B´ezout’stheorem
Given two plane curves C1 and C2 of degree m1 and m2, then #(C1 ∩ C2) = m1m2. Three problems with this statement: parallel lines missing point tangential points not enough points self-intersections Etienne´ Bezout´ (1730 - 1783) too many points n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve? n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n n−1 = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ KP = (K − {0})/K n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} plane curves −→ projective curves (solve problem of parallel lines)
Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). Lie Theory and Algebraic Geometry Loci What is... a projective curve?
n-dim projective space: space of lines in (n + 1) dimensions n def n+1 ∗ n n−1 KP = (K − {0})/K = K ∪ KP n = 1 : projective line = {slopes} = affine line ∪ {∞}
RP1 : CP1 : n = 2 : projective plane (background picture) eqo in n variables −→ homogeneous eqo in n + 1 variables x 2 − y 3 = 1 becomes x 2z − y 3 = z3 {x − y = 1} is parallel to {x − y = 2} {x − y = z} and {x − y = 2z} intersect at (1, 1, 0). plane curves −→ projective curves (solve problem of parallel lines) Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }: 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 6= {y = 0 = x}
funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0) • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points
Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C Lie Theory and Algebraic Geometry Loci What is... an intersection number?
2 Consider the intersection of C1 = {y = 0} with C2 = {y = x }:
Modern algebraic geometry: varieties are determined by their function rings (physics: states ↔ observables) 2 C1 ∩ C2 = {y = 0 = x } 6= {y = 0 = x} funtions on C1 ∩ C2:
2 2 C[x, y]/(y = x = 0) = C[x]/(x = 0)
this is of dimension 2 over C • varieties −→ schemes
• invented by Alexander Grothendieck
• solve problem of tangential points We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X ) a self-intersection = Maps(• •, X ) ••
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X ) a self-intersection = Maps(• •, X ) ••
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
a self-intersection = Maps(• •, X ) ••
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X ) We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X ) a self-intersection = Maps(• •, X ) •• We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X )
• a self-intersection = Maps( • •, X ) •• We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X )
self-intersection = Maps( • , X ) Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g.
Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X )
self-intersection = Maps( • , X ) We then have a (derived) Lie group associated to any variety. Lie Theory and Algebraic Geometry Loci Solving the third problem... derived algebraic geometry
Recent (≥ 2000) extension of algebraic geometry Bertrand Toen¨ & Grabriele Vezzosi Jacob Lurie Main feature: all fiber products (e.g. intersections) are smooth
Example: (derived) self-intersection of the diagonal X ⊂ X × X .
X = Maps(•, X ) and X × X = Maps(••, X )
self-intersection = Maps( • , X ) We then have a (derived) Lie group associated to any variety. Calaque-Caldararu-Tu: analogy between closed embeddings X ,→ Y of algebraic varieties and inclusions of Lie algebras h ⊂ g. This is analogous to L´evi-Mal’tsevdecomposition theorem CY are difficult to list while we know a very few examples of IHS (2 families, and 2 isolated examples due to Kieran O’grady)
We would like to understand this analogy in more rigourous terms
Lie Theory and Algebraic Geometry Loci Towards a dictionary
Bogomolov decomposition
Every compact Kahler manifold X with c1 = 0 admits an ´etale cover by N × T × (L1 × · · · × Lk )
with N Calabi-Yau (CY), T a torus and Li irreducible holomorphic symplectic (IHS). We would like to understand this analogy in more rigourous terms
Lie Theory and Algebraic Geometry Loci Towards a dictionary
Bogomolov decomposition
Every compact Kahler manifold X with c1 = 0 admits an ´etale cover by N × T × (L1 × · · · × Lk )
with N Calabi-Yau (CY), T a torus and Li irreducible holomorphic symplectic (IHS).
This is analogous to L´evi-Mal’tsevdecomposition theorem CY are difficult to list while we know a very few examples of IHS (2 families, and 2 isolated examples due to Kieran O’grady) Lie Theory and Algebraic Geometry Loci Towards a dictionary
Bogomolov decomposition
Every compact Kahler manifold X with c1 = 0 admits an ´etale cover by N × T × (L1 × · · · × Lk )
with N Calabi-Yau (CY), T a torus and Li irreducible holomorphic symplectic (IHS).
This is analogous to L´evi-Mal’tsevdecomposition theorem CY are difficult to list while we know a very few examples of IHS (2 families, and 2 isolated examples due to Kieran O’grady)
We would like to understand this analogy in more rigourous terms Lie Theory and Algebraic Geometry Loci
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