Chapters 8-12

Question 1

Define an ideal

Answer 1

Question 2

Define the semi direct sum of Lie algebras

Answer 2

Question 3

Define the derived Lie algebra

Answer 3

Question 4

Define the lower central series of a Lie algebra

Answer 4

Question 5

Define the derived series of a Lie algebra

Answer 5

Question 6

Define a nilpotent algebra

Answer 6

An algebra is nilpotent if the lower central series stabilises at 0

Question 7

Define a solvable algebra

Answer 7

An algebra is solvable if the derived series stabilises at 0

Question 8

Define a semi simple Lie algebra

Answer 8

An algebra is semi simple if it contains no non-zero solvable ideals

Question 9

Define a simple Lie algebra

Answer 9

A Lie algebra is simple if has no non-zero proper ideals and has dimension > 1

Question 10

Define the radical

Answer 10

Question 11

Define a Levi subalgebra

Answer 11

Question 12

Levi’s decomposition

Answer 12

Question 13

Define the Killing form

Answer 13

Question 14

Cartan’s solvability criterion

Answer 14

Question 15

Cartan’s semisimplicity criterion

Answer 15

Question 16

Lema that a simple algebra is semi simple

Answer 16

Question 17

Define the complexification of a vector space

Answer 17

Question 19

Define the complexification of a Lie algebra

Answer 19

Question 20

Define a compact Lie algebra

Answer 20

Question 21

Define an irreducible representation of a Lie algebra

Answer 21

Question 22

Theorem about rep of SL2C

Answer 22

Question 23

Weyl’s theorem on complete reducibility

Answer 23

Question 24

Lie algebra quotient an ideal

Answer 24

Question 25

Non degenerate form

Answer 25

Question 26

Compact Lie algebras theorems

Answer 26

If K is negative-definite then K is non-degenerate and therefore a compact Lie algebra is necessarily semisimple. **Theorem 1** Let h be a complex semisimple Lie algebra. Then there exists a unique, up to isomorphism, real compact Lie algebra g such that h is isomorphic to gC. In other words, every complex semisimple Lie algebra has a unique compact real form. **Theorem 2** Let g be a real compact Lie algebra. Then it is the Lie algebra of a compact simply-connected Lie group G. Conversely, the Lie algebra of a compact simply- connected Lie group is compact.

Question 27

Rank of a Lie algebra

Answer 27

The common dimension of Cartan subalgebras of g is called the rank of g

Question 28

Representation of a Lie algebra

Answer 28