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Definition/Summary
A simple Lie algebra ("Lee") is a nonabelian Lie algebra with no nontrivial ideals.
Every finite simple Lie algebra is known, and every semisimple Lie algebra is a combination of simple ones.
Every one has a maximal though non-unique abelian subalgebra, its Cartan algebra. That subalgebra's size is called the rank of the algebra. The Cartan subalgebra can be used in a fashion analogous to the projected angular momentum in quantum mechanics, with the other elements being analogous to raising and lowering operators. In fact, 3D angular-momentum operators form the smallest simple Lie algebra.
One can find invariant combinations of the algebra elements that commute with them, much like the total angular momentum. That quantity generalizes to the Casimir invariant and higher-power invariants.
One can find representation basis sets in analogy with angular momentum. However, in many cases, some Cartan-subalgebra eigenvalues will be degenerate, with several basis-set elements sharing some values.
Equations
The Casimir invariant is
[itex]C = g^{ij} L_i L_j[/itex]
where the L's are the elements and [itex]g^{ij}[/itex] is the inverse of the metric tensor
[itex]g_{ij} = f_{ia}^b f_{jb}^a[/itex]
with [itex][L_i,L_j] = f_{ij}^k L_k[/itex]
One can generalize it with the help of the matrix
[itex]K_i{}^j = f_{ia}^j g^{ab} L_b[/itex]
For power p,
[itex]I_p = \text{Tr} K^p[/itex]
It is easy to show that I(2) = C, the Casimir invariant. The number of independent invariants is equal to the rank of the algebra; all the others can be expressed in terms of the independent ones.
For semisimple algebras, each I(p) decomposes into a sum of I(p)'s for each simple-algebra component.
It's possible to construct similar invariants for non-semisimple algebras, though because their algebra metrics is non-invertible, it is less straightforward.
Thus, for U(1) ~ T, there is only one operator, L, with I(p) = L^p. I(1) is nonzero here, though it is zero for all the (semi)simple algebras.
Extended explanation
The classification:
The seven families of finite semisimple Lie algebras are denoted by A(n), B(n), C(n), D(n), E(n), F(n), and G(n), where n is their rank, a positive integer. The first four are infinite families related to the special unitary, SU(n), the special linear, SL(n), the special orthogonal, SO(n), and the symplectic, Sp(2n) algebras. The others are the 5 "exceptional algebras": G2, F4, E6, E7, E8.
They can be graphed with "Dynkin diagrams", with a node for each element of their Cartan subalgebras, or root.
The algebras have several isomorphisms:
A(n) ~ SU(n+1) ~ SL(n+1)
B(n) ~ SO(2n+1)
C(n) ~ Sp(2n)
D(n) ~ SO(2n)
Note that even and odd special-orthogonal algebras are handled differently.
A1 ~ B1 ~ C1
C2 ~ B2
D1 ~ U(1) -- nilpotent, not simple
D2 ~ A1 * A1 -- semisimple, not simple
D3 ~ A3
E3 ~ A2 * A1 -- semisimple, not simple
E4 ~ A4
E5 ~ D5
Their sizes and independent invariant powers:
A(n) -- n(n+2) -- 2, 3, 4, ..., n+1
B(n) -- n(2n+1) -- 2, 4, 6, ..., 2n
C(n) -- n(2n+1) -- 2, 4, 6, ..., 2n
D(n) -- n(2n-1) -- 2, 4, 6, ..., 2n-2, n
E6 -- 78 -- 2, 5, 6, 8, 9, 12
E7 -- 133 -- 2, 6, 8, 10, 12, 14, 18
E8 -- 248 -- 2, 8, 12, 14, 18, 20, 24, 30
F4 -- 52 -- 2, 6, 8, 12
G2 -- 14 -- 2, 6
For D(n), there is an invariant with power n instead of with power 2n, because a power-2n one may contain the square of that power-n one.
Some of these algebras are physically important.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
A simple Lie algebra ("Lee") is a nonabelian Lie algebra with no nontrivial ideals.
Every finite simple Lie algebra is known, and every semisimple Lie algebra is a combination of simple ones.
Every one has a maximal though non-unique abelian subalgebra, its Cartan algebra. That subalgebra's size is called the rank of the algebra. The Cartan subalgebra can be used in a fashion analogous to the projected angular momentum in quantum mechanics, with the other elements being analogous to raising and lowering operators. In fact, 3D angular-momentum operators form the smallest simple Lie algebra.
One can find invariant combinations of the algebra elements that commute with them, much like the total angular momentum. That quantity generalizes to the Casimir invariant and higher-power invariants.
One can find representation basis sets in analogy with angular momentum. However, in many cases, some Cartan-subalgebra eigenvalues will be degenerate, with several basis-set elements sharing some values.
Equations
The Casimir invariant is
[itex]C = g^{ij} L_i L_j[/itex]
where the L's are the elements and [itex]g^{ij}[/itex] is the inverse of the metric tensor
[itex]g_{ij} = f_{ia}^b f_{jb}^a[/itex]
with [itex][L_i,L_j] = f_{ij}^k L_k[/itex]
One can generalize it with the help of the matrix
[itex]K_i{}^j = f_{ia}^j g^{ab} L_b[/itex]
For power p,
[itex]I_p = \text{Tr} K^p[/itex]
It is easy to show that I(2) = C, the Casimir invariant. The number of independent invariants is equal to the rank of the algebra; all the others can be expressed in terms of the independent ones.
For semisimple algebras, each I(p) decomposes into a sum of I(p)'s for each simple-algebra component.
It's possible to construct similar invariants for non-semisimple algebras, though because their algebra metrics is non-invertible, it is less straightforward.
Thus, for U(1) ~ T, there is only one operator, L, with I(p) = L^p. I(1) is nonzero here, though it is zero for all the (semi)simple algebras.
Extended explanation
The classification:
The seven families of finite semisimple Lie algebras are denoted by A(n), B(n), C(n), D(n), E(n), F(n), and G(n), where n is their rank, a positive integer. The first four are infinite families related to the special unitary, SU(n), the special linear, SL(n), the special orthogonal, SO(n), and the symplectic, Sp(2n) algebras. The others are the 5 "exceptional algebras": G2, F4, E6, E7, E8.
They can be graphed with "Dynkin diagrams", with a node for each element of their Cartan subalgebras, or root.
The algebras have several isomorphisms:
A(n) ~ SU(n+1) ~ SL(n+1)
B(n) ~ SO(2n+1)
C(n) ~ Sp(2n)
D(n) ~ SO(2n)
Note that even and odd special-orthogonal algebras are handled differently.
A1 ~ B1 ~ C1
C2 ~ B2
D1 ~ U(1) -- nilpotent, not simple
D2 ~ A1 * A1 -- semisimple, not simple
D3 ~ A3
E3 ~ A2 * A1 -- semisimple, not simple
E4 ~ A4
E5 ~ D5
Their sizes and independent invariant powers:
A(n) -- n(n+2) -- 2, 3, 4, ..., n+1
B(n) -- n(2n+1) -- 2, 4, 6, ..., 2n
C(n) -- n(2n+1) -- 2, 4, 6, ..., 2n
D(n) -- n(2n-1) -- 2, 4, 6, ..., 2n-2, n
E6 -- 78 -- 2, 5, 6, 8, 9, 12
E7 -- 133 -- 2, 6, 8, 10, 12, 14, 18
E8 -- 248 -- 2, 8, 12, 14, 18, 20, 24, 30
F4 -- 52 -- 2, 6, 8, 12
G2 -- 14 -- 2, 6
For D(n), there is an invariant with power n instead of with power 2n, because a power-2n one may contain the square of that power-n one.
Some of these algebras are physically important.
- Rotational symmetry of n-space is SO(n).
- Its complex extension, like for wavefunctions of n identical particles, is SU(n); the overall phase is U(1).
- Angular momentum in n space dimensions is SO(n).
- Hydrogenlike atom with inverse-square force law in n space dimensions is SO(n+1); the Laplace-Runge-Lenz vector gets added to the angular momentum.
- Protons and neutrons have a flavor symmetry called "isotopic spin" or "isospin" which is SU(2) ~ SO(3). Up and down quarks also have that flavor symmetry.
- The three light quarks, up, down, and strange, have flavor symmetry SU(3), though it is more approximate than for up and down alone.
- The unbroken Standard Model has gauge symmetry SU(3)*SU(2)*U(1) -- quantum chromodynamics, weak isospin, and weak hypercharge. It gets broken to SU(3)*U(1) -- QCD and electromagnetism.
- A variety of gauge-symmetry groups have been proposed for Grand Unified Theories; SU(5), SU(6), SO(6)*SO(4) ~ SU(4)*SU(2)*SU(2), SO(10), SU(3)^3, E6, E8.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!