This article deals with Lie algebra , for the group see
Special Linear Group .![{\ mathfrak {sl}} (2, C)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ded600e15b581e1b97149bb9ef0fcefaa3fdb3cb)
![SL (2, C)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e956562207af771af0b6e275a6832ad2ee13d4b8)
In mathematics , the Lie algebra is the prototype of a complex simple Lie algebra . This is a three-dimensional, complex, simple Lie algebra. These properties already clearly identify it as a Lie algebra.
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
That is the three-dimensional Lie algebra of the special linear group . It is defined using the complex number field and has two real forms , the Lie algebra and the Lie algebra .
![{\ displaystyle SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e4b51615b649a6f78ced10d5313c86da67033b)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![{\ mathfrak {su}} (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9)
![{\ mathfrak {sl}} (2, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6194fdd6c417135b0b5e424b4560b8ec55628b1)
The group plays a role particularly in the special theory of relativity , since it is the simply connected superposition of the actual orthochronous Lorentz transformations .
![SO _ {{0}} (3.1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6893b113b4aaad1d154727901620ff1890b89d9)
Commutator relations
We consider the vector space spanned by the base x, y, h . This is then determined by the following commutator relations:
![{\ displaystyle g = \ langle \ {x, y, h \} \ rangle _ {\ mathbb {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10b2ce34b0855367e3c4bc0265184c960934c1af)
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![[x, y] = h, \ quad [h, x] = 2x, \ quad [h, y] = - 2y](https://wikimedia.org/api/rest_v1/media/math/render/svg/78a9776f0df5dd357c3ae3d1dc866309dccea6f9)
A frequently used implementation takes place using the following non-marking 2 × 2 matrices:
![{\ displaystyle x = {\ begin {pmatrix} 0 & 1 \\ 0 & 0 \ end {pmatrix}}, \ quad y = {\ begin {pmatrix} 0 & 0 \\ 1 & 0 \ end {pmatrix}}, \ quad h = {\ begin {pmatrix} 1 & 0 \\ 0 & -1 \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c8bd41b9869ce06b6d0ec4e3b1c86efbdacc67)
Alternative realization through the cross product
By defining the cross product in and the following vectors
![{\ displaystyle \ mathbb {C} ^ {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15f34065b7fcb29527d42a2c449e643d8ec8a083)
![x = (1, {\ mathrm i}, 0), \ quad y = (- 1, {\ mathrm i}, 0), \ quad h = (0,0,2 {\ mathrm i})](https://wikimedia.org/api/rest_v1/media/math/render/svg/d964899950b2bc417e346aa1f394c9bb2fa4cd72)
the result is the same algebra:
![{\ displaystyle x \ times y = h, \ quad h \ times x = 2x, \ quad h \ times y = -2y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22834cdbbbce52ed8252f59061ff34c3d529161d)
properties
is a simple (especially semi-simple ) Lie algebra.
Proof: Be a nontrivial ideal in and be with . If , then , so and so . So we can assume or o. B. d. A . From then follows and with it , so again .
![{\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b)
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![ax + bh + cy \ in {\ mathfrak {a}} \ setminus 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a27d120a455c20b7f21092aa68e22916772c2e4c)
![{\ displaystyle a, b, c \ in \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e1f8d22f53c54175e7eb1d6030785d656df818)
![a = c = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/7891052606b6a2bd524041b1316b7da904b70257)
![h \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b95d91c3e0856720e2145a50641344a340524bd)
![2x = \ left [h, x \ right] \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf119f54ded23ea293ff754dff31a8f6233e9e)
![2y = \ left [h, y \ right] \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32dfb5eadd54e7e1d69afdebbfa360baced5c26d)
![{\ displaystyle {\ mathfrak {a}} = {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54379291251b10a3f4457992857e22f21f04a753)
![a \ not = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1747dd02983210d1089ab1c3009ba2e9d6fad085)
![c \ not = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/2af50b388ae756aa51ca3bcfc66f00aca231506b)
![a \ not = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1747dd02983210d1089ab1c3009ba2e9d6fad085)
![\ left [y, \ left [y, ax + bh + cy \ right] \ right] = \ left [y, -ah + 2by \ right] = -2ay](https://wikimedia.org/api/rest_v1/media/math/render/svg/2264fac4b265cc0cab2d07dd3746cb53a5981b93)
![y \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81599d7c1aa4fcf48c971e19ec2735a93824b52d)
![h = \ left [x, y \ right] \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c987795f27a49f0bb1710516894c145e90a8f06e)
![{\ displaystyle {\ mathfrak {a}} = {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54379291251b10a3f4457992857e22f21f04a753)
Structure of the Lie algebra sl (2, C)
Killing form
The killing form of can be made explicit by the formula
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![B (v, w) = 4 \, \ operatorname {trace} (vw)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9dad32a87a43178e6d91cbdd6c468fc26027a0)
calculate so it is
![B (x, x) = B (y, y) = 0, \ B (h, h) = 8](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3c896d8e78976b5be8ff49413b06da016ff66d)
![{\ displaystyle B (x, y) = 4, \ B (x, h) = B (y, h) = 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6be21f38704d361681f51c9110a3705d7337e0c)
Cartan Involution
A maximally compact subgroup of the Lie group is , its Lie algebra is spanned by and .
![{\ displaystyle SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e4b51615b649a6f78ced10d5313c86da67033b)
![K = SU (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/225620238f7aa3d590e17c45e46a75726bf2b7d1)
![{\ mathfrak {k}} = {\ mathfrak {su}} (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecf614278cb2d0556142fc479abb93c3ef6c5c9)
![i (x + y), \ xy](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aeef2536b426973e73376b80d8f93dd77451aca)
![ih](https://wikimedia.org/api/rest_v1/media/math/render/svg/58504ad0d28185f79f7eb95c242025a92c8a4505)
A Cartan involution of is given by
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
-
.
is their own space to eigenvalue . The Cartan decomposition is obtained
-
,
where is the eigenspace to the eigenvalue .
![{\ displaystyle {\ mathfrak {p}} = \ left \ {A \ in {\ mathfrak {sl}} (2, \ mathbb {C}): A = {\ overline {A}} ^ {T} \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f373d727cb94a21b32036f96f8cc3b6702ce9800)
![-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac)
Iwasawa decomposition
An Iwasawa decomposition of is
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C}) = {\ mathfrak {k}} \ oplus {\ mathfrak {a}} \ oplus {\ mathfrak {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4946421e12478f33ba8ebf99069f7b04d9e6753d)
with .
![{\ displaystyle {\ mathfrak {k}} = {\ mathfrak {su}} (2), \ {\ mathfrak {a}} = \ left \ {{\ begin {pmatrix} \ lambda & 0 \\ 0 & - \ lambda \ end {pmatrix}}: \ lambda \ in \ mathbb {R} \ right \}, \ {\ mathfrak {n}} = \ left \ {{\ begin {pmatrix} 0 & n \\ 0 & 0 \ end {pmatrix}} : n \ in \ mathbb {C} \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac87c0cea5a8a7b8fb57805c1d0ed6183cdb74e8)
Real forms
It has two real forms : its compact real form is , its split real form is .
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![{\ mathfrak {su}} (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9)
![{\ mathfrak {sl}} (2, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6194fdd6c417135b0b5e424b4560b8ec55628b1)
Cartan subalgebras
Is a maximal abelian subalgebra
-
.
is a Cartan sub-algebra .
Every Cartan sub-algebra is to be conjugated; i.e., it is of the form
![{\ displaystyle {\ mathfrak {h}} \ subset {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735e2ee97af0de03bebff5168ee1717d314d2174)
![{\ mathfrak {h}} _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75585b35cdb0ee93937f7f69a7f7c20904d0b570)
![{\ mathfrak {h}} = g {\ mathfrak {h}} _ {0} g ^ {{- 1}}: = \ left \ {ghg ^ {{- 1}}: h \ in {\ mathfrak { h}} _ {0} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c471e5a5195ba024e723f089a3ba7e0842817d5)
for a .
![{\ displaystyle g \ in SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84f735bc24091f9e2884dd1b79427507026559da)
Root system
The root system to be
![{\ mathfrak {h}} _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75585b35cdb0ee93937f7f69a7f7c20904d0b570)
-
.
The dual roots are
-
.
The associated root spaces are
-
.
The Weyl group is the symmetrical group .
![S_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f)
See also
Web links
- Nicolas Perrin: The Lie Algebra
PDF
- Abhinav Shrestha: Representations of semisimple Lie algebras PDF