In mathematics , the theorem of greatest weight is a fundamental tenet of representation theory that goes back to Elie Cartan . It says that finite-dimensional representations of Lie algebras or Lie groups are uniquely determined by their highest weight.
Terms used
Let be a Lie algebra , a Cartan subalgebra, and a representation . A linear figure
![\ mathfrak {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![{\ mathfrak {h}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe)
![{\ displaystyle \ pi \ colon {\ mathfrak {g}} \ to {\ mathfrak {gl}} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63b51446e4bc84122f0222d1c85d140861d7de13)
![{\ displaystyle \ lambda \ colon {\ mathfrak {h}} \ to \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72a79794bd104debab8b7cd061dbb842834d873c)
is called weight of if the weight space
![{\ displaystyle V _ {\ lambda} = \ left \ {v \ in V \ colon \ pi (h) v = \ lambda (h) v \ \ forall h \ in {\ mathfrak {h}} \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2649a74f55f26a2423647ab74c7f2e20e9224970)
does not only consist of the zero vector.
The roots of Lie algebra are defined as follows. To be defined by
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle \ alpha \ in {\ mathfrak {h}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/677ca0b6c404ca3a34f2465b5655e5ea26f87d7f)
![{\ displaystyle \ alpha ^ {\ vee} \ in {\ mathfrak {h}} ^ {*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1919e22e079c8c2875d6e6b71aadca768f26b7)
-
,
where is the killing form . Then is a root if and only if is a weight of the adjoint representation .
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![{\ displaystyle ad \ colon {\ mathfrak {g}} \ to {\ mathfrak {gl}} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91611351a27b77f1317d73451d5dfe78ce62fce2)
After choosing a Weyl chamber , the set of positive roots can be defined by
![{\ displaystyle {\ mathfrak {h}} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e86d50501b0fee7fc239c72bc3f68ddb790bbb28)
-
.
This allows a suborder to be defined on the weights of a given representation
-
.
A weight is called the highest weight if there is no greater weight with regard to this partial order.
Furthermore, a linear mapping is called an integral element if
![{\ displaystyle \ lambda \ in {\ mathfrak {h}} ^ {*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d20157e69417feaedfb31a989de7c3fe34a2c8)
![{\ displaystyle \ lambda (\ alpha) \ in \ mathbb {Z} \ \ \ forall \ alpha \ in R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/382d79a7db78a22cd72b9c7c41c925c4a263fdf5)
applies. It is called a dominant integral element when
![{\ displaystyle \ lambda (\ alpha) \ in \ mathbb {N} \ \ \ forall \ alpha \ in R ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3cf69b21afa1ed98027f749ebd958ce486dd50)
is.
Theorem of the highest weight
Let be a semi-simple complex Lie algebra. In the following, all representations are finite-dimensional. Then the theorem of highest weight says:
- Every irreducible representation has a clear highest weight.
- Two irreducible representations with the same highest weight are equivalent .
- The greatest weight of an irreducible representation is a dominant integral element.
- Each dominant integral element carries the greatest weight in an irreducible representation.
Examples
sl (2, C)
A Cartan sub-algebra of is , one can choose as a positive root . For each one has given a dominant integral element through the mapping
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
![{\ displaystyle {\ mathfrak {h}} = \ left \ {\ left ({\ begin {array} {cc} \ lambda & 0 \\ 0 & - \ lambda \ end {array}} \ right): \ lambda \ in \ mathbb {C} \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae4529f1d3c81648856f989bc7ae62b7aa0eead)
![{\ displaystyle \ alpha = \ left ({\ begin {array} {cc} 1 & 0 \\ 0 & -1 \ end {array}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d68bb3c15a1859654cd1d8b55278eb6edb2f7a1a)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
![\ lambda _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/093ee22c3daf31b92ff5fa04ba0ce7862283e90c)
-
.
This corresponds to the known -dimensional irreducible representation (see representation theory of sl (2, C) ) as , where denotes the defining 2-dimensional representation of .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle Sym ^ {n-1} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f01b075dc3532b388dbcd7adb566343faa2c2216)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle {\ mathfrak {sl}} (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80425b262aec9e7559040696a7443754c4bb3b5e)
sl (3, C)
A Cartan subalgebra of is
![{\ displaystyle {\ mathfrak {sl}} (3, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b13740328f101e80d4f91e2ae4c1e8ddfc48b4e7)
-
,
as positive roots you can
choose and . For each pair one has given a dominant integral element through the mapping
![{\ displaystyle \ alpha _ {1} = \ left ({\ begin {array} {ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \ end {array}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae03c1a9577937228bd3c96d3121c66c6153da02)
![{\ displaystyle \ alpha _ {2} = \ left ({\ begin {array} {ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \ end {array}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da0ef66f81367a9b4b0176ca03bcff2b9fb27d26)
![{\ displaystyle (m, n) \ in \ mathbb {N} \ times \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddacee343d614fca701fdcd6f13ed020f0aade22)
![{\ displaystyle \ lambda _ {m, n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9730189e5c526542023baf749edd9e474b693fd)
-
.
The associated representation is a sub-representation of , wherein the defining 3-dimensional representation denotes. More precisely agrees with for the through
![\ pi _ {{m, n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c0fcb49982800985c85b57725c559a8429e486)
![{\ displaystyle Sym ^ {m} (V) \ otimes Sym ^ {n} (V ^ {*})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fade49fcfbe87dfdbc9eb9ad62a2e16c09eb4df)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle {\ mathfrak {sl}} (3, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b13740328f101e80d4f91e2ae4c1e8ddfc48b4e7)
![\ pi _ {{m, n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c0fcb49982800985c85b57725c559a8429e486)
![{\ displaystyle Ker (\ iota _ {m, n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c64af98c4ac26ef35930f6326c4b4a89260435e)
![{\ displaystyle \ iota _ {m, n} (v_ {1} \ ldots v_ {m} \ otimes v_ {1} ^ {*} \ ldots v_ {n} ^ {*}) = \ sum _ {i, j} v_ {j} ^ {*} (v_ {i}) v_ {1} \ ldots {\ widehat {v_ {i}}} \ ldots v_ {m} \ otimes v_ {1} ^ {*} \ ldots {\ widehat {v_ {j} ^ {*}}} \ ldots v_ {n} ^ {*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c94bbd531e6b658ef7287cfa54ce19872f21dbc)
defined contraction .
Representations of Lie groups
Each representation of a Lie group can be assigned a representation of its Lie algebra , see Representation (Lie algebra) # Representations induced by Lie group representations . In particular, you can define a maximum weight for representations of Lie groups.
Irreducible, finite-dimensional representations of a compact, connected (not necessarily semi-simple) Lie group are classified by their highest weight. This fact is also often referred to as the principle of greatest weight .
literature
- Brian Hall: Lie groups, Lie algebras, and representations. An elementary introduction. Second edition. Graduate Texts in Mathematics, 222. Springer, Cham 2015. ISBN 978-3-319-13466-6