Casimir element

Casimir operator

A central element of special form in the universal enveloping algebra of a semi-simple Lie algebra. Such operators were first introduced, for a particular case, by H. Casimir [1].

Let $ \mathfrak g $ be a semi-simple finite-dimensional Lie algebra over a field of characteristic $ 0 $, and let $ B $ be an invariant symmetric bilinear form on $ \mathfrak g $( that is, $ B ( [ x , y ] , z ) =B ( x , [ y , z ] ) $ for all $ x , y , z \in \mathfrak g $) which is non-degenerate on a Cartan subalgebra $ \mathfrak g _ {0} \subset \mathfrak g $. Then a Casimir element of the Lie algebra $ \mathfrak g $ with respect to the form $ B $ is an element of the universal enveloping algebra $ U ( \mathfrak g ) $ that is representable in the form

$$ b = \sum _ { i=1 } ^ { k } e _ {i} f _ {i} . $$

Here $ \{ e _ {i} \} $, $ \{ f _ {i} \} $ are dual bases of $ \mathfrak g _ {0} $ with respect to $ B $, that is, $ B ( e _ {i} , f _ {i} ) = \delta _ {ij} $, $ i = 1 \dots k $, where $ \delta _ {ij} $ is the Kronecker symbol and $ k = \mathop{\rm dim} \mathfrak g _ {0} $. The element $ b $ does not depend on the choice of the dual bases in $ \mathfrak g _ {0} $ and belongs to the centre of $ U ( \mathfrak g _ {0} ) $. If $ \mathfrak g $ is a simple algebra, then a Casimir element of $ \mathfrak g $ defined by the Killing form $ B $ is the unique (up to a scalar multiplier) central element in $ U ( \mathfrak g ) $ that is representable as a homogeneous quadratic polynomial in the elements of $ \mathfrak g $.

Every linear representation $ \phi $ of a semi-simple algebra $ \mathfrak g $ in a finite-dimensional space $ V $ defines an invariant symmetric bilinear form

$$ B _ \phi ( x , y ) = \ \mathop{\rm Tr} ( \phi (x) \phi (y) ) $$

on $ \mathfrak g $, which is non-degenerate on the subalgebra $ \mathfrak g _ {0} \subset \mathfrak g $ complementary to $ \mathop{\rm ker} \phi $, and therefore also defines some Casimir element $ b _ \phi \in U ( \mathfrak g ) $. If $ \phi $ is an irreducible representation, then the extension of $ \phi $ onto $ U ( \mathfrak g ) $ takes $ b _ \phi $ into $ ( k / \mathop{\rm dim} V ) E $.

References

[1]	H. Casimir, B.L. van der Waerden, "Algebraischer Beweis der Vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen" Math. Ann. , 111 (1935) pp. 1–2
[2]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[5]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)
[6]	J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French)

Comments

The Casimir element $ b _ \phi $ determined by $ \phi $ is called the Casimir element of the linear representation $ \phi $.

An additional good reference is [a1].

References

[a1]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)