# Casimir element

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)

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