Casimir operator

The Casimir operator (also Casimir invariant , named after the physicist Hendrik Casimir ) is investigated in the mathematical sub-area of algebra and differential geometry . It is a special element from the center of the universal enveloping algebra of a Lie algebra . A typical example is the squared angular momentum operator , which is a Casimir invariant of the three-dimensional rotation group .

definition

Assume is a -dimensional semi-simple Lie algebra . Be ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle n}$

${\displaystyle \{X_{i}\}_{i=1}^{n}}$

any basis of and ${\displaystyle {\mathfrak {g}}}$

${\displaystyle \{X^{i}\}_{i=1}^{n}}$

be the dual basis of with respect to a fixed invariant bilinear form (e.g. the killing form ) . The square Casimir element is that by the formula ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \Omega }$

${\displaystyle \Omega =\sum _{i=1}^{n}X_{i}X^{i}}$

given element of universal enveloping algebra . Although the definition of the Casimir element refers to the direct choice of a basis in Lie algebra, it is easy to show that the element generated is independent of it. Furthermore, the invariance of the bilinear form, which was used in the definition, implies that the Casimir element commutes with all elements of Lie algebra and is therefore at the center of the universal enveloping algebra . ${\displaystyle U({\mathfrak {g}})}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle U({\mathfrak {g}})}$

Let be an arbitrary representation of Lie's algebra on a (possibly infinite-dimensional) vector space . Then the corresponding quadratic Casimir invariant is through ${\displaystyle \rho }$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle V}$ ${\displaystyle \rho (\Omega )}$

${\displaystyle \rho (\Omega )=\sum _{i=1}^{n}\rho (X_{i})\rho (X^{i})}$

given linear operator on . ${\displaystyle V}$

Applications

A special case of this construction plays an important role in differential geometry or global analysis . If a connected Lie group with an associated Lie algebra operates on a differentiable manifold , then the elements of are described by differential operators of the first order on . Let the representation be based on the space of the smooth functions . In this case the Casimir invariant given by the above formula is the -invariant differential operator of the second order on . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle M}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle M}$ ${\displaystyle \rho }$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle M}$

Even more general Casimir invariants can be defined; this happens, for example, when investigating pseudo-differential operators in Fredholm's theory.

literature

• James E. Humphreys: Introduction to Lie Algebras and Representation Theory , 2nd revised edition, Graduate Texts in Mathematics, 9th Springer-Verlag, New York, 1978. ISBN 0-387-90053-5