Casimir element




distinguished element of a Lie algebra's center

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.


The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.[1]




Contents






  • 1 Definition


    • 1.1 Quadratic Casimir element


    • 1.2 Casimir invariant of a linear representation and of a smooth action


    • 1.3 General case




  • 2 Properties


    • 2.1 Uniqueness


    • 2.2 Relation to the Laplacian on G


    • 2.3 Generalizations




  • 3 Example: so(3){displaystyle {mathfrak {so}}(3)}{mathfrak {so}}(3)


  • 4 Eigenvalues


  • 5 See also


  • 6 References


  • 7 Further reading





Definition


The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order; their definition is given last.



Quadratic Casimir element


Suppose that g{displaystyle {mathfrak {g}}}{mathfrak {g}} is an n{displaystyle n}n-dimensional semisimple Lie algebra. Let B be a nondegenerate bilinear form on g{displaystyle {mathfrak {g}}}{mathfrak {g}} that is invariant under the adjoint action of g{displaystyle {mathfrak {g}}}{mathfrak {g}} on itself, meaning that B(adXY,Z)+B(Y,adXZ)=0{displaystyle B(ad_{X}Y,Z)+B(Y,ad_{X}Z)=0}B(ad_XY,Z)+B(Y,ad_XZ)=0 for all X,Y,Z in g{displaystyle {mathfrak {g}}}{mathfrak {g}}. (The most typical choice of B is the Killing form.)
Let


{Xi}i=1n{displaystyle {X_{i}}_{i=1}^{n}}{X_i}_{i=1}^n

be any basis of g{displaystyle {mathfrak {g}}}{mathfrak {g}}, and


{Xi}i=1n{displaystyle {X^{i}}_{i=1}^{n}}{X^i}_{i=1}^n

be the dual basis of g{displaystyle {mathfrak {g}}}{mathfrak {g}} with respect to B. The Casimir element Ω{displaystyle Omega }Omega for B is the element of the universal enveloping algebra U(g){displaystyle U({mathfrak {g}})}U({mathfrak {g}}) given by the formula


Ω=∑i=1nXiXi.{displaystyle Omega =sum _{i=1}^{n}X_{i}X^{i}.}Omega = sum_{i=1}^n X_i X^i.

Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that Ω is independent of this choice. On the other hand, Ω does depend on the bilinear form B. The invariance of B implies that the Casimir element commutes with all elements of the Lie algebra g{displaystyle {mathfrak {g}}}{mathfrak {g}}, and hence lies in the center of the universal enveloping algebra U(g){displaystyle U({mathfrak {g}})}U({mathfrak {g}}).[2]



Casimir invariant of a linear representation and of a smooth action


Given a representation ρ of g{displaystyle {mathfrak {g}}}{mathfrak {g}} on a vector space V, possibly infinite-dimensional, the Casimir invariant of ρ is defined to be ρ(Ω), the linear operator on V given by the formula


ρ(Ω)=∑i=1nρ(Xi)ρ(Xi).{displaystyle rho (Omega )=sum _{i=1}^{n}rho (X_{i})rho (X^{i}).}rho(Omega) = sum_{i=1}^n rho(X_i)rho(X^i).

Here we are assuming that B is the Killing form, otherwise B must be specified.


A specific form of this construction plays an important role in differential geometry and global analysis. Suppose that a connected Lie group G with Lie algebra g{displaystyle {mathfrak {g}}}{mathfrak {g}} acts on a differentiable manifold M. Consider the corresponding representation ρ of G on the space of smooth functions on M. Then elements of g{displaystyle {mathfrak {g}}}{mathfrak {g}} are represented by first order differential operators on M. In this situation, the Casimir invariant of ρ is the G-invariant second order differential operator on M defined by the above formula.


Specializing further, if it happens that M has a Riemannian metric on which G acts transitively by isometries, and the stabilizer subgroup Gx of a point acts irreducibly on the tangent space of M at x, then the Casimir invariant of ρ is a scalar multiple of the Laplacian operator coming from the metric.


More general Casimir invariants may also be defined, commonly occurring in the study of pseudo-differential operators in Fredholm theory.



General case


The article on universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties. In particular, all Casimir operators correspond to symmetric homogeneous polynomials in the symmetric algebra of the adjoint representation adg.{displaystyle {mbox{ad}}_{mathfrak {g}}.}{displaystyle {mbox{ad}}_{mathfrak {g}}.} That is, in general, one has that any Casimir operator will have the form


C(m)=κij⋯kXi⊗Xj⊗⋯⊗Xk{displaystyle C_{(m)}=kappa ^{ijcdots k}X_{i}otimes X_{j}otimes cdots otimes X_{k}}{displaystyle C_{(m)}=kappa ^{ijcdots k}X_{i}otimes X_{j}otimes cdots otimes X_{k}}

where m is the order of the symmetric tensor κij⋯k{displaystyle kappa ^{ijcdots k}}{displaystyle kappa ^{ijcdots k}} and the Xi{displaystyle X_{i}}X_{i} form a vector space basis of g.{displaystyle {mathfrak {g}}.}{mathfrak {g}}. This corresponds to a symmetric homogeneous polynomial


c(m)=κij⋯ktitj⋯tk{displaystyle c_{(m)}=kappa ^{ijcdots k}t_{i}t_{j}cdots t_{k}}{displaystyle c_{(m)}=kappa ^{ijcdots k}t_{i}t_{j}cdots t_{k}}

in m indeterminate variables ti{displaystyle t_{i}}t_{i} in the polynomial algebra K[ti,tj,⋯,tk]{displaystyle K[t_{i},t_{j},cdots ,t_{k}]}{displaystyle K[t_{i},t_{j},cdots ,t_{k}]} over a field K. The reason for the symmetry follows from the PBW theorem and is discussed in much greater detail in the article on universal enveloping algebras.


Not just any symmetric tensor (symmetric homogeneous polynomial) will do; it must explicitly commute with the Lie bracket. That is, one must have that


[C(m),Xi]=0{displaystyle [C_{(m)},X_{i}]=0}{displaystyle [C_{(m)},X_{i}]=0}

for all basis elements Xi.{displaystyle X_{i}.}{displaystyle X_{i}.} Any proposed symmetric polynomial can be explicitly checked, making use of the structure constants


[Xi,Xj]=fijkXk{displaystyle [X_{i},X_{j}]=f_{ij}^{;;k}X_{k}}{displaystyle [X_{i},X_{j}]=f_{ij}^{;;k}X_{k}}

in order to obtain


fijkκjl⋯m+fijlκkj⋯m+⋯+fijmκkl⋯j=0{displaystyle f_{ij}^{;;k}kappa ^{jlcdots m}+f_{ij}^{;;l}kappa ^{kjcdots m}+cdots +f_{ij}^{;;m}kappa ^{klcdots j}=0}{displaystyle f_{ij}^{;;k}kappa ^{jlcdots m}+f_{ij}^{;;l}kappa ^{kjcdots m}+cdots +f_{ij}^{;;m}kappa ^{klcdots j}=0}

This result is originally due to Israel Gelfand.[3] The commutation relation implies that the Casimir operators lie in the center of the universal enveloping algebra, and, in particular, always commute with any element of the Lie algebra. It is due to this property of commutation that allows a representation of a Lie algebra to be labelled by eigenvalues of the associated Casimir operators.


Note also that any linear combination of the symmetric polynomials described above will lie in the center as well: therefore, the Casimir operators are, by definition, restricted to that subset that span this space (that provide a basis for this space). For a semisimple Lie algebra of rank r, there will be r Casimir invariants.



Properties



Uniqueness


Since for a simple lie algebra every invariant bilinear form is a multiple of the Killing form, the corresponding Casimir element is uniquely defined up to a constant. For a general semisimple Lie algebra, the space of invariant bilinear forms has one basis vector for each simple component, and hence the same is true for the space of corresponding Casimir operators.



Relation to the Laplacian on G


If G{displaystyle G}G is a Lie group with Lie algebra g{displaystyle {mathfrak {g}}}{mathfrak {g}}, the choice of an invariant bilinear form on g{displaystyle {mathfrak {g}}}{mathfrak {g}} corresponds to a choice of bi-invariant Riemannian metric on G{displaystyle G}G. Then under the identification of the universal enveloping algebra of g{displaystyle {mathfrak {g}}}{mathfrak {g}} with the left invariant differential operators on G{displaystyle G}G, the Casimir element of the bilinear form on g{displaystyle {mathfrak {g}}}{mathfrak {g}} maps to the Laplacian of G{displaystyle G}G (with respect to the corresponding bi-invariant metric).



Generalizations


The Casimir operator is a distinguished quadratic element of the center of the universal enveloping algebra of the Lie algebra. In other words, it is a member of the algebra of all differential operators that commutes with all the generators in the Lie algebra. In fact all quadratic elements in the center of the universal enveloping algebra arise this way.
However, the center may contain other, non-quadratic, elements.


By Racah's theorem,[4] for a semisimple Lie algebra the dimension of the center of the universal enveloping algebra is equal to its rank. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but this way of counting shows that there may be no unique analogue of the Laplacian, for rank > 1.


By definition any member of the center of the universal enveloping algebra commutes with all other elements in the algebra. By Schur's Lemma, in any irreducible representation of the Lie algebra, the Casimir operator is thus proportional to the identity. This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group). Physical mass and spin are examples of these constants, as are many other quantum numbers found in quantum mechanics. Superficially, topological quantum numbers form an exception to this pattern; although deeper theories hint that these are two facets of the same phenomenon.[according to whom?].



Example: so(3){displaystyle {mathfrak {so}}(3)}{mathfrak {so}}(3)


The Lie algebra so(3){displaystyle {mathfrak {so}}(3)}{mathfrak {so}}(3) is the Lie algebra of SO(3), the rotation group for three-dimensional Euclidean space. It is simple of rank 1, and so it has a single independent Casimir. The Killing form for the rotation group is just the Kronecker delta, and so the Casimir invariant is simply the sum of the squares of the generators Lx,Ly,Lz{displaystyle L_{x},,L_{y},,L_{z}}L_x,, L_y,, L_z of the algebra. That is, the Casimir invariant is given by


L2=Lx2+Ly2+Lz2.{displaystyle L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}.}L^2=L_x^2+L_y^2+L_z^2.

Consider the irreducible representation of so(3){displaystyle {mathfrak {so}}(3)}{mathfrak {so}}(3) in which the largest eigenvalue of Lz{displaystyle L_{z}}L_{z} is ℓ{displaystyle ell }ell , where the possible values of ℓ{displaystyle ell }ell are 0,1/2,1,3/2,…{displaystyle 0,1/2,1,3/2,ldots }{displaystyle 0,1/2,1,3/2,ldots }. The invariance of the Casimir operator implies that it is a multiple of the identity operator I. This constant can be computed explicitly, giving the following result[5]


L2=Lx2+Ly2+Lz2=ℓ(ℓ+1)I.{displaystyle L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=ell (ell +1)I.}{displaystyle L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=ell (ell +1)I.}

In quantum mechanics, the scalar value ℓ{displaystyle ell }ell is referred to as the total angular momentum. For finite-dimensional matrix-valued representations of the rotation group, ℓ{displaystyle ell }ell always takes on integer values (for bosonic representations) or half-integer values (for fermionic representations).


For a given value of ℓ{displaystyle ell }ell , the matrix representation is (2ℓ+1){displaystyle (2ell +1)}(2ell+1)-dimensional. Thus, for example, the three-dimensional representation for so(3){displaystyle {mathfrak {so}}(3)}{mathfrak {so}}(3) corresponds to ℓ=1{displaystyle ell ,=,1}ell,=,1, and is given by the generators


Lx=i(00000−1010);Ly=i(001000−100);Lz=i(0−10100000),{displaystyle L_{x}=i{begin{pmatrix}0&0&0\0&0&-1\0&1&0end{pmatrix}};quad L_{y}=i{begin{pmatrix}0&0&1\0&0&0\-1&0&0end{pmatrix}};quad L_{z}=i{begin{pmatrix}0&-1&0\1&0&0\0&0&0end{pmatrix}},}{displaystyle L_{x}=i{begin{pmatrix}0&0&0\0&0&-1\0&1&0end{pmatrix}};quad L_{y}=i{begin{pmatrix}0&0&1\0&0&0\-1&0&0end{pmatrix}};quad L_{z}=i{begin{pmatrix}0&-1&0\1&0&0\0&0&0end{pmatrix}},}

where the factors of i{displaystyle i}i are needed for agreement with the physics convention (used here) that the generators should be self-adjoint operators.


The quadratic Casimir invariant can then easily be computed by hand, with the result that


L2=Lx2+Ly2+Lz2=2(100010001){displaystyle L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=2{begin{pmatrix}1&0&0\0&1&0\0&0&1end{pmatrix}}}L^2=L_x^2+L_y^2+L_z^2= 2<br /> begin{pmatrix}<br /> 1& 0& 0\<br /> 0& 1& 0\<br /> 0& 0& 1<br /> end{pmatrix}

as ℓ(ℓ+1)=2{displaystyle ell (ell +1),=,2}ell(ell+1),=,2 when ℓ=1{displaystyle ell ,=,1}ell,=,1. Similarly, the two dimensional representation has a basis given by the Pauli matrices, which correspond to spin 1/2, and one can again check the formula for the Casimir by direct computation.



Eigenvalues


Given that Ω{displaystyle Omega }Omega is central in the enveloping algebra, it acts on simple modules by a scalar. Let ⟨,⟩{displaystyle langle ,rangle }langle ,rangle be any bilinear symmetric non-degenerate form, by which we define Ω{displaystyle Omega }Omega . Let L(λ){displaystyle L(lambda )}L(lambda) be the finite dimensional highest weight module of weight λ{displaystyle lambda }lambda . Then the Casimir element Ω{displaystyle Omega }Omega acts on L(λ){displaystyle L(lambda )}L(lambda) by the constant


⟨λ,λ+2ρ⟩=⟨λ+ρ,λ+ρ⟩−⟨ρ,ρ⟩,{displaystyle langle lambda ,lambda +2rho rangle =langle lambda +rho ,lambda +rho rangle -langle rho ,rho rangle ,}{displaystyle langle lambda ,lambda +2rho rangle =langle lambda +rho ,lambda +rho rangle -langle rho ,rho rangle ,}

where ρ{displaystyle rho }rho is the weight defined by half the sum of the positive roots.[6]


An important point is that if L(λ){displaystyle L(lambda )}L(lambda) is nontrivial (i.e. if λ≠0{displaystyle lambda neq 0}{displaystyle lambda neq 0}), then the above constant is nonzero. After all, since λ{displaystyle lambda }lambda is dominant, if λ≠0{displaystyle lambda neq 0}{displaystyle lambda neq 0}, then ⟨λ,λ⟩>0{displaystyle langle lambda ,lambda rangle >0}{displaystyle langle lambda ,lambda rangle >0} and ⟨λ,ρ⟩≥0{displaystyle langle lambda ,rho rangle geq 0}{displaystyle langle lambda ,rho rangle geq 0}, showing that ⟨λ,λ+2ρ⟩>0{displaystyle langle lambda ,lambda +2rho rangle >0}{displaystyle langle lambda ,lambda +2rho rangle >0}. This observation plays an important role in the proof of Weyl's theorem on complete reducibility. It is also possible to prove the nonvanishing of the eigenvalue in a more abstract way—without using an explicit formula for the eigenvalue—using Cartan's criterion; see Sections 4.3 and 6.2 in the book of Humphreys.



See also



  • Harish-Chandra isomorphism

  • Pauli–Lubanski pseudovector



References





  1. ^ Oliver, David (2004). The shaggy steed of physics: mathematical beauty in the physical world. Springer. p. 81. ISBN 978-0-387-40307-6..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Hall 2015 Proposition 10.5


  3. ^ Xavier Bekaert, "Universal enveloping algebras and some applications in physics" (2005) Lecture, Modave Summer School in Mathematical Physics.


  4. ^ Racah, Giulio (1965). Group theory and spectroscopy. Springer Berlin Heidelberg.


  5. ^ Hall 2013 Proposition 17.8


  6. ^ Hall 2015 Proposition 10.6





  • Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer


  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666



Further reading




  • Humphreys, James E. (1978). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9 (Second printing, revised ed.). New York: Springer-Verlag. ISBN 0-387-90053-5.


  • Jacobson, Nathan (1979). Lie algebras. Dover Publications. pp. 243–249. ISBN 0-486-63832-4.




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