Representation theory of sl (2, C)

The representation theory of Lie algebra ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ is fundamental in mathematics and physics. In mathematics it is the simplest case in the classification of the representations of semi-simple Lie algebras , in physics it plays a central role in quantum mechanics because it classifies the representations of angular momentum algebra .

The Lie algebra

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$is the Lie algebra of - matrices with trace . It is spanned (as a complex vector space) by the matrices ${\displaystyle (2\times 2)}$ ${\displaystyle 0}$

${\displaystyle X={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\;\;\;\;Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\;\;\;\;\;\;\;H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$,

these satisfy the relations

${\displaystyle [X,Y]=H\;\;\;\;[H,X]=2X\;\;\;\;[H,Y]=-2Y}$

In quantum mechanics, the eigenvalues ​​of the angular momentum operator are calculated , where multiplication with the position coordinates and the derivation with respect to the position coordinates denote. Let be the three components of and , then and . After a suitable scaling of the basis vectors, the angular momentum algebra is isomorphic to . ${\displaystyle L={\frac {h}{i}}x\times \nabla }$ ${\displaystyle x}$ ${\displaystyle \nabla }$ ${\displaystyle L_{x},L_{y},L_{z}}$ ${\displaystyle L}$ ${\displaystyle L_{\pm }=L_{x}\pm iL_{y}}$ ${\displaystyle \left[L_{z},L_{\pm }\right]=\pm hL_{\pm }}$ ${\displaystyle \left[L_{+},L_{-}\right]=2hL_{z}}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

Finally dimensional representations

In the following we consider -linear representations, for the classification of -linear representations of , see representation theory of the Lorentz group . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

Because it is a semi-simple Lie algebra , its representations are completely reducible by Weyl's Theorem ; H. each representation can be decomposed as a direct sum of irreducible representations. It is therefore sufficient to classify irreducible representations . ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

Irreducible representations

It turns out that for every natural number there is an irreducible -dimensional representation of which is unique except for isomorphism . This is determined by a base with the following properties: ${\displaystyle m}$ ${\displaystyle (m+1)}$ ${\displaystyle V_{m}}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle \left\{v_{0},\ldots ,v_{m}\right\}}$

• ${\displaystyle hv_{i}=(m-2i)v_{i}}$ For ${\displaystyle i=0,\ldots ,m}$
• ${\displaystyle xv_{0}=0,yv_{m}=0}$
• ${\displaystyle xv_{i}=i(m-i+1)v_{i-1}}$ For ${\displaystyle i=1,\ldots ,m}$
• ${\displaystyle yv_{i}=v_{i+1}}$ For ${\displaystyle i=0,\ldots ,m-1}$

(Here the images from below denote the representation.) ${\displaystyle h,x,y\in {\mathfrak {gl}}(V_{m})\simeq {\mathfrak {gl}}(m+1,\mathbb {C} )}$ ${\displaystyle H,X,Y\in {\mathfrak {sl}}(2,\mathbb {C} )}$

proof

It is easy to calculate that the above properties clearly define a well-defined representation of . We now show that every irreducible representation is of the above form. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

It is an irreducible representation. Because algebraically closed, there is an eigenvector of , so . It then follows from that that is an eigenvector of to eigenvalue . By induction it follows that there is also an eigenvector to the eigenvalue . Because only a finite number of eigenvalues, there must be a minimum of give. Set and for . From it follows that the eigenvector of is to the eigenvalue . So there is again a minimal with and the vectors are linearly independent. From follows and with it the first assertion. The third claim follows by complete induction: ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle v}$ ${\displaystyle h}$ ${\displaystyle hv=\lambda v}$ ${\displaystyle \left[h,x\right]=2x}$ ${\displaystyle hxv=(\lambda +2)xv}$ ${\displaystyle xv}$ ${\displaystyle h}$ ${\displaystyle \lambda +2}$ ${\displaystyle x^{i}v}$ ${\displaystyle \lambda +2i}$ ${\displaystyle h}$ ${\displaystyle i_{0}}$ ${\displaystyle x^{i_{0}}v=0}$ ${\displaystyle v_{0}=x^{i_{0}-1}v}$ ${\displaystyle v_{i}=y^{i}v_{0}}$ ${\displaystyle i\geq 1}$ ${\displaystyle \left[h,y\right]=-2y}$ ${\displaystyle v_{i}}$ ${\displaystyle h}$ ${\displaystyle \lambda +2(i_{0}-i-1)}$ ${\displaystyle m}$ ${\displaystyle v_{m+1}=0}$ ${\displaystyle v_{0},\ldots ,v_{m}}$ ${\displaystyle \operatorname {Spur} (h)=\operatorname {Spur} (\left[x,y\right])=0}$ ${\displaystyle n=\lambda +2(i_{0}-1)}$

${\displaystyle xv_{i+1}=xyv_{i}=hv_{i}+yxv_{i}=(n-2)v_{i}+i(n-i+1)v_{i}=(i+1)(n-i)v_{i}}$.

Because the subspace spanned by is invariant, it must be whole because of the irreducibility of the representation . ${\displaystyle v_{0},\ldots ,v_{m}}$ ${\displaystyle V}$

Explicit description

The -dimensional representation of can be specified explicitly through ${\displaystyle (m+1)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

${\displaystyle \left({\begin{array}{cc}1&0\\0&-1\end{array}}\right)\mapsto \operatorname {diag} (m,m-2,\ldots ,-m)}$,

${\displaystyle \left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\mapsto \operatorname {diag} ^{+}(m,m-1,\ldots ,1)}$,

${\displaystyle \left({\begin{array}{cc}0&0\\1&0\end{array}}\right)\mapsto \operatorname {diag} ^{-}(1,2,\ldots ,m)}$,

where or denotes those matrices whose first above or below diagonal is and whose other entries are zero. ${\displaystyle \operatorname {diag} ^{+}(v)}$ ${\displaystyle \operatorname {diag} ^{-}(v)}$ ${\displaystyle v}$

For example, the trivial representation, the canonical representation of on, and the adjoint representation . ${\displaystyle V_{0}}$ ${\displaystyle V_{1}}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle V_{2}}$

Representations of the Lie group SL (2, C)

According to Lie's second theorem , the representations of the Lie algebra correspond to the representations of the Lie group . ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle SL(2,\mathbb {C} )}$

An explicit description of the -dimensional representation of goes as follows. Let it be the vector space of the complex-valued homogeneous polynomials of degree in two variables, i.e. the complex vector space spanned by. works through . That defines a representation ${\displaystyle (m+1)}$ ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle V_{m}}$ ${\displaystyle m}$ ${\displaystyle x^{m},x^{m-1}y,\ldots ,xy^{m-1},y^{m}}$ ${\displaystyle A\in SL(2,\mathbb {C} )}$ ${\displaystyle V_{m}}$ ${\displaystyle (AP)(x,y):=P(A^{-1}(x,y))}$

${\displaystyle SL(2,\mathbb {C} )\to GL(V_{m})\simeq GL(m+1,\mathbb {C} )}$,

their differential in the single element the representation constructed above

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\to {\mathfrak {gl}}(V_{m})\simeq {\mathfrak {gl}}(m+1,\mathbb {C} )}$

is.

Clebsch-Gordan's theorem

→ Main article : Clebsch-Gordan coefficient

The tensor product of two representations is again a representation of , which can then be broken down into its irreducible summands. The Clebsch-Gordan theorem says in the case of that ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle SL(2,\mathbb {C} )}$

${\displaystyle V_{m}\otimes V_{n}\simeq V_{m+n}\oplus V_{m+n-2}\oplus \ldots \oplus V_{m-n+2}\oplus V_{m-n}}$

holds for all natural numbers . ${\displaystyle m\geq n}$

The Clebsch-Gordan coefficients are used in the coupling of quantum mechanical angular momentum . These are expansion coefficients with which one goes from the basis of the individual angular momentum to the basis of the total angular momentum. They are used to calculate the spin-orbit coupling and in the isospin formalism.

Highest weight

Representations of semi-simple Lie algebras are classified by their highest weight . For representations of , the highest weight is the greatest eigenvalue of . The -dimensional spin representation has the highest weight . ${\displaystyle \pi }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle \pi \left({\begin{array}{cc}1&0\\0&-1\end{array}}\right)}$ ${\displaystyle (m+1)}$ ${\displaystyle -{\frac {m}{2}}-}$ ${\displaystyle m}$

See also

• Representation (Lie algebra)
• Representation theory of the Lorentz group

literature

• Serre, Jean-Pierre: Complex semisimple Lie algebras. Translated from the French by GA Jones. Reprint of the 1987 edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001. ISBN 3-540-67827-1
• Humphreys, James E .: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972.
• Hilgert, Joachim; Neeb, Karl-Hermann: Lie groups and Lie algebras , Vieweg + Teubner Verlag, Wiesbaden, 1991
• Hall, Brian C .: Lie groups, Lie algebras, and representations. An elementary introduction. Graduate Texts in Mathematics, 222. Springer-Verlag, New York, 2003. ISBN 0-387-40122-9
• Erdmann, Karin; Wildon, Mark J .: Introduction to Lie algebras. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2006. ISBN 978-1-84628-040-5 ; 1-84628-040-0
• Gilmore, Robert: Lie groups, physics, and geometry. An introduction for physicists, engineers and chemists. Cambridge University Press, Cambridge, 2008. ISBN 978-0-521-88400-6
• Mazorchuk, Volodymyr: Lectures on sl2 (C) modules. Imperial College Press, London, 2010. ISBN 978-1-84816-517-5 ; 1-84816-517-X
• Henderson, Anthony: Representations of Lie algebras. An introduction through . ${\displaystyle {\mathfrak {gl}}_{n}}$Australian Mathematical Society Lecture Series, 22nd Cambridge University Press, Cambridge, 2012. ISBN 978-1-107-65361-0

Web links

• Wolfgang Ziller: Lie Groups, representation theory and symmetric spaces (Chapter 5)