Spinor representation

spin representation

The simplest faithful linear representation (cf. Faithful representation; Linear representation) of the spinor group $\mathop{\rm Spin} _ {n} ( Q)$, or the linear representation of the corresponding even Clifford algebra $C ^ {+} = C ^ {+} ( Q)$( see Spinor group; $Q$ is a quadratic form). If the ground field $K$ is algebraically closed, then the algebra $C ^ {+}$ is isomorphic to the complete matrix algebra $M _ {2 ^ {m} } ( K)$( where $n = 2 m + 1$) or to the algebra $M _ {2 ^ {m-1} } ( K) \oplus M _ {2 ^ {m-1} } ( K)$( where $n = 2 m$). Therefore there is defined a linear representation $\rho$ of the algebra $C ^ {+}$ on the space of dimension $2 ^ {m}$ over $K$; this representation is called a spinor representation. The restriction of $\rho$ to $\mathop{\rm Spin} _ {n} ( Q)$ is called the spinor representation of $\mathop{\rm Spin} _ {n} ( Q)$. For odd $n$, the spinor representation is irreducible, and for even $n$ it splits into the direct sum of two non-equivalent irreducible representations $\rho ^ \prime$ and $\rho ^ {\prime\prime}$, which are called half-spin (or) representations. The elements of the space of the spinor representation are called spinors, and those of the space of the half-spinor representation half-spinors. The spinor representation of the spinor group $\mathop{\rm Spin} _ {n}$ is self-dual for any $n \geq 3$, whereas the half-spinor representations $\rho ^ \prime$ and $\rho ^ {\prime\prime}$ of the spinor group $\mathop{\rm Spin} _ {2m}$ are self-dual for even $m$ and dual to one another for odd $m$. The spinor representation of $\mathop{\rm Spin} _ {n}$ is faithful for all $n \geq 3$, while the half-spinor representations of $\mathop{\rm Spin} _ {2m}$ are faithful for odd $m$, but have a kernel of order two when $m$ is even.

For a quadratic form $Q$ on a space $V$ over some subfield $k \subset K$, the spinor representation is not always defined over $k$. However, if the Witt index of $Q$ is maximal, that is, equal to $[ n / 2 ]$( in particular, if $k$ is algebraically closed), then the spinor and half-spinor representations are defined over $k$. In this case these representations can be described in the following way if $\mathop{\rm char} k \neq 2$( see [1]). Let $L$ and $M$ be $k$- subspaces of the $k$- space $V$ that are maximal totally isotropic (with respect to the symmetric bilinear form on $V$ associated with $Q$) and let $L \cap M = 0$. Let $C _ {L}$ be the subalgebra of the Clifford algebra $C = C ( Q)$ generated by the subspace $L \subset V$, and let $e _ {M} \in C$ be the product of $m$ vectors forming a $k$- basis of $M$. If $n$ is even, $n = 2m$, then the spinor representation is realized in the left ideal $C e _ {M}$ and acts there by left translation: $\rho ( s) x = s x$( $s \in C ^ {+}$, $x \in C e _ {M}$). Furthermore, the mapping $x \mapsto x e _ {M}$ defines an isomorphism of vector spaces $C _ {L} \rightarrow C e _ {M}$ that enables one to realize the spinor representation in $C _ {L}$, which is naturally isomorphic to the exterior algebra over $L$. The half-spinor representations $\rho ^ \prime$ and $\rho ^ {\prime\prime}$ are realized in the $2 ^ {m-1}$- dimensional subspaces $C _ {L} \cap C ^ {+}$ and $C _ {L} \cap C ^ {-}$.

If $n$ is odd, then $V$ can be imbedded in the $( n + 1 )$- dimensional vector space $V _ {1} = V \oplus k \epsilon$ over $k$. One defines a quadratic form $Q _ {1}$ on $V _ {1}$ by putting $Q _ {1} ( v + \epsilon ) = Q ( v) - \lambda ^ {2}$ for all $v \in V$ and $\lambda \in k$. $Q _ {1}$ is a non-degenerate quadratic form of maximal Witt index defined over $k$ on the even-dimensional vector space $V _ {1}$. The spinor representation of $C ^ {+} ( Q)$( or of $\mathop{\rm Spin} _ {n} ( Q)$) is obtained by restricting any of the half-spinor representations of $C ^ {+} ( Q _ {1} )$( or of $\mathop{\rm Spin} _ {n+} 1 ( Q _ {1} )$) to the subalgebra $C ^ {+} ( Q)$( or $\mathop{\rm Spin} _ {n} ( Q)$, respectively).

The problem of classifying spinors has been solved when $3 \leq n \leq 14$ and $k$ is an algebraically closed field of characteristic 0 (see [4], [8], [9]). The problem consists of the following: 1) describe the orbits of $\rho ( \mathop{\rm Spin} _ {n} )$ in the spinor space by giving a representative of each orbit; 2) calculate the stabilizers in $\mathop{\rm Spin} _ {n}$ of each of these representatives; and 3) describe the algebra of invariants of the linear group $\rho ( \mathop{\rm Spin} _ {n} )$.

The existence of spinor and half-spinor representations of the Lie algebra $\mathfrak s \mathfrak p _ {n}$ of $\mathop{\rm Spin} _ {n}$ was discovered by E. Cartan in 1913, when he classified the finite-dimensional representations of simple Lie algebras [6]. In 1935, R. Brauer and H. Weyl described spinor and half-spinor representations in terms of Clifford algebras [5]. P. Dirac [3] showed how spinors could be used in quantum mechanics to describe the rotation of an electron.

References

[1]	N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1970) pp. Chapt. II. Algèbre linéaire MR0274237 Zbl 0211.02401
[2]	H. Weyl, "Classical groups, their invariants and representations" , Princeton Univ. Press (1946) (Translated from German) MR0000255 Zbl 1024.20502
[3]	P.A.M. Dirac, "Principles of quantum mechanics" , Clarendon Press (1958) MR2303789 MR0116921 MR0023198 MR1522388 Zbl 0080.22005
[4]	V.L. Popov, "Classification of spinors of dimension fourteen" Trans. Moscow Math. Soc. , 1 (1980) pp. 181–232 Trudy Moskov. Mat. Obshch. , 37 (1978) pp. 173–217 MR0514331 Zbl 0443.20038
[5]	R. Brauer, H. Weyl, "Spinors in -dimensions" Amer. J. Math. , 57 : 2 (1935) pp. 425–449 MR1507084
[6]	E. Cartan, "Les groupes projectifs qui ne laissant invariante aucune multiplicité plane" Bull. Soc. Math. France , 41 (1913) pp. 53–96
[7]	C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) MR0060497 Zbl 0057.25901
[8]	V. Gatti, E. Viniberghi, "Spinors in -dimensional space" Adv. Math. , 30 : 2 (1978) pp. 137–155
[9]	J.I. Igusa, "A classification of spinors up to dimension twelve" Amer. J. Math. , 92 : 4 (1970) pp. 997–1028 MR0277558 Zbl 0217.36203