Orthogonal Overview

By | 2017-12-11

Overview

Orthogonal Group is a symmetry group of a vector space equipped with a (non-degenerate) scalar product x\cdot y that is a group of linear transformations a such that a(x)\cdot a(y) = x\cdot y. If we present a as a matrix with respect to some vector basis then a is orthogonal when its coefficients satisfy certain algebraic equations which depend on how the basic vectors were selected. For example, if these were ortho-normal then the orthogonality condition can be written as ^{t}a a = 1 which simply states that the columns of a form an ortho-normal basis. Applying the log to ^{t}a = a^{-1} we obtain an infinitesimal variant of this relation ^{t}A = -A (known as anti-symmetry property). In general, if g is a matrix defined by scalar products between basic vectors (invertible by non-degenerate assumption) then the matrix of scalar products between column vectors of a is the matrix ^{t}aga. The orthogonality condition states that this matrix coincides with g. This can be written as g^{-1}^{t}a g = a^{-1} and the corresponding infinitesimal condition as g^{-1}^{t}A g = -A.

Related Root Systems

This is a simple Lie algebra of rank \ell = dime(V)/2 and type D_\ell when dim(V) is even and of rank \ell = (dime(V)-1)/2 and type B_\ell when dim(V) is odd. Recall the following standard representation of a root system D_\ell in terms of an orthonormal basis \{\epsilon_1,\cdots,\epsilon_\ell\}.
\{\pm\epsilon_i \pm\epsilon_j|1\leq i \neq j\leq\ell\}
where simple roots are defined by

\alpha_1=\epsilon_1-\epsilon_2 \alpha_2=\epsilon_2-\epsilon_3 \cdots \alpha_{\ell-2}=\epsilon_{\ell-2}-\epsilon_{\ell-1} \alpha_{\ell-1}=\epsilon_{\ell-1}-\epsilon_\ell
\alpha_{\ell}=\epsilon_{\ell-1}+\epsilon_\ell

Root Space Decomposition for Witt Basis – Even Dimensions

The relation between orthogonal Lie algebra and its root system is clear if instead of ortho-normal we use a Witt basis. The resulting root system will be of type B in odd-dimensions and of type D in even dimensions. In the even-dimensional case this basis is present if the vector space decomposes into a direct sum V \oplus \bar{V} of isotropic spaces. The Witt basis is then (e_1,\cdots,e_\ell,e_{\bar{\ell}},\cdots,e_{\bar{1}}) where scalar products are 0 for all pairs except e_i\cdot e_{\bar{i}} =1. In this case g=g^{-1} is the matrix with 1’s on the co-diagonal and 0’s elsewhere and g ^{t}a g is the reflection of a with respect to the co-diagonal. We have a Cartan algebra consisting of diagonal matrices with the above symmetry restriction. This algebra has a convenient basis \{H_i = E_{i,i} - E_{\bar{i},\bar{i}}\}. Its dual basis \{\epsilon_i\} is then an orthonormal basis used in the standard representation of D_\ell above. The root-space decomposition is given by

X_{\epsilon_i - \epsilon_j} = E_{i,j} - E_{\bar{j},\bar{i}}
X_{\epsilon_i + \epsilon_j} = E_{i,\bar{j}} - E_{j,\bar{i}}
X_{-\epsilon_i - \epsilon_j} = E_{\bar{i},j} - E_{\bar{j},i}

where 1\leq i < j\leq\ell.

Spinorial Representation

The name ‘Orthogonal Group’ comes form traditional emphasis on the metric-preserving representation. From a purely group-theoretic point of view, it is just one of the fundamental representations. In this section we will examine another fundamental representation of the ‘Orthogonal Group’ not related to orthogonality. The underlying vector space will be the exterior algebra \bigwedge(V). The transformations will be specified by providing a set of vector-space generators. Namely, composites of 2 members where each is either an anti-derivation related to an element of V^* or an exterior multiplication by an element of V. This representation decomposes into a sum of 2 fundamental representations \bigwedge(V)=\bigoplus_{i-even}\bigwedge^i(V)\oplus\bigoplus_{i-odd}\bigwedge^i(V).
A convenient Cartan-subalgebra basis is given by \{H_i = e_i\imath_i - \frac{1}{2}\}. Its dual basis \{\epsilon_i\} is then an orthonormal basis used in the standard representation of D_\ell above. The root-space decomposition is given by

X_{\epsilon_i - \epsilon_j} = e_i\imath_j
X_{\epsilon_i + \epsilon_j} = e_i e_j
X_{-\epsilon_i - \epsilon_j} = \imath_i\imath_j

where 1\leq i < j\leq\ell.

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