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EmbeddingUXU

Dimensions of {$O(n),U(n),Sp(n)$}

Consider the compact Lie groups: orthogonal group {$O(n)$}, unitary group {$U(n)$}, compact symplectic group {$Sp(n)$}.

Look at their dimensions to gain some understanding about the chains of Lie group embeddings:

{$$U(2r)\supset U(r)\times U(r) \supset U(r)$$}

This manifests 2-fold complex Bott periodicity.

{$$O(16r)\supset U(8r)\supset Sp(4r)\supset Sp(2r)\times Sp(2r)\supset Sp(2r)\supset U(2r) \supset O(2r)\supset O(r)\times O(r) \supset O(r)$$}

This manifests 8-fold real Bott periodicity.

How do we generate these chains? Consider a linear complex structure {$J$}. In some basis, it can be written

{$$J=\begin{pmatrix}0 & -1 & & & \\ 1 & 0 & & 0 & \\ & & \ddots & & \\ & 0 & & 0 & -1 \\ & & & 1 & 0\end{pmatrix}$$}

Then the 2-fold Lie group embedding is generated by retaining what commutes with {$iJ_1$} and then a mutually anticommuting {$iJ_2$}, that is, {$(iJ_1)(iJ_2)=-(iJ_2)(iJ_1)$} or simply {$J_1J_2=-J_2J_1$}.

The 8-fold Lie group embedding is generated by retaining what commutes with mutually anticommuting {$J_1,J_2,\dots J_k$}.

When we have an operator {$K^2=I$}, our vector space {$V=V_+\oplus V_-$}

Note that {$J_1^2=–I$}. Whereas {$(iJ_1)^2=i^2J_1^2=(-1)(-1)=+1$}. The operator {$K=iJ_1$} satisfies the equation {$K^2-I=0$}.

If {$\lambda$} is an eigenvalue of {$K$} with eigenvector {$v$}, then {$v=K^2v=\lambda^2v$}, thus {$\lambda =\pm 1$}.

This means that the vector space {$V=V_+\oplus V_-$} where {$Kv=v$} for {$v\in V_+$} and {$Kv=-v$} for {$v\in V_-$}.

Suppose {$A\in U(2n)$} and {$V_+$} and {$V_-$} are both n-dimensional. What can we conclude if {$iJ$} and {$A$} commute?

If {$v_+\in V_+$}, then {$iJAv_+=AiJv_+=Av_+$}. This implies {$A_v\in V_+$}.

If {$v_-\in V_-$}, then {$iJAv_-=AiJv_-=-Av_-$}. This implies {$A_v\in V_-$}.

Consequently, {$A\in U(n)\times U(n)$}. We have {$U(2n)\supset U(n)\times U(n)$}.

Similarly, consider {$K=J_1J_2J_3$}, where {$J_1J_2=-J_2J_1, J_1J_3=-J_3J_1, J_2J_3=-J_3J_2$}.

Then {$(J_1J_2J_3)(J_1J_2J_3) = J_1^2J_2J_3J_2J_3=-J_1^2J_2^2J_3^2=+1$}.

This yields {$O(2n)\supset O(n)\times O(n)$} and {$Sp(2n)\supset Sp(n)\times Sp(n)$}..

Orthogonal group {$O(n)$} consists of {$n\times n$} orthogonal matrices

{$AA^T=I$}

Unitary group {$U(n)$} consists of {$n\times n$} unitary matrices

{$A\bar{A}^T=I$}

Compact symplectic group consists of {$n\times n$} symplectic matrices

{$A\bar{A}^T=I$}

• {$O(n)$} has {$\frac{1}{2}n^2-\frac{1}{2}n = \frac{n(n-1)}{2}$} real dimensions. This is {$n^2-(\frac{n(n-1)}{2}+n)$}.
• {$U(n)$} has {$n^2$} real dimensions. This is {$2n^2-(2\frac{n(n-1)}{2}+n)$}.
• {$Sp(n)$} has {$2n^2+n=n(2n+1)$} real dimensions. This is {$4n^2 - (4\frac{n(n-1)}{2}+n)$}.

The dimension is calculated by calculating the number of independent equations, then subtracting from the total number of variables (matrix entries times field dimension), to get the number of free variables.

In each case, the diagonals manifest a different self-relation, which is the reason for the discrepancies, and the wobbling behind the periodicity.

{$$U(2r)\supset U(r)\times U(r) \supset U(r)$$}

{$\begin{matrix} \mathbf{Lie\; group} & \mathbf{real\; dimensions} \\ U(2r) & 4r^2 \\ U(r)\times U(r) & 2r^2 \\ U(r) & r^2 \\ \end{matrix}$}

{$$O(16r)\supset U(8r)\supset Sp(4r)\supset Sp(2r)\times Sp(2r)\supset Sp(2r)\supset U(2r) \supset O(2r)\supset O(r)\times O(r) \supset O(r)$$}

{$\begin{matrix} \mathbf{Lie\; group} & \mathbf{real\; dimensions} \\ O(16r) & 128r^2-8r \\ U(8r) & 64r^2 \\ Sp(4r) & 32r^2 + 4r \\ Sp(2r)\times Sp(2r) & 16r^2 + 4r\\ Sp(2r) & 8r^2+2r \\ U(2r) & 4r^2\\ O(2r) & 2r^2-r\\ O(r)\times O(r) & r^2-r\\ O(r) & \frac{1}{2}r^2-\frac{1}{2}r \end{matrix}$}

Off-kilter will show how minds are off-kilter, requiring three minds and eight mental states.