Table 3 System algebras of quasifree fermionic systems with d modes satisfying translation-invariance

2

$\mathfrak{u}(1)+\mathfrak{u}(1)$

$\mathfrak{u}(1)+\mathfrak{u}(1)$

3

$\mathfrak{u}(2)+\mathfrak{u}(1)$

$\mathfrak{su}(2)+\mathfrak{u}(1)$

4

$\mathfrak{u}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$

$\mathfrak{su}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$

5

$\mathfrak{u}(2)+\mathfrak{u}(2)+\mathfrak{u}(1)$

$\mathfrak{su}(2)+\mathfrak{su}(2)+\mathfrak{u}(1)$

6

$\mathfrak{u}(2)+\mathfrak{u}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$

$\mathfrak{su}(2)+\mathfrak{su}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$

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2n − 1

${\sum }_{i=1}^{n-1}\mathfrak{u}(2)+\mathfrak{u}(1)$

${\sum }_{i=1}^{n-1}\mathfrak{su}(2)+\mathfrak{u}(1)$

2n

${\sum }_{i=1}^{n-1}\mathfrak{u}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$

${\sum }_{i=1}^{n-1}\mathfrak{su}(2)+\mathfrak{u}(1)+\mathfrak{u}(1)$