Table 1 System algebras of translation-invariant fermionic systems with d modes for (a) nearest-neighbor interactions only and (b) arbitrary translation-invariant interactions

2

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

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

3

${\sum }_{i=1}^2\mathfrak{su}(2)+{\sum }_{i=1}^3\mathfrak{u}(1)$

${\sum }_{i=1}^2\mathfrak{su}(2)+{\sum }_{i=1}^4\mathfrak{u}(1)$

4

${\sum }_{i=1}^5\mathfrak{su}(2)+{\sum }_{i=1}^4\mathfrak{u}(1)$

${\sum }_{i=1}^8\mathfrak{su}(2)+{\sum }_{i=1}^6\mathfrak{u}(1)$

5

${\sum }_{i=1}^2\mathfrak{su}(4)+{\sum }_{i=1}^8\mathfrak{su}(3)+{\sum }_{i=1}^3\mathfrak{u}(1)$

${\sum }_{i=1}^2\mathfrak{su}(4)+{\sum }_{i=1}^8\mathfrak{su}(3)+{\sum }_{i=1}^8\mathfrak{u}(1)$

6

${\sum }_{i=1}^4\mathfrak{su}(6)+{\sum }_{i=1}^8\mathfrak{su}(5)+{\sum }_{i=1}^3\mathfrak{u}(1)$

${\sum }_{i=1}^4\mathfrak{su}(6)+{\sum }_{i=1}^8\mathfrak{su}(5)+{\sum }_{i=1}^{10}\mathfrak{u}(1)$