Table 4 System algebras for d -mode fermionic systems

General systems:

$\mathfrak{su}(2^{d-1})\oplus \mathfrak{su}(2^{d-1})$

Theorem 4

 {T}

$\mathfrak{s}[{⨁}_{\ell =0}^{d-1}\mathfrak{u}({\hat{r}}_{\ell })]\oplus \mathfrak{s}[{⨁}_{\ell =0}^{d-1}\mathfrak{u}({\hat{r}}_{\ell })]$

Theorem 30

 {N}

$\mathfrak{s}({⨁}_{n\mathrm{even}}\mathfrak{u}[\left(\genfrac{}{}{0ex}{}{d}{n}\right)])\oplus \mathfrak{s}({⨁}_{n\mathrm{odd}}\mathfrak{u}[\left(\genfrac{}{}{0ex}{}{d}{n}\right)])$

Proposition 42

Quasifree systems:

$\mathfrak{so}(2d)$ ^b

Proposition 9

 {T}, d odd

$[{\sum }_{i=1}^{(d-1)/2}\mathfrak{u}(2)]+\mathfrak{u}(1)$

Theorem 34

 {T}, d even

$[{\sum }_{i=1}^{(d-2)/2}\mathfrak{u}(2)]+\mathfrak{u}(1)+\mathfrak{u}(1)$

Theorem 34

 {T,R}, d odd

$[{\sum }_{i=1}^{(d-1)/2}\mathfrak{su}(2)]+\mathfrak{u}(1)$

Corollary 35

 {T,R}, d even

$[{\sum }_{i=1}^{(d-2)/2}\mathfrak{su}(2)]+\mathfrak{u}(1)+\mathfrak{u}(1)$

Corollary 35

 {N}

$\mathfrak{u}(d)$

Proposition 43