What is the underlying vector space of the Super-Poincaré algebra?

Question

The Poincare algebra $P$ is the direct sum $$P=\mathbb{R}^4\oplus so(1,3)$$ which is a real vector space of dimension $10$.

The $\mathcal{N}$-extended supersymmetry algebra is a graded Lie algebra, which enhances the Poincare algebra with fermionic generators $$Q_a^I, \quad \bar{Q}^J_{\dot{a}}$$ which transform under Lorentz transformations in the $(1/2,0)$ and $(0,1/2)$ representations respectively. The indices $a,\dot{a}$ both take values $1,2$ and $I,J$ take values $1,2,\dots,\mathcal{N}$.

The full vector space is then the graded vector space $$P\oplus g_1$$ where $$g_1={\rm span}_\mathbb{R} \{Q_a^I, \bar{Q}^J_{\dot{a}}\}$$ is the odd part.

Question: What is the dimension of this vector space? Naively I would guess $4\mathcal{N}$ but the $\bar{Q}$ are related to the $Q$ by conjugation so perhaps just $2\mathcal{N}$? However, in $\mathcal{N}=4$ SYM I have know that the superconformal algebra is $psu(2,2|4)$, implying the dimension of the odd part is $4$, and not $8$ as my reasoning suggests.

This post imported from StackExchange Physics at 2016-10-20 14:04 (UTC), posted by SE-user ryanp16

Comments

I think the answer is $4\mathcal{N}$. The superconformal algebra is not simply the super-Poincaré algebra, it also includes other generators such as dilations and special conformal transformations (also the precise nature of the conformal transformations depends on dimension), so this might be a source of discrepancy.