Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

Question

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. But I see in the literature that they write $Vir\otimes \overline{Vir}$ instead. The same happens in the more general case of a symmetry algebra $A\otimes \overline{A}$. Why is this?

Answer 1

Ignoring fineprints, the take-home message is that there are basically only 2 correct notations:

1. $G\times H$ for the direct or Cartesian product of Lie groups$^1$ $G$ and $H$.

2. $\mathfrak{g}\oplus\mathfrak{h}$ for the direct sum of Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$.

For details and fine prints, see this related Phys.SE post. ${\rm Vir}$ is an infinite-dimensional Lie algebra, so one should use $\oplus$.

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$^1$ Let us for simplicity assume that the groups are not vector spaces. This excludes additively written Abelian groups.

Answer 2

As a set, the conformal symmetry algebra is $Vir \times \overline{Vir}$. As a vector space, it is $Vir \oplus \overline{Vir}$.

It is also useful to consider the universal enveloping algebra $U(Vir)$, whose generators are products of Virasoro generators of the type $\prod_i L_{m_i}$. This is now an associative algebra, instead of a Lie algebra. Then we have $U(Vir \times \overline{Vir}) = U(Vir) \otimes \overline{U(Vir)}$.

So physicists' writings are right and consistent, provided you accept that $Vir$ may mean various different things (including $U(Vir)$) depending on the context.

Answer 3

This is just a matter of notation. Suppose $G_1$ and $G_2$ are two (Lie) groups of dimension $d_1$ and $d_2$. We all know the natural way to construct the direct sum of these groups: it's a group of dimension $d_1 + d_2$ that mathematicians usually denote as $G_2 \oplus G_2$. However, physicists usually just write $G_1 \times G_2$. The same holds for (Lie) algebras. Moreover, in the physics literature it is rare to distinguish the Lie algebra from its group, unless there is a possible ambiguity or confusion that could arise.