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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?

Qmechanic
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soap
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    Related: https://physics.stackexchange.com/q/206840/50583 and its linked questions. The issue is that physicists don't use $\otimes$ consistently, but use it for any of (direct product, direct sum, tensor product). – ACuriousMind Jun 24 '19 at 16:46
  • Which literature? Which page? – Qmechanic Jun 25 '19 at 16:02
  • @Qmechanic For example, page 71 of "Basic Concepts in String Theory", by Blumenhage, Lust and Theisen. There they refer to the extended symmetry algebra $A\otimes\bar A$. – soap Jun 27 '19 at 09:51
  • If $A$ and $\overline{A}$ are supposed to be Lie algebras (Lie groups), then the correct notation is $A\oplus\overline{A}$ ($A\times \overline{A}$), respectively. – Qmechanic Jun 27 '19 at 10:16

3 Answers3

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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.

Sylvain Ribault
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  • Can you tell me why one may want to consider that universal enveloping algebra, or point me to some text? – soap Jun 27 '19 at 09:52
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    Elements of $U(Vir)$ are used for building descendent states, as in $L_{-1}L_{-2}|\text{primary}\rangle$. Also, the Sugawara construction expresses the Virasoro algebra as a subalgebra of the universal enveloping algebra of your affine Lie algebra. Also, W-algebras are in general not Lie algebras, the commutator of two generators needs not be a linear combination of generators. – Sylvain Ribault Jun 27 '19 at 12:24
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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 fineprints, 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 Lie groups are not vector spaces, which are often the case.

Qmechanic
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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.

Hans Moleman
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    Lie groups don't necessarily have a vector space structure, so the direct sum is meaningless. Whereas algebras are vector spaces by definition and the direct sum is well defined. – MannyC Jun 25 '19 at 19:35