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

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?

• Related: 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 at 16:46
• Which literature? Which page? – Qmechanic Jun 25 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 at 9:51
• 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 at 10:16

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.

• 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 at 19:35

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.

• Can you tell me why one may want to consider that universal enveloping algebra, or point me to some text? – Soap Jun 27 at 9:52
• 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 at 12:24

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.

• In the math literature one sometimes find $G\oplus H$ as a notation for the direct product of two groups, with underlying set the Cartesian product, and with the obvious binary operation. (This notation is more common with rings than with groups though). – AccidentalFourierTransform Jun 25 at 17:08
• Which math literature? Which page? – Qmechanic Jun 25 at 17:26
• Homology groups, for example. See e.g. Künneth theorem. – AccidentalFourierTransform Jun 25 at 17:28
• (Direct products and direct sums are regarded as being the same, except if the number of factors/summands is infinite, in which case the direct sum is by definition the set of sequences of compact support). – AccidentalFourierTransform Jun 25 at 17:30
• Fair enough. I just wanted to point out that $G\oplus H$ is technically not incorrect, and is indeed sometimes used. – AccidentalFourierTransform Jun 25 at 17:34