Chapter 4 of 10

§Basic structure of VOAs

We review subalgebras, homomorphisms, ideals, quotients, extensions, and tensor products before turning to orbifolds and cosets in separate chapters. Throughout, tensor products are over $\mathbb C$ $C$ .

§4.1 Vertex subalgebras and conformal vectors

Definition 4.1 (Vertex subalgebra)

A vertex subalgebra of a vertex algebra $V$ is a linear subspace $U\subseteq V$ such that

\mathbf1\in U,\qquad Y_V(u,x)v\in U((x)) \quad(u,v\in U).

Its fields are the restrictions $Y_U(u,x)=Y_V(u,x)|_U$ $Y_{V} (u, x) ∣_{U}$ .

Proposition 4.2 (Inherited vertex-algebra structure)

A vertex subalgebra $U\subseteq V$ is a vertex algebra with vacuum $\mathbf1$ and translation operator $T|_U$ $T ∣_{U}$ . If $V$ is a VOA and $\omega_V\in U$ $U$ , then $U$ is a VOA with the same conformal vector and central charge, graded by

U_h=U\cap V_h.

If $V$ is of CFT type, so is $U$ $U$ .

Exercise. Prove Proposition 4.2. Hint: Use $Y(u,x)\mathbf1=e^{xT}u$ to check $T$ $T$ -stability, and $L_0$ $L_{0}$ -stability to recover the grading.

Definition 4.3 (VOA subalgebra)

In this chapter, a VOA subalgebra $(U,\omega_U)$ of $V$ means a graded vertex subalgebra $U=\bigoplus_h(U\cap V_h)$ with $\omega_U\in U\cap V_2$ making $U$ a VOA, whose grading agrees with the ambient grading:

L_0^U=L_0^V|_U.

It is full if $\omega_U=\omega_V$ $ω_{V}$ .

The translation operators necessarily agree on $U$ $U$ , since both are determined by $\left.\partial_xY_U(u,x)\mathbf1\right|_{x=0}$ $\partial_{x} Y_{U} (u, x) 1 ∣_{x = 0}$ . The conformal vectors need not agree. Keeping them distinct is essential for the coset construction, where their difference will supply the complementary conformal structure under suitable hypotheses. Terminology for subalgebras varies; the convention above follows Yamauchi, §3.8.

§4.2 Homomorphisms and automorphisms

Definition 4.4 (Homomorphisms and automorphisms)

A vertex-algebra homomorphism $f:U\to V$ is a linear map satisfying

f(\mathbf1_U)=\mathbf1_V,\qquad f\bigl(Y_U(u,x)v\bigr)=Y_V(f(u),x)f(v) \quad(u,v\in U).

For VOAs, a VOA homomorphism additionally satisfies $f(\omega_U)=\omega_V$ $ω_{V}$ . A bijective VOA homomorphism $V\to V$ is a VOA automorphism; these form the group $\operatorname{Aut}(V)$ $Aut (V)$ .

Proposition 4.5 (Compatibility with translation and grading)

A vertex-algebra homomorphism satisfies $fT_U=T_Vf$ $T_{V} f$ . A VOA homomorphism also satisfies

fY_U(\omega_U,x)=Y_V(\omega_V,x)f, \qquad f(U_h)\subseteq V_h, \qquad c_U=c_V.

Consequently, for $g\in\operatorname{Aut}(V)$ $Aut (V)$ ,

gY(a,x)g^{-1}=Y(ga,x),\qquad [g,Y(\omega,x)]=0,\qquad g(V_h)=V_h.

Exercise. Prove Proposition 4.5. Hint: Apply the homomorphism identity to $Y(u,x)\mathbf1$ and $Y(\omega,x)\omega$ $Y (ω, x) ω$ .

The inclusion of a VOA subalgebra is always a vertex-algebra homomorphism; it is a VOA homomorphism in the above sense precisely when the subalgebra is full.

§Inner automorphisms

Proposition 4.6 (Exponentiating weight-one zero modes)

Suppose $V$ is of CFT type. For $a\in V_1$ $V_{1}$ , the operator $a_{(0)}$ preserves every $V_h$ and satisfies

[a_{(0)},Y(b,x)]=Y(a_{(0)}b,x)\quad(b\in V), \qquad a_{(0)}\mathbf1=a_{(0)}\omega=0.

Its exponential, defined on each finite-dimensional $V_h$ by the ordinary matrix exponential, is a VOA automorphism:

e^{a_{(0)}}Y(b,x)e^{-a_{(0)}}=Y(e^{a_{(0)}}b,x).

Proof.

The grading assertion follows from the weight of $a$ $a$ , and the commutator identity follows from the Jacobi identity. Creation gives $a_{(0)}\mathbf1=0$ $0$ . To check the conformal vector, skew-symmetry gives

a_{(0)}\omega=-Ta+T L_0a-\frac12T^2L_1a=0:

here $L_0a=a$ $a$ , $L_1a\in\mathbb C\mathbf1$ $C 1$ , and $L_ja=0$ for $j\ge2$ $2$ . Exponentiating the commutator identity proves the field identity. All exponentials are taken on finite-dimensional graded pieces, and the vacuum and conformal vector are fixed.

Definition 4.7 (Inner automorphism group)

For a VOA of CFT type, its inner automorphism group is

G_{\mathrm{inn}}(V) :=\big\langle e^{a_{(0)}}\mid a\in V_1\big\rangle \subseteq\operatorname{Aut}(V).

More precisely, the Lie algebra action $a\mapsto a_{(0)}$ integrates on each $V_h$ to the simply connected complex Lie group $\widetilde G$ with Lie algebra $V_1$ $V_{1}$ . These actions give a homomorphism $\rho:\widetilde G\to\operatorname{Aut}(V)$ with image $G_{\mathrm{inn}}(V)$ $G_{inn} (V)$ . Thus

G_{\mathrm{inn}}(V)\cong\widetilde G/\ker\rho.

The action need not be faithful: even a nonzero $a\in V_1$ may have $a_{(0)}=0$ on all of $V$ $V$ . Consequently, one should distinguish $V_1$ from its image in the derivations of $V$ $V$ , and $\widetilde G$ from its image in $\operatorname{Aut}(V)$ $Aut (V)$ . See van Ekeren–Möller–Scheithauer, §1.6.

§4.3 Ideals and simplicity

Definition 4.8 (Ideal and simple VOA)

An ideal of a VOA $V$ is a linear subspace $I\subseteq V$ such that

Y(a,x)b\in I((x))\qquad(a\in V,\ b\in I).

A nonzero VOA $V$ is simple if its only ideals are $0$ and $V$ $V$ .

Proposition 4.9 (Properties of ideals)

An ideal $I\subseteq V$ is stable under the Virasoro operators, in particular under $T$ $T$ , and is graded:

I=\bigoplus_h(I\cap V_h).

It is also two-sided:

Y(b,x)a\in I((x))\qquad(b\in I,\ a\in V).

An ideal containing $\mathbf1$ equals $V$ $V$ . Sums and intersections of ideals are ideals.

Proof.

Apply the ideal condition to $Y(\omega,x)$ $Y (ω, x)$ ; stability under the diagonalizable operator $L_0$ gives the grading. Skew-symmetry and $T$ $T$ -stability give

Y(b,x)a=e^{xT}Y(a,-x)b\in I((x)).

If $\mathbf1\in I$ $I$ , creation gives $a=\left.Y(a,x)\mathbf1\right|_{x=0}\in I$ for every $a\in V$ $V$ . The assertions about sums and intersections follow from linearity.

Remark 4.10 (The ordering of ideals)

Ideals are partially ordered by inclusion. By Proposition 4.9, they form a complete lattice: the greatest lower bound of a family is its intersection, and its least upper bound is its sum,

\bigwedge_\alpha I_\alpha=\bigcap_\alpha I_\alpha, \qquad \bigvee_\alpha I_\alpha=\sum_\alpha I_\alpha.

The least and greatest ideals are $0$ and $V$ $V$ . Among proper ideals, maximal means that no strictly larger proper ideal contains it, whereas largest means that it contains every proper ideal. A largest proper ideal need not exist merely because sums of ideals exist: their sum might be $V$ $V$ .

§4.4 Quotients and simple quotients

Proposition 4.11 (Quotient VOA)

Let $I\subsetneq V$ be an ideal and write $[a]=a+I$ $I$ . Then $V/I$ is a VOA with

Y_{V/I}([a],x)[b]=[Y_V(a,x)b],\qquad \mathbf1_{V/I}=[\mathbf1],\qquad \omega_{V/I}=[\omega].

Its grading and central charge are

(V/I)_h\simeq V_h/(I\cap V_h),\qquad c_{V/I}=c_V.

The projection $V\to V/I$ is a surjective VOA homomorphism. If $V$ is of CFT type, so is $V/I$ $V / I$ .

Exercise. Use two-sidedness to check independence of representatives, then descend the VOA axioms. Properness ensures $[\mathbf1]\ne0$ $0$ .

Proposition 4.12 (Kernels and the first isomorphism theorem)

For a VOA homomorphism $f:V\to W$ $W$ , the kernel is an ideal and $f$ induces a VOA isomorphism

V/\ker f\xrightarrow{\sim}\operatorname{im}f,\qquad [a]\mapsto f(a).

In particular, every VOA homomorphism from a simple VOA to a nonzero VOA is injective.

Exercise. Apply $f$ to $Y(a,x)b$ for $b\in\ker f$ $ker f$ , and use preservation of the vacuum. For the general quotient and kernel constructions, see Joyce, lectures 1–2.

Proposition 4.13 (Largest proper ideal and simple quotient for CFT type)

Let $V$ be a nonzero VOA of CFT type, so that $V=\bigoplus_{h\geq0}V_h$ and $V_0=\mathbb C\mathbf1$ $C 1$ . Then

J(V)=\sum_{\substack{I\subsetneq V\\I\text{ an ideal}}}I

is the largest proper ideal of $V$ $V$ : every proper ideal $I$ satisfies $I\subseteq J(V)$ $J (V)$ . In particular, $J(V)$ is the unique maximal proper ideal, and $V/J(V)$ is the unique simple quotient, up to the isomorphism induced by a quotient map. Moreover, $J(V)=0$ if and only if $V$ is simple.

Proof.

By Proposition 4.9, every proper ideal is graded and excludes $\mathbf1$ $1$ . Since $V_0=\mathbb C\mathbf1$ $C 1$ , every proper ideal is contained in $\bigoplus_{h>0}V_h$ $⨁_{h > 0} V_{h}$ . Their sum $J(V)$ is therefore an ideal with zero weight-zero part, so it is proper. By construction it contains every proper ideal.

The inverse image of a nonzero proper ideal of $V/J(V)$ would be a proper ideal strictly containing $J(V)$ $J (V)$ , a contradiction. Thus $V/J(V)$ is simple. Conversely, the kernel of any surjective VOA homomorphism from $V$ onto a nonzero simple VOA is a maximal proper ideal, hence equals $J(V)$ $J (V)$ . The uniqueness follows from Proposition 4.12. Finally, $J(V)=0$ precisely when $V$ has no nonzero proper ideals.

Remark 4.14 (What CFT type guarantees)

This argument applies to every nonzero VOA of CFT type, including universal Virasoro VOAs; it is not specific to minimal-model central charges. A unique largest proper ideal may itself contain smaller nonzero ideals, so the proposition does not assert that there is only one nonzero proper ideal.

For comparison, let $W$ be a nonzero simple VOA of CFT type and equip $V=W\oplus W$ with componentwise fields, vacuum $(\mathbf1_W,\mathbf1_W)$ $(1_{W}, 1_{W})$ , and conformal vector $(\omega_W,\omega_W)$ $(ω_{W}, ω_{W})$ . The proper ideals $W\oplus0$ and $0\oplus W$ sum to $V$ $V$ , so no largest proper ideal exists. Here $\dim V_0=2$ $2$ , and $V$ is not of CFT type.

§4.5 Extensions

Definition 4.15 (VOA extension)

An extension of a VOA $U$ is a VOA $V$ together with an injective VOA homomorphism $\iota:U\hookrightarrow V$ $V$ . Identifying $U$ with its image, this means that $U$ is a full VOA subalgebra of $V$ $V$ :

\mathbf1_U=\mathbf1_V,\qquad \omega_U=\omega_V,\qquad Y_V(u,x)|_U=Y_U(u,x)\quad(u\in U).

The extension is proper if $U\ne V$ $V$ .

By Proposition 4.5, an extension has $U_h=U\cap V_h$ and $c_U=c_V$ $c_{V}$ . Thus a proper extension adds states while retaining the original conformal structure.

§4.6 Tensor products

Proposition 4.16 (Tensor product of VOAs)

Let $U,W$ be VOAs. Their tensor product is a VOA with

\begin{aligned} \mathbf1_{U\otimes W}&=\mathbf1_U\otimes\mathbf1_W,\\ Y_{U\otimes W}(u\otimes v,x)(u'\otimes v') &=Y_U(u,x)u'\otimes Y_W(v,x)v',\\ \omega_{U\otimes W}&=\omega_U\otimes\mathbf1_W +\mathbf1_U\otimes\omega_W. \end{aligned}

The product on the second line uses the same formal variable in both factors. Its grading and central charge are

\begin{aligned} (U\otimes W)_h&=\bigoplus_{r+s=h}U_r\otimes W_s,\\ c_{U\otimes W}&=c_U+c_W. \end{aligned}

If both factors are of CFT type, so is their tensor product.

Proof.

For fixed $u,u',v,v'$ $u, u^{'}, v, v^{'}$ , both $Y_U(u,x)u'$ and $Y_W(v,x)v'$ are lower-bounded Laurent series. Their tensor product in the same variable is therefore well-defined and lower truncated. The vacuum and creation axioms follow factorwise. Translation covariance follows from the Leibniz rule with

T_{U\otimes W}=T_U\otimes\operatorname{id}_W +\operatorname{id}_U\otimes T_W.

For locality, choose exponents $N$ and $M$ for the pairs $u,u'$ and $v,v'$ $v, v^{'}$ , respectively. Multiplication by $(x-y)^{N+M}$ allows us to exchange the two $U$ $U$ -fields using the first factor $(x-y)^N$ $y)^{N}$ , and then the two $W$ $W$ -fields using $(x-y)^M$ $y)^{M}$ . Thus the tensor-product fields are local.

The conformal field is

Y_{U\otimes W}(\omega_{U\otimes W},x) =Y_U(\omega_U,x)\otimes\operatorname{id}_W +\operatorname{id}_U\otimes Y_W(\omega_W,x).

The two summands act on separate factors, so their Virasoro commutators add, with central charge $c_U+c_W$ $c_{W}$ ; their translation operators add to the operator above. The sum of the two $L_0$ operators gives the grading. Since both gradings are bounded below, only finitely many pairs $r+s=h$ occur at each weight, and each summand is finite-dimensional. The CFT-type assertion follows from the same grading formula.

Remark 4.17 (The two tensor factors)

The maps $u\mapsto u\otimes\mathbf1_W$ and $v\mapsto\mathbf1_U\otimes v$ identify the factors with VOA subalgebras carrying their own conformal vectors. Their fields commute:

[Y(u\otimes\mathbf1_W,x),Y(\mathbf1_U\otimes v,y)]=0.

Indeed, these fields act on separate tensor factors. They recover every pure tensor through

\left.Y(u\otimes\mathbf1_W,x)(\mathbf1_U\otimes v)\right|_{x=0} =u\otimes v.

The ambient conformal vector is the sum of those of the two subalgebras.

§4.7 References

Christoph A. Keller, Introduction to Vertex Operator Algebras, §5.3: tensor products.
Hiroshi Yamauchi, A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra, §3.8: subalgebras with their own conformal vectors and cosets.
Dominic Joyce, Vertex Algebras, lectures 1–2: ideals, quotients, and morphisms.

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