Question as the title, I am looking for concret examples of infinite-dimensional Lie algebras without semi-infinite structures. A semi-infinite structure is a necessary condition to do semi-infinite cohomology. A reference for semi-infinite structure and semi-infinite cohomology of Lie algebras is the paper ** Semi-infinite cohomology and string theory** by I. B. Frenkel, H. Garland and G. J. Zuckerman in 1986.

For convenience, I give a brief introduction here.

**Basic assumption**:
$\mathfrak g=\oplus_{n\in \mathbb{Z}} \mathfrak g_n$ is a tame $\mathbb{Z}$-graded Lie algebra, i.e., $\dim \mathfrak g_n< \infty$ and $[\mathfrak g_n, \mathfrak g_m]\subset \mathfrak g_{m+n}$ for all $m, n\in \mathbb{Z}$.

Let $\mathfrak n=\oplus_{n>0}\mathfrak g_n$ and $\mathfrak b=\oplus_{n\leq 0}\mathfrak g_n$ with homogeneous basis $\{e_i~|~i\leq 0\}$ and $\{e_i~|~i>0\}$ respectively. Homogeneous means that each $e_i\in \mathfrak g_m$ for some $m\in \mathbb{Z}$. We also require that whenever $e_i\in \mathfrak g_m$, then $e_{i+1}\in \mathfrak g_m$ or $e_{i+1}\in \mathfrak g_{m+1}$. Let $\mathfrak g'=\oplus_{n\in \mathbb{Z}}\mathfrak g'_n$ be the restricted dual of $\mathfrak g$, where $\mathfrak g'_n:=Hom_{\mathbb{C}}(\mathfrak g_{-n}, \mathbb{C})$, and $\{e_i^*~|~i\in \mathbb{Z}\}$ the dual basis of $\mathfrak g'$ such that $\langle e_i^*, e_j\rangle=\delta_{ij}.$

**Definition**
The space of semi-infinite forms on $\mathfrak g$ is the vector space $\Lambda_{\infty}^*\mathfrak g'$ spanned by the monomials
$$\omega=e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots $$
of infinite wedge products of $\mathfrak g'$ satisfying that for each $\omega$, there exists an integer $N(\omega)$ such that when $k>N(\omega)$, we have $i_{k+1}=i_k-1.$

The Clifford algebra $Cl(\mathfrak g\oplus \mathfrak g')$ associated to $\mathfrak g\oplus \mathfrak g'$ is the unital associative algebra generated by $\{e_i, e_i^*~|~i\in \mathbb{Z}\}$ satisfying the relations: \begin{align} e_ie_j+e_je_i=e_i^*e_j^*+e_j^*e_i^*=0,\,\ e_i^*e_j+e_je_i^*=\delta_{ij}, \, \forall~ i, j\in \mathbb{Z}. \end{align} The space $\Lambda_{\infty}^*\mathfrak g'$ is an irreducible $Cl(\mathfrak g\oplus \mathfrak g')$-module with the actions given by \begin{align*} \varepsilon(e_{i_0}^*)\cdot e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots &=e_{i_0}^*\wedge e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots , \\ \iota(e_{i_o})\cdot e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots &=\sum_{k\geq 1}(-1)^{k-1}\langle e_{i_0}, e_{i_k}^*\rangle e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots \wedge \hat{e}_{i_k}^*\wedge \cdots, \end{align*} where the hat over $e_{i_k}^*$ means ommiting this term.

We want to define a $\mathfrak g$-action on $\Lambda_{\infty}^*\mathfrak g'$ (at the moment we just call it an action but not necessarily a Lie algebra action.)

For $x\in \mathfrak g_n$ with $n\neq 0$, we denote by $\rho(x)$, the natural action on $\Lambda_{\infty}^*\mathfrak g'$, \begin{align*} \rho(x)\cdot e_{i_1}^*\wedge e_{i_2}^*\wedge\cdots =\sum_{k\geq 1}e_{i_1}^*\wedge \cdots \wedge ad^*~x(e_{i_k}^*) \wedge\cdots, \end{align*} where $ad^*$ is the coadjoint action of $\mathfrak g$ on $\mathfrak g'$. The above sum is finite hence well-defined. It is easy to verify the following relations (considered as operators on $\Lambda_{\infty}^*\mathfrak g'$): $~\forall~ x, y\in \mathfrak g, y'\in \mathfrak g',$ \begin{equation} [\rho(x), \iota(y)]=\iota(ad~ x(y)), \,\ [\rho(x), \varepsilon(y')]=\varepsilon(ad^*~x(y')). ~~ (\divideontimes) \end{equation}

For elements in $\mathfrak g_0$, we could not use the above definition because it may occur as an infinite sum. We choose a special monomial (there are other choices) $$\omega_0:=e_0\wedge e_{-1}\wedge e_{-2}\wedge \cdots,$$ which could be considered as the maximal form on $\mathfrak b$. It is characterized up to scalar by the property that $\omega_0$ is killed by $\iota(x)$ for all $x\in \mathfrak n$ and $\varepsilon(x')$ for all $x'\in \mathfrak b^*.$

Choose $\beta\in \mathfrak g'_0$ and consider it as a function on $\mathfrak g$. Then for $x\in \mathfrak g_0$, we define its action on $\omega_0$ as $\beta(x)$ and then extend to an action on $\Lambda_{\infty}^*\mathfrak g'$ by requiring the relations $(\divideontimes)$. This can be done because $\Lambda_{\infty}^*\mathfrak g'$ is irreducible and generated by $\omega_0$ as a module of the Clifford algebra $Cl(\mathfrak g\oplus \mathfrak g')$.

To give an explicit expression of $\rho(x)$, we introduce the normal ordering of two operators, \begin{align*} :\iota(e_i)\varepsilon(e_i^*):=\begin{cases} \iota(e_i)\varepsilon(e_i^*),~~ i\leq 0, \\ -\varepsilon(e_i^*)\iota(e_i), ~~ i>0. \end{cases} \end{align*}

Then for any $x\in \mathfrak g$, the following operator acts well on $\Lambda_{\infty}^*\mathfrak g'$, \begin{align}\label{rho} \rho^{\beta}(x)&=\sum_{i\in \mathbb{Z}}:\varepsilon(ad^*x(e_i^*))\iota(e_i):+\beta(x) ~~(\divideontimes\divideontimes) \end{align} and it realizes the action of $x$ on $\Lambda_{\infty}^*\mathfrak g'$ and satisfies the relations $(\divideontimes)$.

Let $$\gamma^{\beta}(x, y)=[\rho^{\beta}(x), \rho^{\beta}(y)]-\rho^{\beta}([x, y]).$$

It can be proved that $\gamma^{\beta}(\cdot, \cdot)$ is a 2-cocycle of $\mathfrak g$ satisfying $\gamma(\mathfrak g_m, \mathfrak g_n)=0$ whenever $m+n\neq 0$.

**Definition**
We call $\mathfrak g$ admits a semi-infinite structure through $\rho^{\beta}$ as defined in $(\divideontimes\divideontimes)$ if the 2-cocycle $\gamma^{\beta}(\cdot, \cdot)\equiv 0$. And we call $\mathfrak g$ admits a semi-infinite structure if $\mathfrak g$ admits a semi-infinite structure through $\rho^{\beta}$ for some $\beta\in \mathfrak g'_0$.

It is obvious that $\mathfrak g$ admits a semi-infinite structure through $\rho^{\beta}$ if and only if the $\mathfrak g$-module $\Lambda_{\infty}^*\mathfrak g'$ is a Lie algebra module under the action $\rho^{\beta}(x)$.

**Example**
When $H^2(\mathfrak g)=0$, every 2-cocycle is a coboundary hence $\mathfrak g$ admits a semi-infinite structure. For example, the affine Kac-Moody algebras and the Virasoro algebra. If $\mathfrak g$ is abelian, it always admits a semi-infinite structure.

**Question:** Are there examples of tame $\mathbb{Z}$-graded Lie algebras without semi-infinite structures?

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