2. Definitions and preliminary results

We fix a field K, all algebras and vector spaces we consider will be over K. If A is an associative algebra one defines on the vector space of A the Lie bracket $[a,b]=ab-ba$ . Denote by $A^{(-)}$ the Lie algebra thus obtained, the Poincaré–Birkhoff–Witt theorem yields that every Lie algebra is a subalgebra of some $A^{(-)}$ .

Let L be an algebra (not necessarily associative) and let G be a group. A G-grading on L is a vector space decomposition

(2·1) \begin{align}\Gamma\colon L=\oplus_{g\in G}L_g\end{align}

such that $L_gL_h\subseteq L_{gh}$ , for all g, $h\in G$ . In this case one says that L is G-graded. The subspaces $L_g$ are the homogeneous components of the grading and a non-zero element a of L is homogeneous if $a\in L_g$ for some $g\in G$ ; we denote this by $\|a\|_G=g$ (or simply $\|a\|=g$ when the group G is clear from the context). The support of the grading is the set $\mathrm{supp}\ L=\{g\in G \mid L_g\neq 0\}$ . A subalgebra (an ideal, a subspace) B of A is a graded subalgebra (respectively ideal, subspace) if $B=\oplus_{g\in G} (A_g\cap B)$ .

The first example is the Witt algebra $W_1=Der(K[t])$ . It is the Lie algebras of the derivations of the polynomial ring K[t]. The elements $e_n=t^{n+1}d/dt$ , $n\geq -1$ , form a basis of $W_1$ . The Lie algebra structure on the vector space $W_1$ is given by the multiplication

(2·2) \begin{equation}[e_i, e_j] = (j-i)e_{i+j}.\end{equation}

When dealing with a field of positive characteristic p, we take the differences $(j-i)$ modulo p.

The algebra $W_1$ has a $\mathbb{Z}$ -grading, $W_1=\oplus_{i\in\mathbb{Z}}L_i$ , where $L_i=0$ whenever $i\leq -2$ , and $L_i$ is the (one-dimensional) span of $e_i$ if $i\geq -1$ . Thus the element $e_n$ is homogeneous of degree n.

Another example of graded algebras is the algebra $U_1$ . Let $A=K[t,t^{-1}]$ be the algebra of Laurent polynomials in one variable t. Then $U_1$ is the Lie algebras of the derivations of A. The elements $e_n=t^{n+1}d/dt$ , $n\in\mathbb{Z}$ , form a basis of $U_1$ ; the multiplication in $U_1$ is also given by (2·2). The algebra $U_1$ is $\mathbb{Z}$ -graded, $U_1=\bigoplus_{i\in\mathbb{Z}}L_i$ where $L_i$ is the span of $e_i$ , for each $i\in\mathbb{Z}$ . This means that $U_1$ has full support on $\mathbb{Z}$ , in other words $\mathrm{supp}\, U_1= \mathbb{Z}$ .

Since we shall work with the above two graded algebras we will refrain from giving other examples.

Let $ X=\cup_{i\in\mathbb{Z}}X_{i}$ be the disjoint union of infinite countable sets of variables $X_{i}=\{x_{1}^{i},x_{2}^{i},\ldots\}$ , $i\in\mathbb{Z}$ . Assuming that for each $i\in\mathbb{Z}$ the elements of the set $X_{i}$ are of $\mathbb{Z}$ -degree i, the free associative algebra $K\langle X_\mathbb{Z}\rangle$ has a natural $\mathbb{Z}$ -grading $\oplus_{i\in\mathbb{Z}} K\langle X_\mathbb{Z}\rangle^{i}$ . Here $K\langle X_\mathbb{Z}\rangle^{i}$ is the vector subspace of $K\langle X_\mathbb{Z}\rangle$ spanned by all monomials of $\mathbb{Z}$ -degree i. The subalgebra $L\langle X_\mathbb{Z} \rangle$ of $K\langle X_\mathbb{Z}\rangle^{(-)}$ generated by the set $X_{\mathbb{Z}}$ is the free $\mathbb{Z}$ -graded Lie algebra, freely generated by $X_\mathbb{Z}$ . Note that $L\langle X_\mathbb{Z} \rangle$ is a graded subspace of $K\langle X_\mathbb{Z} \rangle$ and that the corresponding decomposition gives a $\mathbb{Z}$ -grading on $L\langle X_\mathbb{Z} \rangle$ . The elements of $L\langle X_\mathbb{Z} \rangle$ are called $\mathbb{Z}$ -graded polynomials (or simply polynomials). The degree of a polynomial f in $x_i^{a_i}$ , denoted by $\deg_{x_i^{a_i}}f$ is defined in the usual way. The definitions of multilinear and multihomogeneous polynomials are the natural ones. We define the (left-normed) commutator $[l_1,\ldots, l_n]$ of $n\geq 2$ elements $l_1,\ldots,l_n$ in a Lie algebra L inductively, $[l_1,\ldots, l_n]=[[l_1,\ldots,l_{n-1}],l_n]$ for $n>2$ .

Let $L=\oplus_{i\in \mathbb{Z}}L_i$ be a Lie algebra with a $\mathbb{Z}$ -grading. An admissible substitution for the polynomial $f(x_{1}^{a_1},\ldots, x_{n}^{a_n})$ in L is an n-tuple $(l_1,\ldots, l_n)\in L\times \cdots \times L$ such that $l_i\in L_{a_i}$ , for $i=1$ ,…, n. If $f(l_1,\dots,l_n)=0$ for every admissible substitution $(l_1,\dots, l_n)$ we say that $f(x_{1}^{a_1},\dots, x_{n}^{a_n})$ is a graded identity for L. The set of $\mathbb{Z}$ -graded polynomial identities of L will be denoted by $T_\mathbb{Z}(L)$ . It is a $T_\mathbb{Z}$ -ideal, that is an ideal invariant under the endomorphisms of $L\langle X_\mathbb{Z} \rangle$ as a graded algebra. The intersection of a family of $T_\mathbb{Z}$ -ideals in $L\langle X_\mathbb{Z} \rangle$ is a $T_\mathbb{Z}$ -ideal; given a set of polynomials $S\subseteq L\langle X_\mathbb{Z}\rangle$ we denote by $\langle S \rangle_\mathbb{Z}$ the intersection of the $T_\mathbb{Z}$ -ideals of $L\langle X_\mathbb{Z} \rangle$ that contain S. We call $\langle S \rangle_\mathbb{Z}$ the $T_\mathbb{Z}$ -ideal generated by S, and refer to S as a basis of this $T_\mathbb{Z}$ -ideal. It is well known that in characteristic 0, every $T_\mathbb{Z}$ -ideal $T_\mathbb{Z}(L)$ is generated by its multilinear polynomials. Over an infinite field of positive characteristic one has to take into account the multihomogeneous polynomials instead of the multilinear ones.

In this paper, unless otherwise stated, K denotes a field of characteristic two. We do not require any further restrictions on K. Our main result is the following theorem which provides a basis of the $\mathbb{Z}$ -graded identities for $U_1$ .

Theorem 2·1. Let K be a field of characteristic two. The ideal of the graded identities of $U_1$ is generated, as a $T_\mathbb{Z}$ -ideal, by the polynomials

(2·3) $$[x_1^a,x_2^b] \equiv 0,$$

where a and b are integers of the same parity, that is $a\equiv b\pmod{2}$ .

We deduce, for the graded identities of $W_1$ , the following theorem.

Theorem 2·2. Let K be a field of characteristic two. The $\mathbb{Z}$ -graded identities

$${x^c} \equiv 0\quad (c \le - 2),\qquad [x_1^a,x_2^b] \equiv 0,$$

where a and b are integers greater than $-2$ , and of the same parity, form a basis for the $\mathbb{Z}$ -graded identities of the Lie algebra $W_1$ over K.

3. $\mathbb{Z}$ -Graded identities of $U_1$

Here we prove Theorem 2·1. To this end, we need a series of results.

Proof. The proof is immediate by the multiplication rules (2·2).

In analogy with the associative case we will call monomials the commutators in the free graded Lie algebra. We recall that a basis for the free Lie algebra $L\langle X_G\rangle$ consists of left-normed monomials, see for more details [Reference KUKIN21]. Additionally, the Lie algebras $U_1$ and $W_1$ are metabelian in characteristic 2 (they are not metabelian otherwise). Thus, without loss of generality, we will consider all monomials to be left-normed.

Proof. We denote by I the $T_G$ -ideal generated by the polynomials (2·3). We prove the claim by induction on the length n of the monomial. The result is obvious for $n=2$ , so we suppose $n\geq 3$ . Let $M^{\prime}=[x_{1}^{a_1},\ldots, x_{n-1}^{a_{n-1}}]$ . If $M^{\prime}\in T_\mathbb{Z}(U_1)$ then it lies in I, by the induction hypothesis, and hence $M\in I$ . We assume now that $M^{\prime}\notin T_\mathbb{Z}(U_1)$ . Let $a=\| M^\prime\|$ , then the result of every admissible substitution in $M^{\prime}$ is a scalar multiple of $e_{a}$ . Therefore, $M\in T_\mathbb{Z}(U_1)$ if and only if $[x_{1}^{a},x_{2}^{a_n}]\in T_\mathbb{Z}(U_1)$ . The commutator $[x_{1}^{a},x_{2}^{a_n}]$ lies in $T_\mathbb{Z}(U_1)$ if and only if a and $a_n$ have the same parity, and hence M lies in I. Thus in all cases $M\in I$ , as required.

It is well known that every $T_\mathbb{Z}$ -ideal is generated by its regular polynomials. Recall that a polynomial $f\in L\langle X_\mathbb{Z}\rangle$ is regular if every one of its variables appears in every monomial of f, not necessarily with the same degree.

Recall that I is the $T_\mathbb{Z}$ -ideal generated by the polynomials in (2·3). The next lemma is a key step in the proof of our main theorem.

Proof. Since $M\notin T_\mathbb{Z}(U_1)$ either $a_0$ or $a_1$ is an odd integer. Suppose there exist i and j, with $0\leq i< j\leq n$ , such that $ a_i $ and $ a_j $ are odd integers. By exchanging the first two variables, if necessary, we can assume that $i = 0$ and $j = n$ , and all other $a_k$ are even integers. (If some additional $a_k$ is odd we simply consider the initial part of the commutator, from $a_0$ to $a_k$ .) Then $M^\prime=[x_{i_0}^{a_0}, x_{i_1}^{a_1}, \ldots, x_{i_{n-1}}^{a_{n-1}}]$ is a monomial in $L\langle X_\mathbb{Z} \rangle$ that is not a graded identity. This implies $\sum_{i=0}^{n-1}a_i$ is odd, that is $M\in T_\mathbb{Z}(U_1)$ which is absurd. Now we apply identity (2·3), together with the Jacobi identity, several times on the variables of even $\mathbb{Z}$ -degree.

As mentioned earlier, we cannot claim in general that a $T_\mathbb{Z}$ -ideal is generated by its multihomogeneous elements; this is certainly true if the base field is infinite. But, in our specific case, we are in a position to deduce this fact.

Proof. The result is well known if K is infinite. So we assume K is a finite field. Let $f=f(x_0,x_1,\ldots,x_n)$ be a regular polynomial in $T_\mathbb{Z}(U_1)$ . By Lemma 3·3, only one of the variables in f is in an odd component. Suppose $x_0$ in an odd component and the remaining variables are in even components. Moreover, f is linear in $x_0$ , and $x_0$ appears in the first position on each monomial in f. We induct on the degree of the polynomial f. Suppose that f is not multihomogeneous. As f is regular, we write $f=\sum_i\alpha_iM_i$ where $M_i$ is the commutator

\begin{equation*}[x_0,x_1,\ldots,x_1,x_2,\ldots,x_{k-1},\underbrace{x_{k},\ldots,x_{k}}_{m^k_i\mbox{ times}},x_{k+1},\ldots,x_n] = x_0 (ad\,x_1)^{m_i^1}\ldots (ad\,x_n)^{m_i^n}.\end{equation*}

Here $m^k_i$ are non-negative integers depending on the variable $x_i$ and on the monomial $M_i$ , and $ad\,x$ is, as usual, the linear transformation $y\, ad\,x \mapsto [y,x]$ in a Lie algebra. For each variable $x_k$ we put $r_k=\min\{m_{i}^k\}$ . By the regularity of f we have $r_k>0$ . Applying identity (2·3), if needed, there exists an element f ^′ which is a sum of regular polynomials: $f^\prime=f^\prime(x_0,x_1,\ldots,x_n)$ in $T_\mathbb{Z}(U_1)$ , not necessarily having the same variables as f, such that

\begin{equation*}f\equiv f^\prime (ad\, x_1)^{r_1} (ad\, x_2)^{r_2} \cdots (ad\, x_n)^{r_n} \pmod{I}.\end{equation*}

Note that the degree of the polynomial $f^\prime$ is lower than the degree of f. Since $I\subseteq T_{\mathbb{Z}}(U_1)$ , by induction the result follows if $f^\prime\in T_\mathbb{Z}(U_1)$ . Suppose that $f^\prime\notin T_\mathbb{Z}(U_1)$ . Then there exists an admissible substitution for the polynomial f, say $(l_0,l_1,\ldots, l_n)$ , of elements in $U_1$ such that $f^\prime(l_0,l_1,\ldots,l_n)\neq 0$ , but $f(l_0,l_1,\ldots,l_n)=0$ . Recall that $U_1=\oplus_{i\in\mathbb{Z}}L_i$ is the natural $\mathbb{Z}$ -grading on $U_1$ . Define $A_0$ as the sum of all even components $L_i$ , $i\equiv 0\pmod{2}$ , and $A_1$ as the sum of the odd ones. Of course, $f^\prime(l_0,l_1,\dots,l_n)\in A_1$ , that is

\begin{equation*}f^\prime(l_0,l_1,\dots,l_n)=\sum_{i\in\mathbb{Z}}\lambda_ie_{2i+1},\end{equation*}

where the set $\{i\mid \lambda_i\neq 0\}$ is finite. As each homogeneous component of $U_1$ is one-dimensional, we have

\begin{equation*}0=f(l_0,l_1,\dots,l_n)= f^\prime(l_0,l_1,\dots,l_n) (ad\, e_{a_1})^{r_1} \cdots (ad\, e_{a_n})^{r_n} =\sum_{i\in \mathbb{Z}}\alpha_i \lambda_i e_{(2i+1)+t}.\end{equation*}

Here $t=\sum_{j=1}^{n}r_ja_j$ . It is clear that each $\alpha_i$ equals 1, this implies that each $\lambda_i=0$ , which contradicts the fact that the substitution in $f^\prime$ is not zero. Therefore the graded identity f is equivalent to a multihomogeneous graded identity.

The corollary to the above lemma is not needed in the proof of the main theorem. If K is an infinite field of characteristic different from 2, it was obtained in [Reference FIDELIS and KOSHLUKOV8, theorem 4·4]. We mention here that the proof given there is essentially characteristic-free, and also works when K is a finite field. We decided to include it here since, as a rule, the multilinear identities do not determine the ideals of identities if the base field is of positive characteristic.

Proof. By Lemma 3·4, we can consider the multihomogeneous graded identities. Let $f=f(x_{1}^{a_1},\ldots, x_{r}^{a_r})$ be a multihomogeneous graded identity of $U_1$ . As $\dim_KL_i\leq 1$ for every $i\in \mathbb{Z}$ , we have that each admissible substitution $\varphi$ can be taken to map $x_{i}^{a_i}$ to $\xi_ie_{a_i}$ where $\xi$ ’s are commutative and associative (independent) variables over K. Here, as above, $e_{a_i}$ spans the vector space $L_{a_i}$ . Hence we have

(3·1) $$\varphi (f(x_1^{{a_1}}, \ldots ,x_r^{{a_r}})) = \xi _1^{{n_1}} \cdots \xi _r^{{n_r}}f({e_{{a_1}}}, \ldots ,{e_{{a_r}}}).$$

Fix some $a_i$ , $\deg_{x_i^{a_i}}f=n_i\geq 1$ . Take new variables $x_{i,j}^{a_i}$ where $1\leq j\leq n_i$ , and consider a multihomogeneous graded polynomial $h(x_{1,1}^{a_1},\ldots,x_{1,n_1}^{a_1},x_{2,1}^{a_2},\ldots, x_{r,n_r}^{a_r})$ such that (here we put, in order to simplify the notation, $i=1$ )

$$h(\underbrace {x_1^{{a_1}}, \ldots ,x_1^{{a_1}}}_{{n_1}},x_2^{{a_2}}, \ldots ,x_r^{{a_r}}) = f(x_1^{{a_1}}, \ldots ,x_r^{{a_r}}).$$

But $x_{i,j}^{a_i}$ can be evaluated to $e_{a_i}$ for each $1\leq i\leq r$ and $1\leq j\leq n_i$ . Then equation. (3·1) implies that h is a graded identity and that f is a consequence of h. Moreover h is linear in each of the new $n_i$ variables. Continuing in this way for the remaining variables we prove the statement.

Now we have all the ingredients for the proof of the main result in the paper.

Proof of Theorem 2·1. As done in the proof of Lemma 3·4, every multihomogeneous polynomial f, modulo I, is equivalent, up to a scalar factor, to the monomial

\begin{equation*}[x_0,x_1,\ldots,x_1,x_2,\ldots,x_{k-1},\underbrace{x_{k},\ldots,x_{k}}_{m^k_i\mbox{ times}},x_{k+1},\ldots,x_n] = x_0 (ad\,x_1)^{m_i^1}\cdots (ad\,x_n)^{m_i^n},\end{equation*}

where $x_1$ ,…, $x_n$ are of even degree. Hence $f\in T_\mathbb{Z}(U_1)$ if and only if $f\in I$ . This implies Theorem 2·1.

The above results are easily adaptable to the case of the Lie algebra $W_1$ . Recall that in [Reference FREITAS, KOSHLUKOV and KRASILNIKOV9] the authors considered the base field of characteristic 0, and this was important in their proofs. In [Reference FIDELIS and KOSHLUKOV8], the result was extended to $W_1$ considered over an infinite field of characteristic different from 2.

4. Independence of graded identities

In this section we use ideas from [Reference FIDELIS and KOSHLUKOV8, Reference FREITAS, KOSHLUKOV and KRASILNIKOV9]. As above K is an arbitrary field of characteristic two.

Denote by $f_{r,s}=[x^r_1, x^s_2] \in L\langle X_\mathbb{Z}\rangle$ the graded polynomials from (2·3), and assume $r\le s$ .

Proof. Let r, $s\in\mathbb{Z}$ be of the same parity, and let $H=UT(3, K)$ be the Lie algebra of strictly upper triangular $3\times 3$ matrices over K. Define the vector subspaces $H_k$ , $k\in\mathbb{Z}$ , in H as follows:

1. (i) if $r\neq s$ we set $H_k=0$ for all $k\neq r$ , s and $r+s$ ; $H_r$ is the span of $E_{12}$ , $H_s$ is the span of $E_{23}$ and $H_{r+s}$ is the span of $E_{13}$ ;

2. (ii) if $r=s=0$ then $H_0=H$ , and $H_k=0$ for every $k\neq 0$ ;

3. (iii) if $r=s\neq 0$ then $H_r$ is spanned by $E_{12}$ and $E_{23}$ , $H_{2r}$ is spanned by $E_{13}$ , and $H_k=0$ for every $k\ne r$ , 2r.

Here $E_{ij}$ is the matrix that has 1 at position (i, j) and 0 elsewhere. It is clear that $H=\oplus_{i\in\mathbb{Z}}H_i$ is a $\mathbb{Z}$ -graded Lie algebra. Since $[E_{12},E_{23}]=E_{13}\neq0$ , the graded identity $[x^r_1, x^s_2]$ is not satisfied in H. On the other hand, one can easily see that H satisfies all graded identities (2·3) as well as all identities $f_{u,v}$ when $(r,s)\neq (u,v)$ . The result follows.

A set I of (graded) polynomials is an independent set of (graded) identities if neither of them lies in the ideal of (graded) identities generated by the remaining ones.

The above statements, together with Theorem 2·1, yield the following theorem.

Theorem 4·3. Over a field of characteristic two, the graded identities

$$[x_1^a,x_2^b] \equiv 0,\quad a \le b,$$

where a and b are of the same parity, form a minimal basis for the $\mathbb{Z}$ -graded identities of $U_1$ .

For the Lie algebra $W_1$ , we add to the list of identities the variables $x^c$ with $c< -1$ . Note that the identity $f_{-1,-1}=[x^{-1}_1, x^{-1}_2]$ is a consequence of $x^{-2}$ .

Proof. Let $d\in\mathbb{Z}$ and let H be the 1-dimensional Lie algebra over K. The algebra $H=\oplus_{i\mathbb{Z}}H_i$ is $\mathbb{Z}$ -graded with ith homogeneous component $H_i$ equal to H if $i =d$ and 0 otherwise. It is clear that H satisfies the graded identities (2·3) as well as all graded identities $x^c$ where $c\neq d$ but does not satisfy the identity $x^d$ .

The next corollary is a direct consequence of the above theorem together with Theorem 4·3.