2.1 Gradings

We start off with the classical notion of grading by a group. Let $G=\{g_1,\ldots ,g_s\}$ be any group of finite order s, and let F be a field. If A is an F-algebra, we say that A is a G-graded algebra if there are subspaces $A_{g}$ for each $g\in G$ such that

$$\begin{align*}A=\bigoplus_{g\in G}A_{g} \ \textrm{and} \ A_{g}A_{h}\subseteq A_{gh}.\end{align*}$$

If $0\neq a\in A_{g}$ , we say that a is homogeneous of G-degree g or G-graded homogeneous of G-degree g, and we write $\deg (a)=g$ .

Assume that the characteristic of F does not divide m and F contains primitive mth roots of the unit. Let $\alpha $ be an automorphism of A of order m (i.e., $\alpha $ has order m as an element of the group Aut $(A)$ of the automorphisms of A). It is well known that $\alpha $ induces a $\mathbb {Z}_m$ -grading on A, that is,

$$ \begin{align*}A = A_0 \oplus A_1 \oplus \cdots \oplus A_{m-1},\end{align*} $$

where, for a fixed primitive m-root of unit $\gamma $ ,

$$ \begin{align*}A_k = \{ a \in A \mid \alpha(a) = \gamma^k a \}.\end{align*} $$

Moreover, there is a one-to-one correspondence (duality) between gradings by a group and the action of a group of automorphisms under some restrictions (see [Reference Giambruno and Zaicev23, Theorem 3.2.1]).

2.2 Taft’s algebras

Let F be a field containing an mth root of the unit $\gamma $ for some positive integer m. Let $H_{m^2}(\gamma )$ be the bialgebra defined by generators c and d with relations

$$ \begin{align*}c^m=1, \quad d^m = 0, \quad dc = \gamma cd.\end{align*} $$

The coalgebra structure is given by

$$ \begin{align*}\Delta(c) = c \otimes c, \quad \Delta(d) = c \otimes d + d \otimes 1,\end{align*} $$

$$ \begin{align*}\varepsilon(c) = 1, \quad \varepsilon(d) = 0.\end{align*} $$

As an F-vector space, $H_{m^2}(\gamma )$ has dimension $m^2$ with basis $ \{ c^ix^j \mid 0 \leq i, j, \leq m-1 \} $ . The bialgebra $H_{m^2}(\gamma )$ becomes a Hopf algebra if we define the antipode as $S(c) = c^{-1}$ and $S(d) = -c^{-1}d$ . This Hopf algebra is known as the mth Taft’s Hopf algebra

2.3 H-module algebras

Let F be a field of characteristic zero, and let H be a Hopf algebra over F. We remand to the books [Reference Dăscălescu, Năstăsescu and Raianu16, Reference Montgomery36, Reference Radford37, Reference Sweedler41] for basic definitions, examples, and further information about Hopf algebras. An algebra A is an H-module algebra if A is endowed with a left H-action $h \otimes a \mapsto ha$ or, equivalently, with a homomorphism $H \to \text {End}_F(A)$ such that:

1. 1. $h(ab) = (h_{(1)}a)(h_{(2)}b)$ .

2. 2. $h(1_A) = \varepsilon (h) 1_A$ , for all $ h \in H, a, b \in A$ .

Here, we use Sweedler’s notation $\Delta (h) = h_{(1)} \otimes h_{(2)}$ , where $\Delta $ and $\varepsilon $ are the comultiplication and the counit in H, respectively.

Let A be a finite dimensional algebra over a field F; assume that the characteristic of F does not divide m. Since $c^m = 1$ , then c acts as an automorphism of A of order m, d acts as a c-derivation, that is, it satisfies $d(ab) = d(a)b + c(a)d(b)$ , for all $a,b \in A$ , and the actions of c and d are related by $dc = \gamma cd$ . Thus, an action of $H_m$ on A is completely determined by a choice of:

1. (1) an automorphism $\alpha $ of A of order m and

2. (2) an $\alpha $ -derivation d of A such that $d^m=0$ and $d\alpha = \gamma \alpha d$ ,

or equivalently (see [Reference Centrone and Yasumura14, Proposition 4]), a choice of:

1. (1) a $\mathbb {Z}_m$ -grading $A = \bigoplus _{i \in \mathbb {Z}_m} A_i$ and

2. (2) an $\alpha $ -derivation d (where $\alpha $ defines the $\mathbb {Z}_m$ -grading) such that $d(A_i) \subseteq A_{i-1}$ and $d^m =0$ .

The proof of [Reference Giambruno and Zaicev23, Proposition 3.3.6] gives us a linear basis $\mathcal {B}_1 = \{ \chi _1, \dots , \chi _m \}$ of the subalgebra $\langle c \rangle $ of $H_m$ generated by c, such that each $\chi _i$ corresponds to a projection $\phi _i \colon A \to A_i$ with respect to the decomposition $A = \bigoplus _{i \in \mathbb {Z}_m} A_i$ . Then $\{ d^j\chi _i \mid i,j = 0, 1, \dots , m-1 \}$ turns out to be a basis of $H_m$ . Let $D_m = \textrm{span}_F\{ 1, d, d^2, \dots , d^{m-1} \}$ , then by [Reference Bahturin and Yasumura3] we have that the variables $d^j\chi _i \cdot x$ correspond to graded variables under the action of $d^j$ . Then $H_m$ -polynomials correspond to $\mathbb {Z}_m$ -graded polynomials with the action of $D_m$ and their polynomial identities coincide, that is, $\operatorname {\mathrm {Id}}^{H_m}(A) = \operatorname {\mathrm {Id}}^{\mathbb{Z}_m,Dm} (A)$ (see [Reference Centrone and Yasumura14, Proposition 14]).

2.4 Free H-module algebras

Let $F \langle X \rangle $ be the free F-algebra on the set of countable noncommutative variables $X = \{x_1, x_2, \ldots \}$ and consider the vector space $V = F \langle X \rangle \otimes _F H $ . The free H-module algebra over X, denoted by $F^H \langle X \rangle $ , is the tensor algebra over V. Any element of $F^H \langle X \rangle $ will be called H-polynomial. In what follows, we shall use the notation

Now, let H be finite dimensional, and let $\{b_1, \dots , b_m \}$ be a basis (as a vector space) of H. It follows that $F^H \langle X \rangle $ is isomorphic to the free algebra over F with free formal (noncommutative) generators $x^{b_j}$ , $ j \in \{1, \ldots , m \}$ , $x \in X$ . Notice that $F^H \langle X \rangle $ has a structure of left H-module algebra by defining the next H-action

$$ \begin{align*}h (x_{i_1}^{h_1} x_{i_2}^{h_2} \cdots x_{i_n}^{h_n}) = x_{i_1}^{h_{(1)}h_1} x_{i_2}^{h_{(2)}h_2} \cdots x_{i_n}^{h_{(n)}h_n},\end{align*} $$

where $h_{(1)} \otimes h_{(2)} \otimes \cdots \otimes h_{(n)}$ is the image of $h\in H$ under the comultiplication $\Delta $ of H applied $(n-1)$ times. Thus, $F^H \langle X \rangle $ is the free H-module algebra on X. This means that, for any H-module algebra $ W$ and for every function $ \alpha \colon X \to W$ , there exists a unique homomorphism of algebras and H-modules (we call these kinds of homomorphisms simply H-homomorphisms) $\beta \colon F^H \langle X \rangle \to W$ extending $\alpha $ . In what follows, we shall identify X with the set $\{ x^{1_H} | x \in X \} \subset F^H \langle X \rangle $ .

Given any H-module algebra W, we say that an H-polynomial $f \in F^H \langle X \rangle $ is an H-identity for W if, for every H-homomorphism $ \varphi \colon F^H \langle X \rangle \to W$ , the polynomial f is in the kernel of $\varphi $ . In other words, $f(x_1, \ldots , x_n) \in F^H \langle X \rangle $ is an H-identity of W if and only if $f(w_1, \ldots , w_n) = 0$ , for all $w_1 \ldots , w_n \in W$ . The set $\operatorname {\mathrm {Id}}^H(W)$ of all identities satisfied by W is an ideal of $F^H \langle X \rangle $ , and it is invariant under all H-endomorphisms of $F^H \langle X \rangle $ . The ideals having such a property are called $T^H$ -ideals. Moreover, all $T^H$ -ideals are of this form: in fact, it is not difficult to see that, given a $T^H$ -ideal I of $F^H \langle X \rangle $ , then $\operatorname {\mathrm {Id}}^H(F^H \langle X \rangle / I )= I$ .

Two H-module algebras $W_1$ and $W_2$ are said to be $T^H$ -equivalent, and we write $W_1 \sim _{T^H} W_2$ , if $\operatorname {\mathrm {Id}}^H(W_1) = \operatorname {\mathrm {Id}}^H(W_2)$ .

Given a nonempty set $S \subseteq F^H\langle X \rangle $ , the class var $^H(S)$ of all H-module algebras W such that f is an H-identity for W for all $f \in S$ is called the variety determined (or generated) by S. Similarly, given an H-module algebra W, the variety of H-module algebras generated by W, denoted by var $^H(W)$ , is the class of all H-module algebras satisfying the H-identities of W. Hence, we say that $A \in $ var $^H(W)$ if and only if $\operatorname {\mathrm {Id}}^H(W) \subseteq \operatorname {\mathrm {Id}}^H(A)$ .

Let $f(x_1, \dots , x_n, Y) \in F^H \langle X \rangle $ be a multilinear H-polynomial, where Y is a set of variables disjoint from $x_1, \dots , x_n$ . We say f is alternating on $\{ x_1, \ldots , x_n \}$ if there exists a multilinear H-polynomial $h(x_1, \ldots , x_n, Y) $ such that

$$ \begin{align*}f(X) = \sum_{\sigma \in S_n} (-1)^\sigma h(x_{\sigma(1)}, \ldots, x_{\sigma(n)}, Y).\end{align*} $$

Equivalently, f is alternating on $\{ x_1, \ldots , x_n \}$ if substituting in $f\ x_i$ by $x_j$ (for every i and j) and vice versa, and we get $-f$ .

2.5 H-cocharacters and H-PI-exponent

From now on, any Hopf algebra is supposed to be finite dimensional over F. Let A be an H-module algebra. Denote by $P_n^H$ the space (of dimension ( $\dim _F H$ ) $^n \cdot n!$ ) of all multilinear H-polynomials in $x_1, \dots , x_n$ , $n \in \mathbb {N}$ , i.e.,

where $S_n$ is the symmetric group on n elements. The space $P_n^H$ has a natural structure of left $S_n$ -module induced by $\sigma \cdot x_1^{h_1} x_2^{h_2} \cdots x_n^{h_n} = x_{\sigma (1)}^{h_1} x_{\sigma (2)}^{h_2} \cdots x_{\sigma (n)}^{h_n} $ , if $\sigma \in S_n$ . Since $P_n^H \cap \operatorname {\mathrm {Id}}^H(A)$ is a subspace invariant under the above action,

is a left $S_n$ -module. This leads us to consider the $S_n$ -character of $P_n^H(A)$ , namely $\chi _n^H(A)$ , which is called nth H-cocharacter of A. The nonnegative integer

is called nth H-codimension of A. Moreover, the sequences $\{ \chi _n^H(A)\}_{n \geq 0} $ and $\{ c_n^H(A) \}_{n \geq 0} $ are called the H-cocharacter sequence of A and the H-codimension sequence of A, respectively.

Given an H-module algebra A, if the limit

$$ \begin{align*}\limsup_{n \to \infty} \sqrt[n]{c_n^H(A)}\end{align*} $$

exists, we shall call it H PI-exponent of A and we shall denote it by exp $^H(A)$ (see [Reference Centrone, Estrada and Ioppolo12, Section 9]).

The existence of the exponent for H-module algebras was studied in [Reference Karasik27] in the case H is finite dimensional and semisimple acting on an associative algebra over a field of characteristic 0. In particular, in [Reference Karasik27], the author proved that the H-exponent exists and is an integer. It is easy to see that Taft’s algebras are not semisimple algebras. In [Reference Gordienko24], the author proved the existence of the exponent for finite dimensional algebras over an algebraically closed field of characteristic 0 that are simple under the action of a Taft’s algebra. We recall Taft’s algebras are noncommutative, noncocommutative, and not semisimple Hopf algebras.

Let $n \geq 1$ be an integer. A partition $\lambda $ of n is a finite sequence of integers $\lambda = (\lambda _1, \dots , \lambda _k)$ such that $\lambda _1 \geq \cdots \geq \lambda _k> 0$ and $\sum _{i=1}^{r}\lambda _i = n$ . In this case, we write $\lambda \vdash n$ or $\left | \lambda \right | = n $ . By representation theory of $S_n$ in characteristic zero, there is a one-to-one correspondence between irreducible $S_n$ -characters and partitions of n see [Reference Curtis and Reiner15] or Sagan’s book [Reference Sagan39]. If $\chi _\lambda $ denotes the irreducible $S_n$ -character corresponding to the partition $\lambda \vdash n$ , then we can write

$$\begin{align*}\chi_n^H(A) = \sum_{\lambda \vdash n} m_\lambda^H \chi_\lambda, \end{align*}$$

where $m_\lambda ^H \geq 0$ are the corresponding multiplicities. The irreducible $S_n$ -submodules of $P_n^H(A)$ can be written as $FS_n e_{T_\lambda }\cdot f$ , where f is some H-polynomial in $P_n^H(A)$ , $T_\lambda $ is some Young tableau of the partition $\lambda \vdash n$ and

$$ \begin{align*}e_{T_\lambda} = \sum_{\sigma \in \mathcal{R}_{T_\lambda}, \tau \in \mathcal{C}_{T_\lambda} } (-1)^\tau \sigma \tau,\end{align*} $$

where $\mathcal {R}_{T_\lambda }$ and $\mathcal {C}_{T_\lambda }$ are the rows and columns stabilizers, respectively. For more details towardH-cocharacters, we remand the reader to Berele’s seminal paper [Reference Berele6].

2.6 Gradings on $UT_2$

Let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over the field F. A detailed description of the G-graded identities satisfied by the algebra $UT_2$ when the characteristic of F is 0 is given in [Reference Valenti42]. In particular, in [Reference Valenti42], the author shows that, up to isomorphism, there is only one nontrivial grading. So any G-grading on $UT_2$ is actually a $\mathbb {Z}_2$ -grading. The algebras with $\mathbb {Z}_2$ -grading are called superalgebras.

2.7 $H_m$ actions on $UT_2$

From now on, F is a field of characteristic zero containing a primitive mth root of the unit; $H_m$ is an mth Taft’s Hopf algebra over F.

Consider the $H_m$ -action on $UT_2$ . Then, by a result of Centrone and Yasumura (see page 738 of [Reference Centrone and Yasumura14]), there exist three structures of $H_m$ -module algebra on $UT_2$ :

1. (i) the trivial grading and d acts trivially: in this case, $\operatorname {\mathrm {Id}}^{H_m}(UT2)$ is merely the ideal of ordinary polynomial identities of $UT_2$ ;

2. (ii) the canonical $\mathbb {Z}_2$ -grading and d acts trivially: in this case, $\operatorname {\mathrm {Id}}^{H_m}(UT2)$ coincides with the ideal of $\mathbb {Z}_2$ -graded polynomial identities of $UT_2$ which was originally calculated by Valenti in [Reference Valenti42]; and

3. (iii) the canonical $\mathbb {Z}_2$ -grading and d acts nontrivially: in this case, necessarily $d =\,{}$ ad $_{\alpha } (a e_{12})$ , for some $0 \neq a \in F$ , that is, if $A = \begin {pmatrix} x_{11} & x_{12} \\ 0 & x_{22} \end {pmatrix} \in UT_2 $ , then

   (1) $$ \begin{align} A^d = \textrm{ad}_{\alpha} (a e_{12}) \left(\begin{pmatrix} x_{11} & x_{12} \\ 0 & x_{22} \end{pmatrix} \right) = \begin{pmatrix} 0 & a(x_{22} - x_{11}) \\ 0 & 0 \end{pmatrix}. \end{align} $$

The Specht properties for (i) and (ii) are particular cases of the Specht property for ordinary PI-algebras [Reference Kemer30] and G-graded PI-algebras [Reference Aljadeff and Kanel-Belov1], respectively. Therefore, we will study the case (iii). Thus, from now on, an $H_m$ -action on $UT_2$ means the canonical $\mathbb {Z}_2$ -grading on $UT_2$ with a nontrivial action of d on $UT_2$ . This forces us to see an action of $H_m$ on the algebra $UT_2$ as an action of $H_2$ on $UT_2$ . It is worth recalling that, in [Reference Gordienko25], the author gives an explicit description of the simple algebras that are module algebra under the action of a Sweedler’s algebra that is a Taft’s algebra of dimension 4.

2.8 The $H_m$ -identities of $UT_2$

Let $F \langle X \rangle $ be the free associative algebra over the countable set $X = \{ x_1, x_2, \dots \}$ . If we write $X = Y \cup Z$ where $Y = \{ y_1, y_2, \dots \}$ is the countable set of variables of degree zero and $Z = \{ z_1, z_2, \dots \}$ is the countable set of variables of degree one, and $Y \cap Z = \emptyset $ , then $F\langle Y\cup Z \rangle $ has a natural structure of free superalgebra on $Y \cup Z$ . The elements of $F\langle Y\cup Z \rangle $ are called graded polynomials.

A graded polynomial $f(y_1,\ldots ,y_t, z_1,\ldots ,z_s) \in F\langle Y\cup Z \rangle $ is a graded identity of the superalgebra $A = A_0 \oplus A_1$ , and we write $f\equiv 0,$ if, for all $a_1, \ldots , a_t \in A_0, b_1, \ldots , b_s \in A_1$ , we have $f(a_1, \ldots , a_t, b_1, \ldots , b_s) = 0.$ We denote by $\operatorname {\mathrm {Id}}^{gr}(A)=\{f\in F\langle Y\cup Z \rangle \mid f\equiv 0 \textrm{ on } A\}$ the ideal of graded identities of A. Notice that $\operatorname {\mathrm {Id}}^{gr}(A)$ is a $T_2$ -ideal of $F\langle Y\cup Z \rangle $ , i.e., an ideal that is invariant under all $\mathbb {Z}_2$ -graded endomorphisms of the free superalgebra $F\langle Y\cup Z \rangle $ . Since the characteristic of F is zero, it is well known that $\operatorname {\mathrm {Id}}^{gr}(A)$ is completely determined by its multilinear graded polynomials.

Now, we construct $F\langle Y\cup Z \mid D_2 \rangle $ the free superalgebra on $X = Y \cup Z$ with action of $D_2 = F \langle d \mid d^2=0 \rangle $ as follows. The algebra $F\langle Y\cup Z \mid D_2 \rangle $ is the algebra freely generated by the set $\{ x^{d_1} = d_1(x) \mid x\in Y $ or $x \in Z, d_1 \in D_2 \} $ . We let $D_2$ act on $F\langle Y\cup Z \mid D_2 \rangle $ by requiring that if $d_1, d_2 \in D_2$ , then $(x^{d_1})^{d_2} = x^{d_1d_2}$ , and then by extending this action on all of $F\langle Y\cup Z \mid D_2 \rangle $ as follows: if $v, w$ are monomials, then define $(vw)^d = v^d w + (-1)^{\deg (v)} v w^d$ and then extend this action by linearity to all of $F\langle Y\cup Z \mid D_2 \rangle $ . The elements of $F\langle Y\cup Z \mid D_2 \rangle $ are called $\mathbb {Z}_2$ – $D_2$ -polynomials.

The algebra $F\langle Y\cup Z \mid D_2 \rangle $ has the following universal property: given any superalgebra $A = A_0 \oplus A_1$ with $D_2$ -action, any set-theoretic map $\varphi \colon Y \cup Z \to A$ such that $\varphi (Y) \subseteq A_0$ and $\varphi (Z) \subseteq A_1$ can be extended to a unique homomorphism of superalgebras $\bar {\varphi } \colon F\langle Y\cup Z \mid D_2 \rangle \to A$ such that $\bar {\varphi }(f^d_1) = \bar {\varphi }(f)^d_1$ , for any $f \in F\langle Y\cup Z \mid D_2 \rangle $ , $d_1 \in D_2$ .

If we let $\Phi $ be the set of all such homomorphisms, then $\operatorname {\mathrm {Id}}^{\mathbb {Z}_2,D_2}(A) = \cap _{\bar {\phi } \in \Phi } \ker \bar {\phi }$ is the ideal of $\mathbb {Z}_2$ – $D_2$ -polynomials identities of A. This means that a $\mathbb {Z}_2$ – $D_2$ -polynomial $f(y_1, \dots , y_s, z_1, \dots , z_t) \in F\langle Y\cup Z \mid D_2 \rangle $ is a $\mathbb {Z}_2$ – $D_2$ -identity for A if $f(a_1, \dots , a_s, b_1, \dots , b_t) = 0$ for all $a_1, \dots , a_s \in A_0$ and $b_1 , \dots , b_t \in A_1$ . We write $f \equiv 0$ on A, in this case.

Assume $\mathbb {Z}_2 = \{ 1, c \}$ .

Proposition 3 [Reference Centrone and Yasumura14, Proposition 14]

$$\begin{align*}\operatorname{\mathrm{Id}}^{H_2}(A) = \operatorname{\mathrm{Id}}^{\mathbb{Z}_2,D_2}(A), \end{align*}$$

and

$$\begin{align*}F^{H_2}\langle X \rangle \cong F\langle Y\cup Z \mid D_2 \rangle. \end{align*}$$

Theorem 4 [Reference Centrone and Yasumura14, Theorem 17]

For each $j = 0,1, \dots , m-1$ , let $\beta _j = \sum _{l=0}^{m-1} \gamma ^{jl} c^l$ , $y_i = x_i^{\beta _0}$ , and $z_i = x_i^{\beta _1}$ . Then the $T^{H_m}$ -ideal of $UT_2$ is generated by the following polynomials:

$$\begin{align*}[y_1, y_2], \, z_1x^hz_2, \, z^d, \, x^{d^2}, \, y_1^d x^h y_2^d, \, x^{\beta_j}, \end{align*}$$

where $h \in H_m$ and $j = 2, \dots , m-1$ .

6.1 Gelfand–Kirillov dimension

In what follows, we shall introduce an H-module algebra version of the GK dimension of the relatively free H-module algebra. Let A be an finitely generated H-module algebra over F, where H is a finite dimensional Hopf algebra over F with F-basis $\{b_1, \dots , b_m \}$ . We shall denote with the symbol $F_k^H(A)$ the relatively free H-module algebra of A in k variables, that is,

$$ \begin{align*} F_k^H(A) := F^H \left \langle x_1, \dots , x_k \right \rangle / ( F^H \left \langle x_1, \dots , x_k \right \rangle \cap T^H(A)). \end{align*} $$

Recall that $F^H \left \langle x_1, \dots , x_k \right \rangle $ is isomorphic to the free algebra over F with free formal generators $x_i^{b_j}$ , where $i \in \{1, \cdots , k \}$ and $j \in \{1, \cdots , m \}$ . Thus,

$$ \begin{align*} F_k^H(A) = \frac{F^H \left \langle x_1^{b_1}, \dots , x_1^{b_m}, \dots , x_k^{b_1}, \dots , x_k^{b_m} \right \rangle}{ F^H \left \langle x_1^{b_1}, \dots , x_1^{b_m}, \dots , x_k^{b_1}, \dots , x_k^{b_m} \right \rangle \cap T^H(A)}. \end{align*} $$

Definition 17 (H-Gelfand–Kirillov dimension in k variables)

Let H be a finite dimensional Hopf algebra over a field F with F-basis $\{b_1, \dots , b_m \}$ , and let A be a finitely generated H-module algebra over F. The H-Gelfand–Kirillov dimension of A in k variables is

In [Reference Belov4], the author studies several properties of $\text {GKdim}_r(A)$ (that is, when H is the trivial Hopf algebra). In particular, it can be proved that $\text {GKdim}_r(A)$ is defined by the complexity type of the algebra A or by a set of semidirect products of matrix algebras over the ring of polynomials from the variety generated by A. See also the paper [Reference Berele7] by Berele for explicit computations of the GK dimension of some remarkable PI-algebras, the paper [Reference Aljadeff and Kanel-Belov2] by Aljadeff and Kanel-Belov the papers [Reference Centrone8, Reference Centrone9] by Centrone, or the surveys [Reference Drensky17] by Drensky and [Reference Centrone10] by Centrone.

Here, we have the next result based on direct computations.

Theorem 18 Let $m,k\geq 2$ integers, and let us consider $H_m$ . Then,

$$\begin{align*}\mathrm{GKdim}_k^{H_m}(UT_2)=2k.\end{align*}$$

Proof From the results showed in Section 2.8, it turns out the monomials

$$\begin{align*}y_1^{a_1}\cdots y_k^{a_k}, y_1^{b_1}\cdots y_k^{b_k}y^dy_1^{c_1}\cdots y_k^{c_k},y_1^{b_1}\cdots y_k^{b_k}zy_1^{c_1}\cdots y_k^{c_k}\end{align*}$$

constitute a linear basis of $F_k^{H_m}(UT_2)$ . Then the growth function of $F_k^{H_m}(UT_2)$ , for sufficiently large n, is

$$\begin{align*}g_V(n):=2\binom{n-1+2k}{2k} +\binom{n+k}{k}+k+2,\end{align*}$$

which grows as a polynomial in n of degree $2k$ , and we are done at light of the definition of GK dimension of an algebra.▪

Because the $H_m$ -exponent of $UT_2$ is 2, the previous result is an experimental confirmation of the following result obtained by one of the authors in [Reference Centrone11] in the environment of graded algebras.

Theorem 19 Let A be a finite dimensional G-graded algebra. Then,

$$\begin{align*}\mathrm{GKdim}_k^{G}(A)=\mathrm{exp}^G(A)k+\alpha,\end{align*}$$

where $\alpha $ is integer and $\mathrm {exp}^G(A)$ denotes the G-graded exponent of A.

We can formulate the next conjecture.

Conjecture 20 Let A be a finite dimensional $H_m$ -module algebra. Then,

$$\begin{align*}\mathrm{GKdim}_k^{H_m}(A)=\mathrm{exp}^{H_m}(A)k+\alpha,\end{align*}$$

where $\alpha $ is integer and $\mathrm {exp}^{H_m}(A)$ denotes the $H_m$ -exponent of A.

6.2 Automata algebras and regular languages

Given a (possibly infinite) set of variables X, we consider the set of monomials $M(X)$ in the free algebra $F\langle X\rangle $ . We define formal language any subset L of $M(X)$ .

A language is said to be regular if it can be generated by a regular grammar. We remand to the book of [Reference Eilenberg19] for a general overview toward grammars, languages, and formal computer science. Notice that regular languages can be obtained from finite languages by means of regular operations. We have the following, well-known, result due to Kleene (see [Reference Kleene32]).

We recall that a monomial algebra over a field F is an algebra that is isomorphic to $F\langle X\rangle /I$ , where I is generated by elements of $M(X)$ . Moreover, a monomial algebra is said to be automata if I is a regular language. We advise the reader to read the book by [Reference Belov, Borisenko and Latyshev5] for a compendium about monomial algebras. With a slight modification of the original definition of automata algebra, we have the notion of automata algebra on relatively free algebra.

Of course, the previous definition can be adapted trivially to graded algebras, module algebras, and so on. It is also convenient to recall a sort of duality between regularity of a monomial ideal and regularity of its quotient algebra.

By the proof of Theorem 18, we get a basis of the relatively free $H_m$ -algebra of $UT_2$ in k variables is monomial and can be written in terms of regular operation as

$$\begin{align*}\{y_1\}^*\cdots \{y_k\}^*\cup\{y_1\}^*\cdots \{y_k\}^*\{y^d\}\{y_1\}^*\cdots \{y_k\}^*\cup\{y_1\}^*\cdots \{y_k\}^*\{z\}\{y_1\}^*\cdots \{y_k\}^*,\end{align*}$$

i.e., in the light of Proposition 24 and Theorem 22, $\hat {I}$ is regular. This is the content of the following.