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Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $V$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $V$ have a name, or does $V$ belong to a more well-known class of algebras?

Background: I want to understand the nilpotent elements of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3}$, where $A$ is a certain non-commutative $\mathbb{F}_p$-algebra. When there is an irreducible binomial $T^3 - \lambda \in \mathbb{F}_p[T]$ (this is the case when $p=3$ or $p \equiv 1 \bmod 3$), we get equations of the type above for any element $a+bT+cT^2$ of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3} = A[T]/\langle T^3 - \lambda \rangle$ that squares to $0$.

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $V$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $V$ have a name, or does $V$ belong to a more well-known class of algebras?

Background: I want to understand the nilpotent elements of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3}$, where $A$ is a certain non-commutative $\mathbb{F}_p$-algebra. When there is an irreducible binomial $T^3 - \lambda \in \mathbb{F}_p[T]$ (this is the case when $p \equiv 1 \bmod 3$), we get equations of the type above for any element $a+bT+cT^2$ of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3} = A[T]/\langle T^3 - \lambda \rangle$ that squares to $0$.

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $A$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $A$ have a name, or does $A$ belong to a more well-known class of algebras?

Background: I want to understand the nilpotent elements of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3}$, where $A$ is a certain non-commutative $\mathbb{F}_p$-algebra. When there is an irreducible binomial $T^3 - \lambda \in \mathbb{F}_p[T]$, we get equations of the type above for any element $a+bT+cT^2$ of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3} = A[T]/\langle T^3 - \lambda \rangle$ that squares to $0$.

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $V$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $V$ have a name, or does $V$ belong to a more well-known class of algebras?

Background: I want to understand the nilpotent elements of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3}$, where $A$ is a certain non-commutative $\mathbb{F}_p$-algebra. When there is an irreducible binomial $T^3 - \lambda \in \mathbb{F}_p[T]$ (this is the case when $p=3$ or $p \equiv 1 \bmod 3$), we get equations of the type above for any element $a+bT+cT^2$ of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3} = A[T]/\langle T^3 - \lambda \rangle$ that squares to $0$.

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $A$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $A$ have a name, or does $A$ belong to a more well-known class of algebras?

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $A$ generated by the $a^i b^j c^k$ with $i,j,k \geq 0$ as an $\mathbb{F}_p$-module?

It is enough to show that $b a^i b^j$ and $c a^i b^j$ lie in the span. I tried induction on $i+j$, but I am going in circles. Even for $i+j=2$ already. Maybe it is not true actually? The corresponding question for $2$ generators has a positive answer, though.

Also, does $A$ have a name, or does $A$ belong to a more well-known class of algebras?

Background: I want to understand the nilpotent elements of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3}$, where $A$ is a certain non-commutative $\mathbb{F}_p$-algebra. When there is an irreducible binomial $T^3 - \lambda \in \mathbb{F}_p[T]$, we get equations of the type above for any element $a+bT+cT^2$ of $A \otimes_{\mathbb{F}_p} \mathbb{F}_{p^3} = A[T]/\langle T^3 - \lambda \rangle$ that squares to $0$.