Separable deformations of the generalized quaternion group algebras

TL;DR

This paper constructs separable deformations of group algebras of generalized quaternion groups over certain fields, matching their simple component dimensions with those over complex numbers, thus solving the Donald-Flanigan conjecture for these groups.

Contribution

It introduces explicit separable deformations of generalized quaternion group algebras over fields containing specific finite fields, confirming the conjecture for these groups.

Findings

Deformed algebras are separable over $k((t))$.

Dimensions of simple components match those over $\,\mathbb{C}$.

Provides strong solutions to the Donald-Flanigan conjecture for generalized quaternion groups.

Abstract

The group algebras $kQ_{2^n}$ of the generalized quaternion groups $Q_{2^n}$ over fields $k$ which contain $\mathbb{F}_{2^{n-2}}$, are deformed to separable $k((t))$-algebras $[kQ_{2^n}]_t$. The dimensions of the simple components of $\overline{k((t))}\otimes_{k((t))}[kQ_{2^n}]_t$ over the algebraic closure $\overline{k((t))}$, and those of $\mathbb{C} Q_{2^n}$ over $\mathbb {C}$ are the same, yielding strong solutions of the Donald-Flanigan conjecture for the generalized quaternion groups.

Equations

$$[\beta ]:1\to C_{2^{n-1}}\to Q_{2^n}\to C_2\to 1,$$

$$\eta (\bar{\tau }):\sigma \mapsto {\sigma }^{2^{n-1}-1}(={\sigma }^{-1}),$$

$$\beta (1,1)=\beta (1,\bar{\tau })=\beta (\bar{\tau },1)=1,\beta (\bar{\tau },\bar{\tau })={\sigma }^{2^{n-2}}.$$

$$g(y):=y^2-{\sigma }^{2^{n-2}}\in k{C}_{2^{n-1}}[y;\eta ].$$

$$\displaystyle\begin{array}[]{rcl}kC_{2^{n-1}}&\xrightarrow{\simeq}&k[x]/\langle\,x^{2^{n-1}}-1\,\rangle\\ \sigma&\mapsto&{\bar{x}}:=x+\langle\,x^{2^{n-1}}-1\,\rangle.\end{array}$$

$$[k{Q}_{2^n}{]}_t:=[k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]/⟨g_t(y)⟩$$

$$\overline{k((t))}{\otimes }_{k((t))}[k{Q}_{2^n}{]}_t\simeq \overline{k((t))}{\otimes }_{k((t))}[k{D}_{2^n}{]}_t\simeq \underset{i=1}{\overset{4}{⨁}}\overline{k((t))}\oplus \underset{i=1}{\overset{2^{n-2}-1}{⨁}}{M}_2(\overline{k((t))}).$$

$$\mathbb{C}{Q}_{2^n}\simeq \mathbb{C}{D}_{2^n}\simeq \underset{i=1}{\overset{4}{⨁}}\mathbb{C}\oplus \underset{i=1}{\overset{2^{n-2}-1}{⨁}}{M}_2(\mathbb{C}).$$

$$U:=\{1+tk[[t]]\}\subseteq k[[t]]$$

$${\pi }_t(x):=(x+c_1)(x+c_2)\prod _{a\in {\mathbb{F}}_{2^{n-2}}^{\ast }}(x^2+atx+1)\in k((t))[x].$$

$$[k{C}_{2^{n-1}}{]}_t:=k((t))[x]/⟨{\pi }_t(x)⟩$$

$$e_{c_1}\in ⟨\frac{{\pi }_t(\bar{x})}{\bar{x}+c_1}⟩,e_{c_2}\in ⟨\frac{{\pi }_t(\bar{x})}{\bar{x}+c_2}⟩,e_a\in ⟨\frac{{\pi }_t(\bar{x})}{{\bar{x}}^2+at\bar{x}+1}⟩,\forall a\in {\mathbb{F}}_{2^{n-2}}^{\ast }.$$

$$k_1:=[k{C}_{2^{n-1}}{]}_t{e}_{c_1},k_2:=[k{C}_{2^{n-1}}{]}_t{e}_{c_2}$$

$$K_a:=[k{C}_{2^{n-1}}{]}_t{e}_a\simeq k((t))[x]/⟨x^2+atx+1⟩$$

$$[k{C}_{2^{n-1}}{]}_t=\underset{a\in {\mathbb{F}}_{2^{n-2}}^{\ast }}{⨁}{K}_a\oplus k_1\oplus k_2.$$

$$({\bar{x}}^{2^{n-2}}+t^{2^{n-2}-1}\bar{x})e_a=d_a{e}_a.$$

$$p_t(x):=(d{x}^2+btx+1)\prod _{a\in {\mathbb{F}}_{2^{n-2}}^{\ast }}(x^2+atx+1)\in k((t))[x]$$

$$p_{t=0}(x)=x^{2^{n-1}}+1.$$

$$f_t(x):=p_t(x)+x^2+1\in k((t))[x]$$

$$f_t(c_i)=p_t(c_i)+c_i^2+1=0,i=1,2.$$

$${\tilde{\eta }}_t(x):=\frac{p_t(x)+1}{x}\in k((t))[x].$$

$$\displaystyle\tilde{\eta_{t}}:\begin{array}[]{ccc}k((t))[x]&\rightarrow&k((t))[x]\\ x&\mapsto&\tilde{\eta}_{t}(x)\end{array}.$$

$$\displaystyle\eta_{t}:\begin{array}[]{ccc}k((t))[x]/\langle\,\pi_{t}(x)\,\rangle&\rightarrow&k((t))[x]/\langle\,\pi_{t}(x)\,\rangle\\ {\bar{x}}&\mapsto&\overline{\tilde{\eta}_{t}(x)}\end{array},$$

$${\eta }_t(e_{c_1})=e_{c_1},{\eta }_t(e_{c_2})=e_{c_2},{\eta }_t(e_a)=e_a,\forall a\in {\mathbb{F}}_{2^{n-2}}^{\ast },$$

$${\eta }_t(\bar{x}{e}_a)=(\bar{x}+at)e_a.$$

$${\eta }_{t=0}:\bar{x}\mapsto \frac{p_{t=0}(\bar{x})+1}{\bar{x}}={\bar{x}}^{2^{n-1}-1}.$$

$$[k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]=(k((t))[x]/⟨{\pi }_t(x)⟩)[y;{\eta }_t]$$

$$[k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]=\underset{a\in {\mathbb{F}}_{2^{n-2}}^{\ast }}{⨁}[k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]e_a\oplus [k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]e_{c_1}\oplus [k{C}_{2^{n-1}}{]}_t[y;{\eta }_t]e_{c_2}.$$

$$g_t(y):=y^2+z(e_{c_1}+e_{c_2})y+{\bar{x}}^{2^{n-2}}+t^{2^{n-2}-1}\bar{x}\in [k{C}_{2^{n-1}}{]}_t[y;{\eta }_t].$$

$$z(e_{c_1}+e_{c_2}{)}_{t=0}=0$$

Full text

The Donald-Flanigan (DF) conjecture [2] says that any group algebra $kG$ of a finite group $G$ over a field $k$ admits a separable deformation. It was verified in [3, 4, 5, 6, 7, 8, 11, 12, 13] for certain families of finite groups. In [1] a separable deformation was constructed for the quaternion group $G=Q_{8}$, turning the generalized quaternion group $Q_{16}$ to a current minimal unsolved case. In this note, we extend the strategy of [1] in order to deal, as promised therein, with the family of generalized quaternion groups $Q_{2^{n}}$. As a by-product we establish separable deformations for the family of dihedral 2-groups $D_{2^{n}}$ (K. Erdmann and M. Schaps have already found separable deformations for this family in [4]). Certainly, for both families of 2-groups, the interesting case is where the field $k$ is of characteristic 2. For our considerations, $k$ is further assumed to contain the Galois field $\mathbb{F}_{2^{n-2}}$ of $2^{n-2}$ elements. The solutions to the DF conjecture for $Q_{2^{n}}$ and $D_{2^{n}}$ established here are so called strong.

1. Background

Let $A$ be a $k$-algebra (all the algebras throughout are associative), let $k[[t]]$ be the ring of formal power series over $k$, and let $k((t))$ be its field of fractions. Suppose that the free $k[[t]]$-module $\Lambda_{t}:=k[[t]]\otimes_{k}A$ admits a multiplication such that there is an isomorphism $A\simeq\Lambda_{t}/\langle t\Lambda_{t}\rangle$ of $k$-algebras. Then the $k((t))$-algebra $A_{t}:=k((t))\otimes_{k[[t]]}\Lambda_{t}$ is called a deformation of $A$. The algebra $A$ specializes $\Lambda_{t}$ at $t=0$. Even though $t$ is invertible in a deformation $A_{t}$, we adopt an abuse of language saying that $A$ is a specialization also of $A_{t}$ at $t=0$.

Consider the extension

[TABLE]

with an action $\eta:C_{2}\to\text{Aut}(C_{2^{n-1}})$ of $C_{2}=\langle\,\bar{\tau}\,\rangle$ on $C_{2^{n-1}}=\langle\,\sigma\,\rangle$ via

[TABLE]

and an associated 2-cocycle $\beta:C_{2}\times C_{2}\to C_{2^{n-1}}$ representing (1.1) which is given by

[TABLE]

The group algebra $kQ_{2^{n}}$ can be viewed as follows. First, the group automorphism (1.2) of $C_{2^{n-1}}$ can be extended to an algebra automorphism of the subgroup algebra $kC_{2^{n-1}}$ taking $\sigma$ to its inverse. Next, let $kC_{2^{n-1}}[y;\eta]$ be the skew polynomial ring (see [10, §1.2]) over $kC_{2^{n-1}}$, where we keep the notation $\eta$ for the action of the indeterminate $y$ on $kC_{2^{n-1}}$ via the above extension of (1.2). Then $kQ_{2^{n}}$ is isomorphic to the quotient $kC_{2^{n-1}}[y;\eta]/\langle\,g(y)\,\rangle$ of this skew polynomial ring by the central polynomial

[TABLE]

The base algebra $kC_{2^{n-1}}$ can itself be identified with a quotient

[TABLE]

The above description is good for any field $k$. From now on, the above condition $\mathbb{F}_{2^{n-2}}\subseteq k$ is entailed.

Here is a layout of the paper. In §2.1 the subgroup algebra $kC_{2^{n-1}}$ is deformed to a separable algebra $[k{C_{2^{n-1}}}]_{t}$ which is isomorphic to a direct sum of $2^{n-2}+1$ fields $\bigoplus_{a\in\mathbb{F}^{*}_{2^{n-2}}}K_{a}\oplus k((t))\oplus k((t))$, where $K_{a}$ are separable field extensions of $k((t))$ of degree 2. The next step (§2.2) is to construct an automorphism $\eta_{t}$ of $[k{C_{2^{n-1}}}]_{t}$ which agrees with the action $\eta$ of $C_{2}$ on $kC_{2^{n-1}}$ when specializing $t=0$. This action fixes all the primitive idempotents of $[{kC_{2^{n-1}}}]_{t}$. By that we obtain the skew polynomial ring $[k{C_{2^{n-1}}}]_{t}[y;\eta_{t}]$. In §2.3 we deform the polynomial $g(y)$ (1.3) to a separable polynomial $g_{t}(y)$ of degree 2 in $y$, which lies in the center of $[kC_{2^{n-1}}]_{t}[y;\eta_{t}]$. Factoring out the two-sided ideal generated by $g_{t}(y)$, we establish a deformation

[TABLE]

of $kQ_{2^{n}}={kC_{2^{n-1}}}[y;\eta]/\langle\,g(y)\,\rangle$. The proofs of the claims of §2 are postponed and given in §4. In §3 we show that $[{kQ_{2^{n}}}]_{t}$ as above is separable. In §5 we adapt the above strategy to the $2^{n}$-dihedral group algebra $kD_{2^{n}}$, constructing a separable deformation $[{kD_{2^{n}}}]_{t}$. Moreover, passing to the algebraic closure $\overline{k((t))}$ we have

[TABLE]

These are strong solutions to the DF conjecture since their decompositions to simple components afford the same dimensions as

[TABLE]

2. The deformation

2.1.

We first deform the cyclic group algebra $kC_{2^{n-1}}$ as follows. Let

[TABLE]

be the group of 1-units of $k[[t]]$. For any distinct 1-units $c_{1},c_{2}\in U$ (which are determined more precisely in the next section), define

[TABLE]

Then $\pi_{t=0}(x)=x^{2^{n-1}}+1$, and hence by (1.6), the algebra

[TABLE]

is a deformation of $kC_{2^{n-1}}$. Note that for any element $a\in\mathbb{F}_{2^{n-2}}^{*}(=\mathbb{F}_{2^{n-2}}\setminus\{0\})$, the polynomial $x^{2}+atx+1\in k((t))[x]$ is separable. Furthermore, it does not admit roots mod($t^{2}$), and is hence irreducible. The polynomial $\pi_{t}(x)$ is given as a product (2.1) of distinct irreducible polynomials, each of which is separable, and so is itself separable. The commutative algebra $[kC_{2^{n-1}}]_{t}$ is then a sum of fields, corresponding to the irreducible factors of $\pi_{t}(x)$ as follows. Denote the primitive idempotents of $[kC_{2^{n-1}}]_{t}$ by

[TABLE]

The fields

[TABLE]

are one-dimensional over $k((t))$, and the fields

[TABLE]

are two-dimensional over $k((t))$ for every ${a\in\mathbb{F}^{*}_{2^{n-2}}}$. Write

[TABLE]

As customary, denote ${\bar{x}}:=x+\langle\,\pi_{t}(x)\,\rangle\in[kC_{2^{n-1}}]_{t}$. We record the following claim for a later use, it is proven in §4.1.

Lemma 2.1**.**

For every $a\in\mathbb{F}^{*}_{2^{n-2}}$ there exists $d_{a}\in k[[t]]\setminus\{0\}$ such that

[TABLE]

2.2.

Our next step is to deform the action $\eta$. Let

[TABLE]

for some $d\in U$ and $b\in k[[t]]$ (both are later to be chosen). Then it is not hard to verify that

[TABLE]

Lemma 2.2**.**

With the notation (2.4) there exist $d\in U$ and $b\in k[[t]]$ such that the polynomial

[TABLE]

admits two distinct roots in $U$.

Owing to Lemma 2.2, whose proof can be found in §4.2, we choose $b\in k[[t]]$ and $d\in U$ such that $c_{1},c_{2}\in U$ are distinct roots of $f_{t}(x)$. These are the 1-units in $\pi_{t}(x)$ (2.1). We record

[TABLE]

Note that since $p_{t}(0)=1$, then $p_{t}(x)+1\in xk((t))[x]$ (the ideal generated by $x$ in the polynomial algebra $k((t))[x]$), and hence

[TABLE]

The polynomial (2.8) determines a $k((t))$-algebra endomorphism

[TABLE]

Lemma 2.3**.**

With the notation above, $\tilde{\eta_{t}}$ induces an automorphism

[TABLE]

of order 2, which fixes all the idempotents of $[kC_{2^{n-1}}]_{t}=k((t))[x]/\langle\,\pi_{t}(x)\,\rangle$

[TABLE]

furthermore, for every $a\in\mathbb{F}^{*}_{2^{n-2}}$

[TABLE]

By Lemma 2.3, whose proof can be found in §4.3, $\eta_{t}$ induces automorphisms of order 2 of the fields $K_{a}\subseteq[kC_{2^{n-1}}]_{t}$ while fixing the two fields $k_{1},k_{2}\subseteq[kC_{2^{n-1}}]_{t}$ pointwise. Furthermore, by the definitions (2.8), (2.11), and equation (2.5) we have

[TABLE]

Consequently, the automorphism $\eta_{t=0}$ of $[{kC_{2^{n-1}}}]_{t=0}$ agrees with the automorphism $\eta(\bar{\tau})$ of $kC_{2^{n-1}}$ (with the identification (1.6)). The skew polynomial ring

[TABLE]

is therefore a deformation of $kC_{2^{n-1}}[y;\eta]$. Note that by (2.15), the idempotents $e_{c_{1}},e_{c_{2}},e_{a}(a\in\mathbb{F}^{*}_{2^{n-2}})$ are central in $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}]$ and hence

[TABLE]

2.3.

We complete the construction of $[kQ_{2^{n}}]_{t}$ by deforming the polynomial $g(y)$ (1.3). For any $z\in k[[t]]$ let

[TABLE]

Lemma 2.4**.**

With the notation (2.18), $g_{t}(y)$ is in the center of $[kC_{2^{n-1}}]_{t}[y;\eta_{t}].$ Consequently, $\langle\,g_{t}(y)\,\rangle=g_{t}(y)[kC_{2^{n-1}}]_{t}[y;\eta_{t}]$ is a two-sided ideal.

The proof of Lemma 2.4 is given in §4.4. For every $z\in k[[t]]$ the element $z(e_{c_{1}}+e_{c_{2}})$ lies in $[kC_{2^{n-1}}]_{t}$. Choose a non-zero $z\in k[[t]]$ such that

[TABLE]

($z$ can be taken as $t^{m}$ for sufficiently large $m$). Plugging this choice of $z$ in (2.18) and identifying $\sigma$ and ${\bar{x}}$ as in (1.6) we have

[TABLE]

Lemma 2.4 and equation (2.19) yield that

[TABLE]

is a deformation of $kQ_{2^{n}}$.

3. Separability of $[kQ_{2^{n}}]_{t}$

Separability of the deformed algebra $[kQ_{2^{n}}]_{t}$ is proved in the same fashion as in [1] for the case $n=3$. By (2.3), (2.17) and (2.20),

[TABLE]

We now show that all the direct summands of (3.1) are separable.

Let $a\in\mathbb{F}^{*}_{2^{n-2}}$. The non-zero element $d_{a}\in k[[t]]$ provided in Lemma 2.1, as well as orthogonality of the central idempotents $e_{a}$, $e_{c_{1}}$ and $e_{c_{2}}$ yield

[TABLE]

We obtain

[TABLE]

The rightmost term is the crossed product of the group $C_{2}:=\langle\,\bar{\tau}\,\rangle$ acting faithfully on the field $K_{a}$ via $\eta_{t}$ (2.16), with a twisting determined by the 2-cocycle

[TABLE]

This is a central simple algebra over the subfield of invariants $k((t))$ [9, Theorem 4.4.1]. Evidently, this simple algebra is split by $\overline{k((t))}$, i.e.

[TABLE]

Next, by Lemma 2.3, the action $\eta_{t}$ is trivial on both $k_{1}$ and $k_{2}$, hence we may regard the skew polynomial rings $k_{1}[y;\eta_{t}]$ and $k_{2}[y;\eta_{t}]$ as ordinary polynomial rings $k_{1}[y]$ and $k_{2}[y]$ respectively. Equation (2.2) yields $({\bar{x}}+c_{i})e_{c_{i}}=0$ for $i=1,2,$ in other words

[TABLE]

Orthogonality of the idempotents $e_{c_{1}}$ and $e_{c_{2}}$, together with (3.3) yields

[TABLE]

We obtain

[TABLE]

and

[TABLE]

The polynomials $y^{2}+zy+{c_{1}}^{2^{n-2}}+t^{2^{n-2}-1}{c_{1}}$ and $y^{2}+zy+{c_{2}}^{2^{n-2}}+t^{2^{n-2}-1}{c_{2}}$ in $k[y]$ are separable (since $z$ is non-zero) and split as products of degree-1 polynomials over the algebraic closure $\overline{k((t))}$. Both $k_{1}[y;\eta_{t}]/\langle\,g_{t}(y)e_{c_{1}}\,\rangle$ and $k_{2}[y;\eta_{t}]/\langle\,g_{t}(y)e_{c_{2}}\,\rangle$ are thus separable $k((t))$-algebras and

[TABLE]

By (3.1),(3.2) and (3.6), a strong solution is established

[TABLE]

4. Proofs

4.1. Proof of Lemma 2.1

Since the idempotent $e_{a}$ lies in the ideal generated by $\frac{\pi_{t}({\bar{x}})}{{\bar{x}}^{2}+at{\bar{x}}+1}$ (2.2) it is enough to prove that for every $a\in\mathbb{F}^{*}_{2^{n-2}}$ there exists $d_{a}\in k[[t]]\setminus\{0\}$ such that

[TABLE]

Indeed, for any non-negative integer $j$

[TABLE]

and so

[TABLE]

Putting $j=n-3$ and then $j=n-4$ in (4.2) we obtain for every $a\in\mathbb{F}^{*}_{2^{n-2}}$

[TABLE]

Proceeding the iteration of (4.2) yields

[TABLE]

where $d_{a}$ is a sum of certain powers of $at$, and hence does not depend on $x$. We now make use of the fact that the elements $a\in\mathbb{F}^{*}_{2^{n-2}}$ satisfy $a^{2^{n-2}-1}=1$. Consequently,

[TABLE]

and (4.1) is obtained.∎

4.2. Proof of Lemma 2.2

For the sake of simplicity we denote

[TABLE]

Note that

[TABLE]

By (4.4), using the fact that the product of invertible elements in $\mathbb{F}_{2^{n-2}}$ equals 1, we have

[TABLE]

which we record for a later use.

We prove the lemma by an example of two distinct 1-units which annihilate the polynomial $f_{t}(x)$ for certain

[TABLE]

Our 1-units are $1+\alpha_{1}t^{m}$ and $1+\alpha_{2}t^{m}$, where $m\geq 2^{n-2}$ and where $\alpha_{1},\alpha_{2}\in k^{*}$ are any distinct elements. With the notation (2.4), (2.6) and (4.3), the elements $1+\alpha_{1}t^{m}$ and $1+\alpha_{2}t^{m}$ are roots of $f_{t}(x)$ if and only if for $i=1,2$

[TABLE]

that is, if and only if

[TABLE]

Consider (4.7) as a system of two non-homogeneous linear equations in the variables $b$ and $d$. We show that the system admits a (unique) solution satisfying (4.6). Indeed, solving the system (4.7) for $b$, it can be verified using (4.5) that

[TABLE]

Our choice of $m\geq 2^{n-2}$ ascertains that the left condition in (4.6) is fulfilled. Returning to (4.7), using the fact that $b\in k[[t]]$ was just established, we get

[TABLE]

Since $1+\alpha_{i}^{2}t^{2m}\in U$, we deduce that $d$ satisfies the right condition in (4.6). ∎

4.3. Proof of Lemma 2.3

We show that $\tilde{\eta}_{t}$ takes the ideal generated by each irreducible factor of $\pi_{t}(x)$ to itself. First, by the definition of $\tilde{\eta}_{t}$ (2.8),

[TABLE]

Then by (2.7) it follows that $c_{1}$ and $c_{2}$ annihilate the polynomials $\tilde{\eta}_{t}(x+c_{1})$ and $\tilde{\eta}_{t}(x+c_{2})$ respectively. Hence

[TABLE]

Next, for every $a\in\mathbb{F}^{*}_{2^{n-2}}$ develop

[TABLE]

Computing (4.9) modulo $x^{2}+atx+1$, bearing in mind that by the definition (2.4)

[TABLE]

we get

[TABLE]

Since the polynomial $x^{2}$ is prime to $x^{2}+atx+1$ we obtain

[TABLE]

From equations (4.8) and (4.11) we conclude that $\tilde{\eta_{t}}(\pi_{t}(x))\in\langle\,\pi_{t}(x)\,\rangle$ and thus (2.14) is a well-defined $k((t))$-algebra morphism. Moreover, by (4.8) and (4.11), all the minimal ideals of $[kC_{2^{n-1}}]_{t}$, namely $\langle\,\frac{\pi_{t}({\bar{x}})}{{\bar{x}}+c_{1}}\,\rangle,\langle\,\frac{\pi_{t}({\bar{x}})}{{\bar{x}}+c_{2}}\,\rangle,$ and $\langle\,\frac{\pi_{t}({\bar{x}})}{{\bar{x}}^{2}+at{\bar{x}}+1}\,\rangle_{a\in\mathbb{F}^{*}_{2^{n-2}}}$ are stable under $\eta_{t}$. Hence, every primitive idempotent (see (2.2)) is either fixed or vanishes under $\eta_{t}$. Since these primitive idempotents sum up to 1, and since $\eta_{t}(1)=1$, it follows that all are fixed by $\eta_{t}$ proving (2.15) and, in particular, that $\eta_{t}$ is an automorphism of $[kC_{2^{n-1}}]_{t}$.

Finally, for every $a\in\mathbb{F}^{*}_{2^{n-2}}$

[TABLE]

We apply (2.15), which has just been verified, together with (4.10) and (4.12) to obtain $\eta_{t}({\bar{x}}e_{a})=\frac{1}{{\bar{x}}}\cdot e_{a}=({\bar{x}}+at)e_{a}$ proving (2.16). By equations (2.15) and (2.16) the automorphism $\eta_{t}^{2}$ is an identity on the elements

[TABLE]

which form a $k((t))$-basis for $k((t))[x]/\langle\,\pi_{t}(x)\,\rangle$. Hence $\eta_{t}$ is of order 2. ∎

4.4. Proof of Lemma 2.4

We show that each of the terms of $g_{t}(y)$, namely the leading term, the free term and the term $z(e_{c_{1}}+e_{c_{2}})y$ lie in the center of $[kC_{2^{n-1}}]_{t}[y;\eta_{t}].$ First, the leading term $y^{2}$ is central since by Lemma 2.3 the automorphism $\eta_{t}$ is of order 2. Next, in order to prove that the free (of $y$) term ${\bar{x}}^{2^{n-2}}+t^{2^{n-2}-1}{\bar{x}}\in[kC_{2^{n-1}}]_{t}$ is central, it is enough to show that multiplying it with all the idempotents of $[kC_{2^{n-1}}]_{t}$, i.e., $e_{c_{1}},e_{c_{2}}$ and $e_{a}$ (for every $a\in\mathbb{F}_{2^{n-2}}$) yield $\eta_{t}$-invariant elements in $[kC_{2^{n-1}}]_{t}$. This is clear for $e_{c_{1}},e_{c_{2}}$ since by Lemma 2.3, the subspace Span${}_{k((t))}\{e_{c_{1}},e_{c_{2}}\}=k_{1}\oplus k_{2}$ is $\eta_{t}$-invariant. As for the idempotents $e_{a}$, Lemma 2.1 says that for every $a\in\mathbb{F}^{*}_{2^{n-2}}$ the projection $({\bar{x}}^{2^{n-2}}+t^{2^{n-2}-1}{\bar{x}})e_{a}$ is equal to $d_{a}e_{a}$ for some $d_{a}\in k[[t]]$ (i.e. independent of $x$). Again by Lemma 2.3, these projections are also $\eta_{t}$-invariant.

It is left to check that the term $z(e_{c_{1}}+e_{c_{2}})y$ is central. We show that it commutes with each component of the decomposition (2.17). Indeed, since $e_{c_{1}}$ and $e_{c_{2}}$ are $\eta_{t}$-invariant (2.15), then $z(e_{c_{1}}+e_{c_{2}})y$ commutes both with $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}]e_{c_{1}}$ and $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}]e_{c_{2}}$. Furthermore, for every $a\in\mathbb{F}^{*}_{2^{n-2}}$, $e_{a}$ is orthogonal to $e_{c_{1}}$ as well as to $e_{c_{2}}$. Thus,

[TABLE]

and hence $z(e_{c_{1}}+e_{c_{2}})y$ commutes with $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}]$. ∎

5. Dihedral 2-groups

A slight modification of above construction yields a separable deformation of $kD_{2^{n}}$, where $D_{2^{n}}$ is the dihedral group of order $2^{n}$ (and $\mathbb{F}_{2^{n-2}}\subseteq k$ as before). It is outlined herein only briefly since the DF conjecture for this family of 2-groups has already been solved in [4].

The dihedral group $D_{2^{n}}$ admits a split extension $1\to C_{2^{n-1}}\to D_{2^{n}}\to C_{2}\to 1,$ with the same action $\eta(\bar{\tau}):\sigma\mapsto\sigma^{2^{n-1}-1}$ of $C_{2}=\langle\,\bar{\tau}\,\rangle$ on $C_{2^{n-1}}=\langle\,\sigma\,\rangle$ as in the extension (1.1). Then the dihedral group algebra satisfies

[TABLE]

We deform the group algebra $kC_{2^{n-1}}$ and the action $\eta$ exactly as in §2.1 and §2.2 respectively so as to obtain the deformed skew polynomial algebra $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}].$ The difference from the construction in §2 for the generalized quaternions is manifested in the polynomial $y^{2}+z(e_{c_{1}}+e_{c_{2}})y+1\in[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}],$ where $e_{c_{1}}$ and $e_{c_{2}}$ are the primitive idempotents as in §2.1, and $z\in k[[t]]$ is the same as in §2.3. This polynomial replaces (2.18) having a different free term. Its centrality in $[{kC_{2^{n-1}}}]_{t}[y;\eta_{t}]$ follows from Lemma 4.4 (the free term here is obviously central). Define

[TABLE]

Then it is easy to verify that $[kD_{2^{n}}]_{t}$ is indeed a deformation of $kD_{2^{n}}$. Separability of $[kD_{2^{n}}]_{t}$ is proven similarly to §3, moreover we establish again a strong solution

[TABLE]

Acknowledgement. The author thanks A. Amsalem for useful discussions.