# Generic Algebras

Two of the most fundamental theorems of noncommutative algebras are the Wedderburn-Malcev decomposition and the Artin-Wedderburn theorem. The first states that if $W$ is any Artinian algebra and $J(W)$ is its Jacobson radical, then $W_{ss}\cong{W}/{J(W)}$ is semisimple Artinian, and there is a vector space decomposition $W=J(W)\oplus W_{ss}$. The second states that any semisimple Artinian (and in particular finite dimensional) algebra is isomorphic to a product of matrix algebras over division algebras. In a sense, these two theorems tells us that (up to a radical) the building blocks of noncommutative algebras are the algebras of the form $M_{n}(D)$ where $D$ is a division algebra over our base field $\mathbb{F}$. Our goal is try to understand these algebras.

The best known noncommutative division algebra is the real quaternion algebra $\mathbb{H}_\mathbb{R}$. Recall that we have an embedding of the real quaternions inside $M_2(\mathbb{C})$ as follows

$\mathbb{H}_\mathbb{R}=\mathbb{R} \overbrace{ \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) }^1$ $\oplus \mathbb{R} \overbrace{ \left( \begin{array}{cc} i & 0 \\ 0 &- i \end{array} \right) }^i$ $\oplus \mathbb{R} \overbrace{ \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) }^j$ $\oplus \mathbb{R} \overbrace{ \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right) }^{ij} .$

Since the basis over $\mathbb{R}$ of the quaternions is also a basis over $\mathbb{C}$ of $M_2(\mathbb{C})$, we conclude that the scalar extension $\mathbb{H}_\mathbb{C}:=\mathbb{C}\otimes_\mathbb{R} \mathbb{H}_\mathbb{R}$ is isomorphic to $M_2(\mathbb{C})$.

This is true in general. Every finite dimensional division algebra $D$  central over a field $\mathbb{F}$ becomes a matrix algebra after suitable scalar extension, or equivalently can be embedded “nicely” inside $M_n(\mathbb{F}^{alg})$ for a suitable $n$. Thus, we are left to study these “special” subalgebras of matrix algebras (usually over a smaller field) which are division algebra, or, more generally, central simple algebras (CSA), namely are isomorphic to $M_m(D)$ where $Z(D)=\mathbb{F}$.

In order to get all of these algebras, consider the generic matrices $X^{(k)}=(x_{i,j}^{(k)}), Y^{(k)}=(y_{i,j}^{(k)}), 1\leq i,j \leq n$ where the $x_{i,j}^k,y_{i,j}^k$ are algebraically independent over $\mathbb{F}$. Clearly any central simple algebra over $\mathbb{F}$, is a homomorphic image of $R_n:=\mathbb{F}[X^{(k)},Y^{(k)} \mid k\in\mathbb{Z}]$ – just specialize the variables so that the generic matrices are sent to the generators of the CSA. Moreover, a theorem of Amitsur shows that it is enough to take two generic matrices $X,Y$. There are of course more homomorphic images (for example, the trivial algebra), but it can be shown that after inverting a suitable central element, the only images are CSA of degree $n$ (further details can be found in [1]).

This new algebra is called the generic division algebra, and with a bit more work it can be shown to have a specialization (homomorphic image followed by localization) to any CSA of degree $n$ over any field extension of $\mathbb{F}$ – we denote this class by $\mathcal{C}(\mathbb{F},n)$. This suggests the following method to study this class of algebras – first show that the generic algebra satisfy a “nice” property, and second, show that this property is inherited by specialization and conclude that this property is satisfied by all CSA of degree $n$.

One of these “nice” properties is the structure of a crossed product which we now recall. Let $\mathbb{L}/\mathbb{F}$ be a $G$-Galois extension and $c:G\times G\to\mathbb{L}^{\times}$ be a normalized 2-cocycle, i.e. it satisfies

$\mbox{for all }g_{1},g_{2},g_{3}\in G:\qquad c(g_{1},g_{2})\cdot c(g_{1}g_{2},g_{3}) = g_{1}(c(g_{2},g_{3}))\cdot c(g_{1},g_{2}g_{3}).$
$\mbox{for all }g\in G:\,\qquad\qquad\quad c(g,e)=c(e,g) = 1$

The $G$-crossed product $\mathbb{L}^{c}G$ is defined to be the $\mathbb{L}$-vector space $\bigoplus_{g\in G}\mathbb{L} U_{g}$ where the multiplication is defined by

$\mbox{for all }g,h\in G,\;\alpha\in\mathbb{L}:\qquad U_{g}U_{h}=c(g,h)U_{gh}\qquad g(\alpha)U_{g}=U_{g}\alpha.$

It is well known that any $G$-crossed product is a CSA, but the other direction is not true in general. The first proof for this claim was given by Amitsur who showed that for some suitable $n$ there are CSA $A_i, i=0,1$ which are crossed products only with groups $G_0\not \cong G_1$ respectively. It follows that the generic algebra is not a $G$-crossed product for any group $G$, since otherwise both $A_0,A_1$ would be a crossed product with the same group (see [2]).

While the generic division algebra is not a central simple algebra, it is an Azumaya algebra which is a generalization of CSA. A good intuition about the structure of the generic division algebra (and Azumaya algebras in general), is to think of it as the algebra $M_n(\mathbb{F}[x])$. The simple homomorphic images are matrix algebras over field extensions of $\mathbb{F}$. To classify the homomorphic images is equivalent to classifying the ideals of this algebra. An easy exercise shows that each such ideal is induced by an ideal of the center, so we actually need to understand the center which is a commutative algebra. This property, that ideals comes from the center, is one of the defining properties of Azumaya algebra and hence of the generic division algebra.

Let $R_n(\mathbb{F})$ be the generic division algebra of degree $n$ over $\mathbb{F}$, and let $\mathbb{L}_n$ be the fraction field of the center of $R_n$. As the argument above suggest it is interesting to understand the extension $\mathbb{L}_n/\mathbb{F}$. If the center of $R_n(\mathbb{F})$ is a polynomial ring (or a localized polynomial ring) which is the free object of commutative algebras, then $\mathbb{L}_n/\mathbb{F}$ is a rational (i.e. purely transcendental) extension. Thus, the main question is whether this extension is always rational, and if not how close is it to being rational.

For the base field $\mathbb{F}=\mathbb{Q}$ the rationality for $n=2$  was already proved by Sylvester in 1883. 80 years later, it was  reproved in a much further generality by Procesi who also showed that $\mathbb{L}_n/\mathbb{Q}$ is always unirational (can be embedded in a rational extension). The rationality for the $n=3,4$ cases was proved by Formanek. The stable rationality of $n=5,7$ was proved by Le Bruyn and Bessenrodt and another more elementary proof was given by Beneish. Finally, it was shown independently by Schofield, Katsylo and Saltman that $\mathbb{L}_{ab}$ is stably rational over the field of fractions of $\mathbb{L}_a \otimes \mathbb{L}_b$ whenever $(a,b)=1$, thus reducing the problem to the prime power cases. Finally, Saltman showed that for $n=p$ prime, the extension is retract rational (if rational extensions are the “free” objects, that retract rational extensions are the “projective” objects). For a comprehensive survey on the center of the generic division algebra with the proper definitions and references we refer the reader to [3].

## The generic crossed product

As mentioned before, every crossed product is a CSA. While it is not true in general that every CSA is isomorphic to a crossed product it is always equivalent to one as follows. The Artin-Wedderburn theorem, mentioned at the beginning, also states that if $M_{n}(D_{1})\cong M_{k}(D_{2})$ where $D_{1},D_{2}$ are division algebras, then $n=k$ and $D_{1}\cong D_{2}$. This suggests the equivalence relation $A\sim B$ on central simple algebras if $M_{n}(A)\cong M_{k}(B)$ for some positive integers $n,k$. One can show that if $A,B$ are $\mathbb{F}$ central simple, then so is $A\otimes B$. The set of CSA over $\mathbb{F}$ modulo the equivalence relation $\sim$ and with the product $[A]\cdot [B]=[A\otimes B]$ is a group which is called the Brauer group and is denoted by $Br(\mathbb{F})$. With this notation we have that every CSA is Brauer equivalent to a $G$-crossed product. It follows that up to this equivalence relation, it is enough to consider a “generic crossed products” instead of the generic division algebra.

The idea here is to consider graded generic matrices instead of general matrices. Fix a group $G$ and consider the $g$-generic matrix defined by

$X_{i,j}^{(g)}=\begin{cases} x_{i,j}\qquad & g_{i}^{-1}g_{j}=g\\ 0 & else \end{cases}$

As in the standard case, one can define the generic $G$-crossed product to be the algebra generated by all of the $g$-generic matrices for $g\in G$, after inverting a suitable central element. In this work we consider the field of fractions of the center of the generic $G$-crossed product, which we denote by $\mathbb{F}(G)$, and ask how close it is to being a rational extension of the base field $\mathbb{F}$.
While the center of the generic crossed product (and generic algebra) can be thought of as the matrices invariant under conjugation, an approach going to back to the works of Procesi and Formanek show that we can actually reduce the problem to invariants under a finite group (for full details, see for example [4]).

Let us consider an example of this phenomenon. Let $G=\{0,1,2\}$ be the cyclic group of order 3. A central element is a scalar matrix. Consider the following

$X^{(0)}\cdot X^{(1)}\cdot X^{\left(2\right)}=\left(\begin{array}{ccc} x_{0,0} & 0 & 0\\ 0 & x_{1,1} & 0\\ 0 & 0 & x_{2,2} \end{array}\right)\left(\begin{array}{ccc} 0 & x_{0,1} & 0\\ 0 & 0 & x_{1,2}\\ x_{2,0} & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} 0 & 0 & x_{0,2}\\ x_{1,0} & 0 & 0\\ 0 & x_{2,1} & 0 \end{array}\right)=\left(\begin{array}{ccc} x_{0,0}x_{0,1}x_{1,0} & 0 & 0\\ 0 & x_{1,1}x_{1,2}x_{2,1} & 0\\ 0 & 0 & x_{2,2}x_{2,0}x_{0,2} \end{array}\right)$

The indices of the elements on the diagonal are of the form $(i_1,i_2),(i_2,i_3),(i_3,i_1)$. In other words, considering the full Cayley graph of the group $G$, these pair of indices correspond to cycles in this graph. We have the natural $G$ action defined by $g(x_{h_1,h_2})=g_{gh_1,gh_2}$ and we note that the elements on the diagonal matrix above are exactly a $G$ orbit. It follows that the matrix is central, namely a scalar matrix, if and only if its elements on the diagonal are invariant under the $G$-action.

To make this more precise, consider the field $\mathbb{F}[x_{h_1,h_2}\;\mid\;h_1,h_2\in G]$ and its subfield $\mathbb{L}$ generated by $\mathbb{F}$ and monomial elements that correspond to cycles in the Cayley graph (i.e. $x_{h_1,h_2}x_{h_2,h_3}\cdots x_{h_{n-1},h_n}x_{h_n,h_1}$). This field is a field of rational functions in $|G|^2-|G|+1$ variables over $\mathbb{F}$ (which is the number of edges in the Cayley graph minus the number of edges in a spanning tree, or equivalently the number of edge we need to remove to kill all the cycles). The group $G$ acts on the elements $x_{h_1,h_2}$ as above and induces a $G$ action on $\mathbb{L}$. Under this action it can be shown that the center of the generic $G$-crossed product is exactly the invariants $\mathbb{L}^G$. In the generic division algebra, a similar phenomenon happens, just take the full graph on $n$ vertices with the $S_n$ action on it.

Remark: A very similar object is the generic $G$-Galois extension. This time we start with the field $\mathbb{L}=\mathbb{F}(x_g \mid g\in G)$ with the $G$-action $h(x_g)=x_{hg}$ and the question is whether $\mathbb{L}/\mathbb{L}^G$ is a rational (or close to rational) extension. This is called Noether’s problem and a positive solution to it will produce a positive solution to the inverse Galois problem over the field $\mathbb{F}$. Just to have some intuition, the field $\mathbb{L}$ is the fraction field of the ring $\mathbb{F}[x_g \mid g\in G]$ which can specialize to any $G$-Galois extension of $\mathbb{F}$ – just send the elements $x_g$ to some normal basis (i.e. a basis which is an orbit under the Galois action).

### Known results:

We will assume that $\mathbb{F}$ always contains a primitive root of unity of the order of the exponent of the group $G$. With this assumption it is easily shown that for cyclic groups $G$, the extension $\mathbb{F}(G)/\mathbb{F}$ is always rational. More interesting results were given by Snider in [5] that proved rationality for the Klein four group, and stable rationality for Dihedral groups of order $2n$ where $n$ is odd. Furthermore, Saltman proved in [1] that if $\mathbb{F}(G)/\mathbb{F}$ is retract rational, then all the Sylow subgroups of $G$ are not only abelian, but are a product of at most two cyclic subgroup.

### Our results [6]

Paralleling Saltman’s result for the generic division algebra of prime degree, we have proved the following:

Theorem: Suppose that every Sylow subgroup of $G$ is cyclic. Then $\mathbb{F}(G)/\mathbb{F}$ is a retract rational extension.

Remark: It is known that the property that any Sylow subgroup of $G$ is cyclic, is equivalent to $G$ being isomorphic to a semidirect product of cyclic groups $C_n \rtimes C_m$ where $(n,m)=1$.

Recall that in the generic division algebra it is enough to consider the prime power case. Its parallel for crossed products is:

Theorem: Let $H,K$ be finite groups of coprime orders. Then $\mathbb{F}(H\times K)$ is stably equivalent to the fraction field of $\mathbb{F}(H)\otimes \mathbb{F}(K)$. In particular, if bote $\mathbb{F}(H),\mathbb{F}(K)$ are stably rational over $\mathbb{F}$ then so is $\mathbb{H\times K}/\mathbb{F}$.

Using Snider’s result on the Klein four group, and the rationality for cyclic groups, we conclude

Corollary: Let $m$ be an odd number. Then $\mathbb{F}(C_2\times C_{2m})$ is stably rational over $\mathbb{F}$.

Finally, we generalizes Snider’s result about Dihedral groups.

Theorem: Let $G=\{ \sigma,\tau\;\mid\; \sigma^n=\tau^{2m}=e,\;\tau\sigma\tau^{-1}=\sigma^{-1}\}$ such that $(n,2m)=1$. Then $\mathbb{F}(G)/\mathbb{F}$ is stably rational. If $m=1$ (i.e. $G$ is Dihedral), then the extension is rational.

### Bibliography:

1. D.J. Saltman, Lectures on division algebras, J Vol. 94. American Mathematical Soc., 1999.
2. S. A. Amitsur, On central division algebras, Israel journal of mathematics 12.4 (1972): 408-420.
3. L. Le Bruyn, Centers of generic division algebras, the rationality problem 1965–1990, Israel Journal of Mathematics 76.1-2 (1991): 97-111.
4. E. Formanek, The Polynomial Identities and Invariants of n x n Matrices, American Mathematical Society (1992).
5. R. L. Snider, Is the Brauer group generated by cyclic algebras?, Ring Theory Waterloo 1978 Proceedings, University of Waterloo, Canada, 12–16 June, 1978. Springer Berlin Heidelberg, 1979. 279-301.
6. O. David, The center of the generic $G$-crossed product, arXiv:1401.4717v3 [math.RA]