A note on a separating system of rational invariants for finite dimensional generic algebras

TL;DR

This paper introduces a method for constructing rational invariants to distinguish finite dimensional generic algebras, aiding their classification through associated quadratic forms.

Contribution

It proposes a new approach to classify finite dimensional algebras using rational invariants and quadratic forms, advancing algebraic classification techniques.

Findings

Constructed a separating system of rational invariants.

Linked algebra classification to quadratic forms.

Provided a framework for rough classification of algebras.

Abstract

The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by attaching them some quadratic forms.

Equations

$$\left(\begin{array}[]{cccc}a_{11}B&a_{12}B&...&a_{1q}B\\ a_{21}B&a_{22}B&...&a_{2q}B\\ .&.&\ddot{.}&.\\ a_{p1}B&a_{p2}B&...&a_{pq}B\end{array}\right),\ \mbox{where}\ F-\ \mbox{is a field of characteristic not}\ 2.$$

$$A_e=(A_{ejk}^i{)}_{i,j,k=1,2,...,m}\in Mat(m\times m^2;F),$$

$$\mathbf{u}\cdot \mathbf{v}=e{A}_e(u\otimes v)$$

$$A_{e^{\prime }}=g{A}_e(g^{-1}{)}^{\otimes 2}$$

$$(\sum _{i=1}^m{X}_{i1}^i,\sum _{i=1}^m{X}_{i2}^i,...,\sum _{i=1}^m{X}_{im}^i),(\sum _{i=1}^m{X}_{1i}^i,\sum _{i=1}^m{X}_{2i}^i,...,\sum _{i=1}^m{X}_{mi}^i),$$

$$\tau :(g,A)\mapsto B=gA(g^{-1}\otimes g^{-1}).$$

$$A=(A_1,A_2,...,A_m),A_1,A_2,...,A_m\in Mat(m,F).$$

$$A^{\overline{t}}=\left(\begin{array}[]{c}A_{1}\\ A_{2}\\ \vdots\\ A_{m}\end{array}\right)-\ \ \mbox{column vector over}\ Mat(m,F),\$$

$$A^{\overline{\otimes }2}=(A_1^2,A_1{A}_2,...,A_1{A}_m,A_2{A}_1,A_2^2,...,A_2{A}_m,...,A_m{A}_1,A_m{A}_2,...,A_m^2).$$

$$B=(B_1,B_2,...,B_m)=gA(g^{-1}{)}^{\otimes 2}=(g{A}_1{g}^{-1},g{A}_2{g}^{-1},...,g{A}_m{g}^{-1})(g^{-1}\otimes I),$$

$$(C\otimes I)\overline{\otimes }(D\otimes I)=(C\otimes D)\otimes I$$

$$(B_1,B_2,...,B_m{)}^{\overline{\otimes }k}=(g{A}_1{g}^{-1},g{A}_2{g}^{-1},...,g{A}_m{g}^{-1}{)}^{\overline{\otimes }k}((g^{-1}{)}^{\otimes k}\otimes I),$$

$$\left(\begin{array}[]{c}B_{1}\\ B_{2}\\ \vdots\\ B_{m}\end{array}\right)^{\overline{\otimes}k}(B_{1},B_{2},...,B_{m})^{\overline{\otimes}k}=\left(\begin{array}[]{cccc}B_{1}^{2}&B_{1}B_{2}&\cdots&B_{1}B_{m}\\ B_{2}B_{1}&B_{2}^{2}&\cdots&B_{2}B_{m}\\ \vdots&\vdots&\cdots&\vdots\\ B_{m}B_{1}&B_{m}B_{2}&\cdots&B_{m}^{2}\end{array}\right)^{\overline{\otimes}k},$$

$$\left(\begin{array}[]{cccc}B_{1}^{2}&B_{1}B_{2}&\cdots&B_{1}B_{m}\\ B_{2}B_{1}&B_{2}^{2}&\cdots&B_{2}B_{m}\\ \vdots&\vdots&\cdots&\vdots\\ B_{m}B_{1}&B_{m}B_{2}&\cdots&B_{m}^{2}\end{array}\right)^{\overline{\otimes}k}=$$

$$(((g^{t})^{-1})^{\otimes k}\otimes I)\left(\begin{array}[]{cccc}gA_{1}^{2}g^{-1}&gA_{1}A_{2}g^{-1}&\cdots&gA_{1}A_{m}g^{-1}\\ gA_{2}A_{1}g^{-1}&gA_{2}^{2}g^{-1}&\cdots&gA_{2}A_{m}g^{-1}\\ \vdots&\vdots&\cdots&\vdots\\ gA_{m}A_{1}g^{-1}&gA_{m}A_{2}g^{-1}&\cdots&gA_{m}^{2}g^{-1}\end{array}\right)^{\overline{\otimes}k}((g^{-1})^{\otimes k}\otimes I).$$

$$\tilde{Tr}(\left(\begin{array}[]{cccc}B_{1}^{2}&B_{1}B_{2}&\cdots&B_{1}B_{m}\\ B_{2}B_{1}&B_{2}^{2}&\cdots&B_{2}B_{m}\\ \vdots&\vdots&\cdots&\vdots\\ B_{m}B_{1}&B_{m}B_{2}&\cdots&B_{m}^{2}\end{array}\right)^{\overline{\otimes}k})=$$

$$((g^{t})^{-1})^{\otimes k}\tilde{Tr}(\left(\begin{array}[]{cccc}gA_{1}^{2}g^{-1}&gA_{1}A_{2}g^{-1}&\cdots&gA_{1}A_{m}g^{-1}\\ gA_{2}A_{1}g^{-1}&gA_{2}^{2}g^{-1}&\cdots&gA_{2}A_{m}g^{-1}\\ \vdots&\vdots&\cdots&\vdots\\ gA_{m}A_{1}g^{-1}&gA_{m}A_{2}g^{-1}&\cdots&gA_{m}^{2}g^{-1}\end{array}\right)^{\overline{\otimes}k})(g^{-1})^{\otimes k}=$$

$$((g^{-1})^{\otimes k})^{t}\tilde{Tr}(\left(\begin{array}[]{cccc}A_{1}^{2}&A_{1}A_{2}&\cdots&A_{1}A_{m}\\ A_{2}A_{1}&A_{2}^{2}&\cdots&A_{2}A_{m}\\ \vdots&\vdots&\cdots&\vdots\\ A_{m}A_{1}&A_{m}A_{2}&\cdots&A_{m}^{2}\end{array}\right)^{\overline{\otimes}k})(g^{-1})^{\otimes k},$$

$$\widetilde{Tr}((C\otimes I)D(E\otimes I))=C\widetilde{Tr}(D)E$$

$$\widetilde{Tr}((B^{\overline{t}}B{)}^{\overline{\otimes }k})=((g^{-1}{)}^{\otimes k}{)}^t\widetilde{Tr}((A^{\overline{t}}A{)}^{\overline{\otimes }k})(g^{-1}{)}^{\otimes k}.$$

$$\tilde{Tr}(A^{\overline{t}}A)=\left(\begin{array}[]{cccc}Tr(A_{1}^{2})&Tr(A_{1}A_{2})&\cdots&Tr(A_{1}A_{m})\\ Tr(A_{2}A_{1})&Tr(A_{2}^{2})&\cdots&Tr(A_{2}A_{m})\\ \vdots&\vdots&\cdots&\vdots\\ Tr(A_{m}A_{1})&Tr(A_{m}A_{2})&\cdots&Tr(A_{m}^{2})\end{array}\right),$$

$$\tilde{Tr}(B^{\overline{t}}B)=\left(\begin{array}[]{cccc}Tr(B_{1}^{2})&Tr(B_{1}B_{2})&\cdots&Tr(B_{1}B_{m})\\ Tr(B_{2}B_{1})&Tr(B_{2}^{2})&\cdots&Tr(B_{2}B_{m})\\ \vdots&\vdots&\cdots&\vdots\\ Tr(B_{m}B_{1})&Tr(B_{m}B_{2})&\cdots&Tr(B_{m}^{2})\end{array}\right)$$

$$\widetilde{Tr}({\overline{X}}^{\overline{t}}\overline{X})=(Q(X^{\overline{t}}X{)}^{-1}{)}^t\widetilde{Tr}(X^{\overline{t}}X)Q(X^{\overline{t}}X{)}^{-1}=D(X)$$

$$\overline{B}=\tau (g_0,\overline{A})\text{mbox}forsome{g}_0\in GL(m,F)\text{mbox}forwhich{g}_0^{t}D(B)g_0=D(A).$$

$$\tau (Q(B^{\overline{t}}B)g,A)=\tau (Q(B^{\overline{t}}B)gQ(A^{\overline{t}}A{)}^{-1},\tau (Q(A^{\overline{t}}A),A)=\tau (Q(B^{\overline{t}}B)gQ(A^{\overline{t}}A{)}^{-1},\overline{A}),$$

$$g_0^{t}D(B)g_0=(Q(B^{\overline{t}}B)gQ(A^{\overline{t}}A{)}^{-1}{)}^{t}D(B)Q(B^{\overline{t}}B)gQ(A^{\overline{t}}A{)}^{-1}=$$

$$(Q(A^{\overline{t}}A{)}^{-1}{)}^t(g^t(Q(B^{\overline{t}}B{)}^{t}D(B)Q(B^{\overline{t}}B))g)Q(A^{\overline{t}}A{)}^{-1}=$$

$$(Q(A^{\overline{t}}A{)}^{-1}{)}^t(g^t(\widetilde{Tr}(B^{\overline{t}}B))g)Q(A^{\overline{t}}A{)}^{-1}=(Q(A^{\overline{t}}A{)}^{-1}{)}^t\widetilde{Tr}(A^{\overline{t}}A)Q(A^{\overline{t}}A{)}^{-1}=D(A).$$

$$\tau (g,A)=\tau (Q(B^{\overline{t}}B{)}^{-1}{g}_{0}Q(A^{\overline{t}}A),A)=\tau (Q(B^{\overline{t}}B{)}^{-1}{g}_0,\tau (Q(A^{\overline{t}}A),A)=$$

$$\tau (Q(B^{\overline{t}}B{)}^{-1}{g}_0,\overline{A})=\tau (Q(B^{\overline{t}}B{)}^{-1},\tau (g_0,\overline{A}))=\tau (Q(B^{\overline{t}}B{)}^{-1},\overline{B})=B.$$

Full text

1 Introduction

In B2017 we have offered an approach to classification problem of finite dimensional algebras with respect to basis changes. It has been shown that if one has a special map with some properties then he is able to classify, to list canonical representations, all algebras who’s set of structural constants , with respect to a fixed basis, do not nullify some polynomial. In this case he is also able to provide a separating system of rational invariants for those algebras. It was successfully applied in A2017 to get a complete classification of all $2$-dimensional algebras over algebraically closed fields.

Unfortunately, so far we have no example of such a map in $3$-dimensional case. Therefore in the current paper we deal with a weaker problem, namely with a construction of separating system of rational invariants for finite dimensional generic algebras. The theoretical existence of such system of invariants is known P2014 . By generic algebras we mean the set of all algebras who’s system of structural constants does not nullify a fixed nonzero polynomial in structural variables, over the basic field $F$. In process of dealing with the problem we show a way for a rough classification of finite dimensional algebras by attaching them some quadratic forms.

The next section contains the main results.

2 Main results

Further whenever $A=(a_{ij})\in Mat(p\times q,F)$, $B\in Mat(p^{\prime}\times q^{\prime},F)$ we use $A\otimes B$ for the matrix

[TABLE]

Let us consider any $m$-dimensional algebra $\mathbf{A}$ with multiplication $\cdot$ given by a bilinear map $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}$. If $e=(e_{1},e_{2},...,e_{m})$ is a basis for $\mathbf{A}$ then one can represent the bilinear map by a matrix

[TABLE]

where $e_{j}\cdot e_{k}=e_{1}A^{1}_{ejk}+e_{2}A^{2}_{ejk}+...+e_{m}A^{m}_{ejk}$, $j,k=1,2,...,m$, such that

[TABLE]

for any $\mathbf{u}=eu,\mathbf{v}=ev,$ where $u=(u_{1},u_{2},...,u_{m}),v=(v_{1},v_{2},...,v_{m})$ are column vectors. So the algebra $\mathbf{A}$ (binary operation, bilinear map, tensor) is presented by the matrix $A_{e}\in Mat(m\times m^{2};F)$-the matrix of structure constants (MSC) of $\mathbf{A}$ with respect to the basis $e$.

If $e^{\prime}=(e^{\prime}_{1},e^{\prime}_{2},...,e^{\prime}_{m})$ is also a basis for $\mathbf{A}$, $g\in GL(m,F)$, $e^{\prime}g=e$ then it is well known that

[TABLE]

is valid. Further a basis $e$ is fixed and therefore instead of $A_{e}$ we use $A$, we do not make difference between $\mathbf{A}$ and its matrix $A$. Let $X=(X^{i}_{jk})_{i,j,k=1,2,...,m}$ stand for a variable matrix and $Tr_{1}(X)$, $Tr_{2}(X)$ stand for the row vectors

[TABLE]

respectively.

We use $\tau$ for the representation of $GL(m,F)$ on the $n=m^{3}$ dimensional vector space $V=Mat(m\times m^{2};F)$ defined by

[TABLE]

For simplicity instead of ”$\tau$-equivalent”, ”$\tau$-invariant” we use ”equivalent” and ”invariant”.

We represent each MSC $A$ as a row vector with entries from $Mat(m,F)$ by parting it consequently into elements of $Mat(m,F)$:

[TABLE]

If $C$ is a block matrix with blocks from $Mat(m,F)$ we use notation $C^{\overline{*}}$, where $*$ is the tensor product or transpose operation, to mean that the operation $*$ with $C$ is done ”over $Mat(m,F)$” (not over $F$), for example for the above presented matrix $A$

[TABLE]

[TABLE]

One can see that the equality $B=gA(g^{-1}\otimes g^{-1})$ can be presented as

[TABLE]

where $I$ stands for $m\times m$ size identity matrix. Moreover for any matrices $C$ and $D$ the equality

[TABLE]

holds true. Therefore the following equalities hold true.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Component-wise application of trace to this equality, which is denoted by $\tilde{Tr}$ results in

[TABLE]

[TABLE]

[TABLE]

as far as for any matrices $C$, $D$ and $E$, where $D$ is a block matrix with blocks from $Mat(m,F)$ and $(C\otimes I)D(E\otimes I)$ has meaning, the equality

[TABLE]

is valid. One can represent the above obtained matrix equality in the following compact form

[TABLE]

Note that $\tilde{Tr}((A^{\overline{t}}A)^{\overline{\otimes}k})$ is a symmetric matrix. The obtained equality allows formulation of the following theorem.

Theorem 1. Invariants of the quadratic forms given by the matrix $\tilde{Tr}((X^{\overline{t}}X)^{\overline{\otimes}k})$ are invariants of the $m$-dimensional algebras.

This result can be used for a rough classification of finite dimensional algebras: Two $m$-dimensional algebras $A$, $B$ are rough equivalent if the quadratic forms given by matrices

[TABLE]

[TABLE]

are equivalent.

It is clear that entries of $\tilde{Tr}(X^{\overline{t}}X)$ are polynomials in components of $X$ and there exists nonsingular matrix $Q(X^{\overline{t}}X)$ with rational entries in $X$ such that the matrix

[TABLE]

is a diagonal matrix and $Q(g)=I$ whenever $g$ is a nonsingular diagonal matrix, where $\overline{X}=\tau(Q(X^{\overline{t}}X),X)$.

In algebraically closed field $F$ case it means that one can define a nonempty invariant open subset $V_{0}\subset V$ such that $\tilde{Tr}(\overline{A}^{\overline{t}}\overline{A})=D(A)$ and $D(A)$ is nonsingular whenever $A\in V_{0}$.

Theorem 2. * Two algebras $A,B\in V_{0}$ are equivalent(isomorphic) if and only if*

[TABLE]

Proof. If $B=\tau(g,A)$ then $\overline{B}=\tau(Q(B^{\overline{t}}B),B)=\tau(Q(B^{\overline{t}}B),\tau(g,A))=$

[TABLE]

and for $g_{0}=Q(B^{\overline{t}}B)gQ(A^{\overline{t}}A)^{-1}$ one has

[TABLE]

[TABLE]

[TABLE]

Visa versa if $\overline{B}=\tau(g_{0},\overline{A})$ for some $g_{0}$ for which $g_{0}^{t}D(B)g_{0}=D(A)$ then for

$g=Q(B^{\overline{t}}B)^{-1}g_{0}Q(A^{\overline{t}}A)$ one has

[TABLE]

[TABLE]

Assume that there exists matrix $P(X)$, with rational entries with respect to components of $X$, such that $P(\overline{A})$ is nonsingular for any $A\in V_{0}$ and the equality

[TABLE]

Theorem 3. * For $A$,$B\in V_{0}$ there exists $g_{0}\in GL(m,F)$ such that $g_{0}^{t}D(B)g_{0}=D(A)$ and $\overline{B}=\tau(g_{0},\overline{A})$ if and only if*

[TABLE]

Proof. If $\overline{B}=\tau(g_{0},\overline{A})$ and $g_{0}^{t}D(B)g_{0}=D(A)$ then

[TABLE]

and $(P(\overline{B})^{-1})^{t}D(B)P(\overline{B})^{-1}=((P(\overline{A})g_{0}^{-1})^{-1})^{t}D(B)(P(\overline{A})g_{0}^{-1})^{-1}=$

[TABLE]

Visa versa, if equalities

[TABLE]

are valid then for $g_{0}=P(\overline{B})^{-1}P(\overline{A})$ one has $g_{0}^{t}D(B)g_{0}=D(A)$ and

[TABLE]

So Theorems 2 and 3 imply that the system of entries of matrices

[TABLE]

is a separating system of rational invariants for algebras from $V_{0}$.

The above presented results show importance of construction of matrix $P(X)$ with properties (1). Further we discuss a construction of such matrix by the use of rows $r(\overline{A})$ for which the equality

[TABLE]

is valid, whenever $g^{t}D(\tau(g,A))g=D(A)$. To construct such rows one can use the following approach.

Assume that the equalities

[TABLE]

are true, where $C^{t}=C$ and $C$ is a nonsingular matrix. In this case

[TABLE]

and

[TABLE]

On induction it is easy to see that for any natural $k$ the equality

[TABLE]

holds true. Therefore due to the equalities

[TABLE]

[TABLE]

[TABLE]

[TABLE]

one has

[TABLE]

[TABLE]

The last equality shows that in our algebra case on can try to construct the needed matrix $P(X)$ by the use of rows

[TABLE]

where $i=1,2,\ k=0,1,2,...$.

What is left unjustified here is that one should justify existence, in general, of a linear independent system consisting of $m$ such rows.

Remark. After a rough classification one can classify further each case of the rough classification with respect to the corresponding stabilizer.

The author acknowledges MOHE for a support by grant FRGS14-153-0394.