arXiv subjects

Jaka Cimprič

Publications and source records attributed to Jaka Cimprič.

3 recordsLinked to original sources

arXiv2021

Prime and semiprime submodules of $R^n$ and a related Nullstellensatz for $M_n(R)$

Let $R$ be a commutative ring with $1$ and $n$ a natural number. We say that a submodule $N$ of $R^n$ is semiprime if for every $f=(f_1,\ldots,f_n) \in R^n$ such that $f_i f \in N$ for $i=1,\ldots,n$ we have $f \in N$. Our main result is that every semiprime submodule of $R^n$ is equal to the intersection of all prime submodules containing it. It follows that every semiprime left ideal of $M_n(R)$ is equal to the intersection of all prime left ideals that contain it. For $R=k[x_1,\ldots,x_d]$ where $k$ is an algebraically closed field we can rephrase this result as a Nullstellensatz for $M_n(R)$: For every $G_1,\ldots,G_m,F \in M_n(R)$, $F$ belongs to the smallest semiprime left ideal of $M_n(R)$ that contains $G_1,\ldots,G_m$ iff for every $a \in k^d$ and $v \in k^n$ such that $G_1(a)v=\ldots=G_m(a)v=0$ we have $F(a)v=0$.

math.RA↗

arXiv2020

A local-global principle for linear dependence in enveloping algebras of Lie algebras

For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors $π(a_1)v,\ldots,π(a_n)v$ are linearly dependent for every $π\in \mathcal{C}$ and every $v$ from the representation space of $π$. We answer this question in the following cases: (1) $A=U(L)$ is the enveloping algebra of a finite-dimensional complex Lie algebra $L$ and $\mathbb{C}$ is the class of all finite-dimensional representations of $A$. (2) $A=U(\mathfrak{sl}_2(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$. (3) $A=U(\mathfrak{sl}_3(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$ with sufficiently high weights. In case (1) the answer is: tuples that are linearly dependent over $\mathbb{C}$ while in cases (2) and (3) the answer is: tuples that are linearly dependent over the center of $A$. Similar results have been proved before for free algebras and Weyl algebras.

math.RT↗

arXiv2017

A nullstellensatz for linear partial differential equations with polynomial coefficients

In this paper an equation means a homogeneous linear partial differential equation in $n$ unknown functions of $m$ variables which has real or complex polynomial coefficients. The solution set consists of all $n$-tuples of real or complex analytic functions that satisfy the equation. For a given system of equations we would like to characterize its Weyl closure, i.e. the set of all equations that vanish on the solution set of the given system. It is well-known that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$ where $F$ is either the field of real or complex numbers, $A_m(F)$ (respectively $B_m(F)$) consists of all linear partial differential operators with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and $N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module $A_m(F)^n/N$ has finite rank which used to be a standard assumption. Our approach works also for the real case which was not possible with previous methods. Moreover, our proof is constructive as it depends only on the Riquier-Janet theory.

math.RA↗