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We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.
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The algebras $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>kge 1$, a complex elliptic curve $E$, and a point $tauin E$. The main result in this paper is that $Q_{n,k}(E,tau)$ has the same Hilbert series as the polynomial ring on $n$ variables when $tau$ is not a torsion point. We also show that $Q_{n,k}(E,tau)$ is a Koszul algebra, hence of global dimension $n$ when $tau$ is not a torsion point, and, for all but countably many $tau$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,tau)$ is the image of an operator $R_{tau}(tau)$ that belongs to a family of operators $R_{tau}(z):mathbb{C}^notimesmathbb{C}^ntomathbb{C}^notimesmathbb{C}^n$, $zinmathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.
We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.
The commutation principle of Ramirez, Seeger, and Sossa cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr{e}chet differentiable function $Theta(x)$ and a spectral function $F(x)$ is minimized over a spectral set $Omega$, any local minimizer $a$ operator commutes with the Fr{e}chet derivative $Theta^{prime}(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.
In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.
Fix a pair of relatively prime integers $n>kge 1$, and a point $(eta , | , tau) in mathbb{C} times mathbb{H}$, where $mathbb{H}$ denotes the upper-half complex plane, and let ${{a ; ,b} choose {c , ; d}} in mathrm{SL}(2,mathbb{Z})$. We show that Feigin and Odesskiis elliptic algebras $Q_{n,k}(eta , | , tau)$ have the property $Q_{n,k} big( frac{eta}{ctau+d} ,bigvert , frac{atau+b}{ctau+d} big) cong Q_{n,k}(eta , | , tau)$. As a consequence, given a pair $(E,xi)$ consisting of a complex elliptic curve $E$ and a point $xi in E$, one may unambiguously define $Q_{n,k}(E,xi):=Q_{n,k}(eta , | , tau)$ where $tau in mathbb{H}$ is any point such that $mathbb{C}/mathbb{Z}+mathbb{Z}tau cong E$ and $eta in mathbb{C}$ is any point whose image in $E$ is $xi$. This justifies Feigin and Odesskiis notation $Q_{n,k}(E,xi)$ for their algebras.
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