اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدJordan Algebras of Symmetric Matrices
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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.
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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 Feig
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