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Register a new userNon-commutative Geometry of Homogenized Quantum $mathfrak{sl}(2,mathbb{C})$
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Alexandru Chirv\\u{a}situ L.
Publication date
2016
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English
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This paper examines the relationship between certain non-commutative analogues of projective 3-space, $mathbb{P}^3$, and the quantized enveloping algebras $U_q(mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras $S$, one for each $q in mathbb{C}^times$, having a degree-two central element $c$ such that $S[c^{-1}]_0 cong U_q(mathfrak{sl}_2)$. The non-commutative analogues of $mathbb{P}^3$ are the spaces $operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $operatorname{Proj}_{nc}(S)$, and their incidence relations, correspond to finite dimensional irreducible representations of $U_q(mathfrak{sl}_2)$, Verma modules, annihilators of Verma modules, and homomorphisms between them.
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In this paper we explore the possibility of endowing simple infinite-dimensional ${mathfrak{sl}_2(mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].
The $q$-analog of Kostants weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $mathfrak{sl}_4(mathbb{C})$ and give closed formulas for the $q$-analog of Kostants weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostants partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostants weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $mathfrak{sl}_4(mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.
For every integer $g ,geq, 2$ we show the existence of a compact Riemann surface $Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${mathcal O}^{oplus 2}_{Sigma}$ admits holomorphic connections with $text{SL}(2,{mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Sigma$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.
Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}z^2bar{zeta}+frac{1}{2} bar{z}^2 zeta}{1-zeta bar{zeta}}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone ${x_1^2+x_2^2-x_3^2=0}$. Another representation is the tube $u=frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: [ (z,zeta,w) longmapsto big( f(z,zeta), g(z,zeta), rho,w+h(z,zeta) big) =: (z,zeta,w). ] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiolas formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincare-Moser complete normal form: [ u = frac{zbar{z}+frac{1}{2},z^2bar{zeta} +frac{1}{2},bar{z}^2zeta}{ 1-zetabar{zeta}} + sum_{a,b,c,d atop a+cgeqslant 3}, G_{a,b,c,d}, z^azeta^bbar{z}^cbar{zeta}^d, ] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {rm Im}, G_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0, 2, 3, 0}$, ${rm Re} G_{3,0,1,1}$. With this, a brige Poincare $longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.
We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x SL(2,C). Here C is the field of complex numbers.
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