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Register a new userAlternating quaternary algebra structures on irreducible representations of sl(2,C)
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We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial identities of degree $le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $Lambda^4 V(n) to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.
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We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)oplus oplus_k (Ca_k)$. As a subalgebra of $S^3H$ we have the algebra $C[phi]$ of the Laurent polynomial spinors on $S^3$. Then we have a Lie subalgebra $hat{gl}(n, H)=C[phi] otimes gl(n, C)$ of $S^3gl(n,H)$, as well as its central extension by the 2-cocycles ${c_k}$ and the Euler vector field $d$: $hat{gl}=hat{gl}(n, H) oplus oplus_k(Ca_k)oplus Cd$ . The Lie algebra $hat{sl}(n,H)$ is defined as a Lie subalgebra of $hat{gl}(n,H)$ generated by $C[phi]otimes sl(n,C))$. We have the corresponding central extension of $hat{sl}(n,H)$ by the 2-cocycles ${c_k}$ and the derivation $d$, which becomes a Lie subalgebra $hat{sl}$ of $hat{gl}$. Let $h_0$ be a Cartan subalgebra of $sl(n,C)$ and $hat{h}=h_0 oplus oplus_k(Ca_k)oplus Cd$. The root space decomposition of the $ad(hat{h})$-representation of $hat{sl}$ is obtained. The set of roots is $Delta ={ m/2 delta + alpha ; alpha in Delta_0, m in Z} bigcup {m/2 delta ; m in Z }$ . And the root spaces are $hat{g}_{m/2 delta+ alpha}= C[phi ;m] otimes g_{alpha}$, for $alpha eq 0$ , $hat{g}_{m/2 delta}= C[phi ;m] otimes h_0$, for $m eq 0$, and $hat{g}_{0 delta}= hat{h}$, where $C[phi ;m]$ is the subspace with the homogeneous degree m. The Chevalley generators of $hat{sl}$ are given.
A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painleve Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painleve Series and one given by a Right Painleve Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
We determine the Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of the quantum deformed algebras U_q(sl(2)) and U_q(osp(1|2)). While our results for the former algebra reproduce formulas by Ponsot and Teschner, the expressions for the orthosymplectic algebra are new. Up to some normalization factors, the associated Racah-Wigner coefficients are shown to agree with the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory.
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.
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on P 1 (C).
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