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Register a new userBethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum
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We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $binom {n}{l}-binom{n}{l-1}$ two-dimensional vector subspaces $V subset C[u]$ with a basis $f,gin V$ such that $deg f = l, deg g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.
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In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the gl_N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [MTV4] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (C^N)^{otimes n}.
To any 2x2-matrix K one assigns a commutative subalgebra B^{K}subset U(gl_2[t]) called a Bethe algebra. We describe relations between the Bethe algebras, associated with the zero matrix and a nilpotent matrix.
Let $frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(frak{g}))$ of the quantum group $U_q(frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(frak{g}))$ is a polynomial algebra if and only if $frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $frak{g}$ is of type $E_6$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $frak{g}$ is of type $A_{n}$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences.
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