Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe gl_2 Bethe algebra associated with a nilpotent element
118
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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}.
We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. We also give a Lie theoretic interpretation of this realization in terms of Kac-Moody Lie algebras and Kac-Moody groups.
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 $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.
For a C1-cofinite vertex algebra V, we give an efficient way to calculate Zhus algebra A(V) of V with respect to its C1-generators and relations. We use two examples to explain how this method works.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
