arXiv subjects

Darren Funk-Neubauer

Publications and source records attributed to Darren Funk-Neubauer.

6 recordsLinked to original sources

arXiv2021

Bidiagonal Triads and the Tetrahedron Algebra

We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other two. A triad differs from a triple in the way these bidiagonal actions are defined. We modify a number of theorems about bidiagonal triples to the case of bidiagonal triads. We also describe the close relationship between bidiagonal triads and the representation theory of the tetrahedron Lie algebra.

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arXiv2016

Bidiagonal Triples

We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other two. The concept of bidiagonal triple is a generalization of the previously studied and similarly defined concept of bidiagonal pair. We show that every bidiagonal pair extends to a bidiagonal triple, and we describe the sense in which this extension is unique. In addition we generalize a number of theorems about bidiagonal pairs to the case of bidiagonal triples. In particular we use the concept of a parameter array to classify bidiagonal triples up to isomorphism. We also describe the close relationship between bidiagonal triples and the representation theory of the Lie algebra $sl_2$ and the quantum algebra $U_q(sl_2)$.

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arXiv2012

Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2)

We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. Using this concept of a parameter array we present a classification of bidiagonal pairs up to isomorphism. The statement of this classification does not explicitly mention the Lie algebra $\SL$ or the quantum group $\uq$. However, its proof makes use of the finite-dimensional representation theory of $\SL$ and $\uq$. In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for $\SL$ and $\uq$.

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arXiv2010

On the simplicity of Lie algebras associated to Leavitt algebras

For any field $\K$ and integer $n\geq 2$ we consider the Leavitt algebra $L_\K(n)$; for any integer $d\geq 1$ we form the matrix ring $S = M_d(L_\K(n))$. $S$ is an associative algebra, but we view $S$ as a Lie algebra using the bracket $[a,b]=ab-ba$ for $a,b \in S$. We denote this Lie algebra as $S^-$, and consider its Lie subalgebra $[S^-,S^-]$. In our main result, we show that $[S^-,S^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$ and char$(\K)$ does not divide $d$. In particular, when $d=1$ we get that $[L_\K(n)^-,L_\K(n)^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$.

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arXiv2009

Tridiagonal pairs and the q-tetrahedron algebra

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0 \leq i \leq d$ let $θ_i$ (resp. $θ^*_i$) denote a standard ordering of the eigenvalues of A (resp. A^*). Fix a nonzero scalar q which is not a root of unity. T. Ito and P. Terwilliger have shown that when $θ_i = q^{2i-d}$ and $θ^*_i = q^{d-2i}$ there exists an irreducible $\boxtimes_q$-module structure on V such that the $\boxtimes_q$ generators x_{01}, x_{23} act as A, A^* respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that $θ_i = q^{2i-d}$ and $θ^*_i = q^{2i-d} + c q^{d-2i}$. In this case we associate to A,A^* a polynomial P and prove the following equivalence. The following are equivalent: (i) There exists a $\boxtimes_q$-module structure on V such that x_{01} acts as A and x_{30} + cx_{23} acts as A^*, where x_{01}, x_{30}, x_{23} are standard generators for $\boxtimes_q$. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \neq 0. Suppose (i),(ii) hold. Then the $\boxtimes_q$-module structure on V is unique and irreducible.

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arXiv2006

Raising/lowering maps and modules for the quantum affine algebra $U_q(sl_2hat)$

Let V denote a finite dimensional vector space over an algebraically closed field. Let U_0, U_1,..., U_d denote a sequence of nonzero subspaces whose direct sum is V. Let R:V \to V and L:V \to V denote linear maps with the following properties: for 0 \leq i \leq d, R U_i \subseteq U_{i+1} and L U_i \subseteq U_{i-1} where U_{-1}=0, U_{d+1}=0; for 0 \leq i \leq d/2, the restrictions R^{d-2i}: U_i \to U_{d-i} and L^{d-2i}: U_{d-i} \to U_i are bijections; the maps R and L satisfy the cubic q-Serre relations where q is nonzero and not a root of unity. Let K:V \to V be the linear map such that (K-q^{2i-d}I)U_i=0 for 0 \leq i \leq d. We show that there exists a unique U_q(sl_2hat)-module structure on V such that each of R-e^{-}_1, L-e^{-}_0, K-K_0, and K^{-1}-K_1 vanish on V, where e^{-}_1, e^{-}_0, K_0, K_1 are Chevalley generators for U_q(sl_2hat). We determine which U_q(sl_2hat)-modules arise from our construction.

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