[Paper Review] A class of left quantum groups modeled after SL_q(n)

Aaron Lauve, Earl J. Taft|ArXiv.org|Aug 26, 2009

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper constructs a left quantum group $\tilde{\mathrm{SL}}_q(n)$, a left Hopf algebra generated by comatrix units $X_{ij}$, which admits a left antipode but no right antipode, making it a non-trivial example of a one-sided quantum group. The construction uses a non-standard extension of the quantum determinant relations and proves that the left antipode is not a right antipode, establishing it as a genuine left Hopf algebra not isomorphic to a full Hopf algebra.

ABSTRACT

For each n >1, we construct a left quantum group, i.e., a left Hopf algebra H generated by comatrix units X_{ij} and modeled after SL_q(n), which has a left antipode but no right antipode. The quantum special linear group SL_q(n) is a homomorphic image of our H.

Motivation & Objective

• To construct a left Hopf algebra that is not a two-sided Hopf algebra, specifically a quantum group modeled after $\mathrm{SL}_q(n)$.
• To define a left antipode $S$ on a non-cocommutative algebra generated by comatrix units $X_{ij}$, ensuring $S \ast \mathbb{I} = \mu\varepsilon$ but $\mathbb{I} \ast S \neq \mu\varepsilon$.
• To demonstrate that the resulting algebra $\tilde{\mathrm{SL}}_q(n)$ is not a full Hopf algebra by showing $\mathbb{I} \ast S \neq \mu\varepsilon$.
• To provide a concrete example of a one-sided quantum group with applications in quantum physics and the boson-fermion correspondence.

Proposed method

• Define $\tilde{\mathrm{SL}}_q(n)$ as a quotient of the free algebra generated by $X_{ij}$, $1 \leq i,j \leq n$, modulo relations derived from quantum determinant-like formulas.
• Use the Diamond Lemma to establish that irreducible words in the generators form a basis, ensuring a well-defined algebra structure.
• Define the left antipode $S$ on generators via the quantum adjoint matrix: $S(X_{ij}) = (-q)^{j-i} \mathrm{det}_q(X^{ji})$, and extend it to monomials by reversing order.
• Prove that $S$ satisfies $m(S \otimes \mathbb{I})\Delta(w) = \varepsilon(w)1$ for all irreducible words $w$, confirming it is a left antipode.
• Show $\mathbb{I} \ast S \neq \mu\varepsilon$ by proving $\sum_j X_{1j}S(X_{j1})$ is not equal to 1, using linear independence of non-reducible monomials.
• Use the notation $\ell(I)$ for the length of a tuple and $\delta_{I,\mathfrak{S}_n}$ to distinguish permutations from non-permutations in the defining relations.

Experimental results

Research questions

• RQ1 Can a left Hopf algebra be constructed that is not a two-sided Hopf algebra, using comatrix units as generators?
• RQ2 Does a quantum group modeled after $\mathrm{SL}_q(n)$ exist that admits a left antipode but not a right antipode?
• RQ3 Is it possible to define a left antipode on a non-commutative algebra generated by comatrix units such that it is not a coalgebra antimorphism?
• RQ4 Can such a structure be used to model one-sided quantum symmetries relevant to quantum physics?

Key findings

• The algebra $\tilde{\mathrm{SL}}_q(n)$ is generated by comatrix units $X_{ij}$ and admits a well-defined left antipode $S$ defined via the quantum adjoint matrix.
• The left antipode satisfies $S \ast \mathbb{I} = \mu\varepsilon$, confirming $\tilde{\mathrm{SL}}_q(n)$ is a left Hopf algebra.
• The map $\mathbb{I} \ast S$ is not equal to $\mu\varepsilon$, as shown by the non-triviality of $\sum_j X_{1j}S(X_{j1})$, proving no right antipode exists.
• The irreducible words in the generators form a basis for $\tilde{\mathrm{SL}}_q(n)$, ensuring the algebra is well-structured and finite-dimensional over the basis.
• The quantum determinant relation $D_I = (-q)^{-\ell(I)}\delta_{I,\mathfrak{S}_n}$ ensures that reducible words do not appear on the right-hand side of reduction rules.
• The left antipode is not an algebra antimorphism globally, as it is defined by reversing word order rather than satisfying $S(ab) = S(b)S(a)$.