  The standard state-dependent Heisenberg-Robertson uncertainly-relation lower
bound fails to capture the quintessential incompatibility of observables as the
bound can be zero for some states. To remedy this problem, we establish a class
of tight (i.e., inequalities are saturated)variance-based sum-uncertainty
relations derived from the Lie algebraic properties of observables and show
that our lower bounds depend only on the irreducible representation assumed
carried by the Hilbert space of state of the system. We illustrate our result
for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to
rank 4, and any semisimple compact algebra. We also prove the usefulness of our
results by extending a known variance-based entanglement detection criterion.
