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The normalized Hecke generator requires inverting v plus one

Example

Over $D = C [v, v^{- 1}, (v + 1)^{- 1}]$ , the Jones idempotent generator $e_{i} = (T_{i} + 1) / (v + 1)$ is defined. The specialization $v = - 1$ is not a ring map from $D$ to $C$ , because it sends the inverted element $v + 1$ to zero. Moreover, in the specialized two-strand Hecke algebra the element $T_{1} + 1$ is a nonzero square-zero element, so no scalar multiple of it is an idempotent generator.

Facts & Assumptions

Given: The rank-two Hecke quadratic relation and the normalized generator definition.

[F1]

$T_{i}^{2} = (v - 1) T_{i} + v$ and $1, T_{i}$ are distinct standard-basis elements (The generic type-A Hecke algebra).

[F2]

The element $e_{i} = (T_{i} + 1) / (v + 1)$ is defined in the localization where $v + 1$ is a unit (The Temperley-Lieb quotient and the Jones specialization).

Verification

technique · direct calculation

1.1F2givenalgebra

A unital map from $D$ to $C$ specializing $v$ to $- 1$ would have to send the unit $(v + 1)^{- 1}$ to an inverse of $(- 1) + 1 = 0$ . No element of $C$ is an inverse of zero, so the localization has no such specialization. Thus the formula in [F2] does not define $e_{i}$ at $v = - 1$ .

2.1F1F2step 1.1algebra∎

In the specialized Hecke algebra, put $x = T_{1} + 1$ . The quadratic relation at $v = - 1$ gives $T_{1}^{2} = - 2 T_{1} - 1$ , hence $x^{2} = T_{1}^{2} + 2 T_{1} + 1 = 0$ . The standard basis in [F1] shows $x \neq 0$ because $1$ and $T_{1}$ are linearly independent. If $c \in C$ and $c x$ were idempotent, then $(c x)^{2} = c x$ would force $0 = c x$ ; for $c \neq 0$ this contradicts $x \neq 0$ . Therefore a scalar multiple cannot supply a normalized idempotent at this specialization.

Depends on

Used by

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Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)
Joshua Chen, The Temperley-Lieb Categories and Skein Modules (standard reference, not scraped)