PI degree parity in q-skew polynomial rings

Heidi Haynal

arXiv:0704.0846·math.RA·May 23, 2007·1 cites

PI degree parity in q-skew polynomial rings

Heidi Haynal

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TL;DR

This paper investigates the PI degree of iterated skew polynomial rings with q-skew relations, confirming the quantum Gelfand-Kirillov conjecture for certain quantized coordinate rings and extending results to prime factor algebras.

Contribution

It establishes that the PI degree remains unchanged under specific q-skew extensions and confirms the quantum Gelfand-Kirillov conjecture for various quantized coordinate rings.

Findings

01

PI degree of iterated skew polynomial rings matches that of simpler rings under q-skew relations.

02

Confirmed the quantum Gelfand-Kirillov conjecture for multiple quantized coordinate rings.

03

Extended results to completely prime factor algebras.

Abstract

For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;\tau_1,\delta_1]...b[x_n;\tau_n,\delta_n] agrees with the PI degree of R[x_1;\tau_1]...b[x_n;\tau_n] when each (\tau_i,\delta_i) satisfies a q_i-skew relation for q_i \in k^{\times} and extends to a higher q_i-skew \tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras.

Equations332

Z^{n} (a_{ij}) Z^{n} π (Z / ℓ Z)^{n}

Z^{n} (a_{ij}) Z^{n} π (Z / ℓ Z)^{n}

(m)_{t}

(m)_{t}

(m)!_{t}

(m n)_{t}

(0 n)_{t} = (n n)_{t} = 1 for all n \geq 0

(0 n)_{t} = (n n)_{t} = 1 for all n \geq 0

(m n)_{t} = (m n - 1)_{t} + t^{n - m} (m - 1 n - 1)_{t} = (m - 1 n - 1)_{t} + t^{m} (m n - 1)_{t} for all 0 < m < n

(m n)_{t} = (m n - 1)_{t} + t^{n - m} (m - 1 n - 1)_{t} = (m - 1 n - 1)_{t} + t^{m} (m n - 1)_{t} for all 0 < m < n

δ^{n} (r s)

δ^{n} (r s)

x^{n} r

d_{0} is the identity

d_{0} is the identity

d_{n} (r s) = i = 0 \sum n τ^{n - i} d_{i} (r) d_{n - i} (s) for all r, s \in R and all n

d_{i} τ = q^{i} τ d_{i} for all i .

d_{i} = \frac{δ ^{i}}{( i ) ! _{q}}

d_{i} = \frac{δ ^{i}}{( i ) ! _{q}}

d_{i} (y^{n}) = {(i n)_{q} y^{n - i} when i \leq n, 0 when i > n .

d_{i} (y^{n}) = {(i n)_{q} y^{n - i} when i \leq n, 0 when i > n .

R [[x; τ^{- 1}]] \ignorespaces \ignorespaces \ignorespaces \ignorespaces

R [[x; τ^{- 1}]] \ignorespaces \ignorespaces \ignorespaces \ignorespaces

\Psi(rs)=\sum_{i=0}^{\infty}x^{i}d_{i}(rs)=\sum_{i=0}^{\infty}x^{i}\Big{(}\sum_{m=0}^{i}\tau^{i-m}d_{m}(r)d_{i-m}(s)\Big{)}.

\Psi(rs)=\sum_{i=0}^{\infty}x^{i}d_{i}(rs)=\sum_{i=0}^{\infty}x^{i}\Big{(}\sum_{m=0}^{i}\tau^{i-m}d_{m}(r)d_{i-m}(s)\Big{)}.

\Psi(r)\Psi(s)=\Big{(}\sum_{i=0}^{\infty}x^{i}d_{i}(r)\Big{)}\Big{(}\sum_{i=0}^{\infty}x^{i}d_{i}(s)\Big{)}=\sum_{i=0}^{\infty}x^{i}\Big{(}\sum_{m=0}^{i}\tau^{i-m}d_{m}(r)d_{i-m}(s)\Big{)}.

\Psi(r)\Psi(s)=\Big{(}\sum_{i=0}^{\infty}x^{i}d_{i}(r)\Big{)}\Big{(}\sum_{i=0}^{\infty}x^{i}d_{i}(s)\Big{)}=\sum_{i=0}^{\infty}x^{i}\Big{(}\sum_{m=0}^{i}\tau^{i-m}d_{m}(r)d_{i-m}(s)\Big{)}.

d_{n} (r) 1^{- 1}

d_{n} (r) 1^{- 1}

= τ^{n} (r s^{- 1}) d_{n} (s) 1^{- 1} + \dots + d_{n} (r s^{- 1}) s 1^{- 1} .

d_{n}(rs^{-1})=\Big{[}d_{n}(r)-\sum_{j=0}^{n-1}\tau^{n-j}d_{j}(rs^{-1})d_{n-j}(s)\Big{]}s^{-1}.

d_{n}(rs^{-1})=\Big{[}d_{n}(r)-\sum_{j=0}^{n-1}\tau^{n-j}d_{j}(rs^{-1})d_{n-j}(s)\Big{]}s^{-1}.

R S^{- 1} [[x; τ^{- 1}]] \ignorespaces \ignorespaces \ignorespaces \ignorespaces

R S^{- 1} [[x; τ^{- 1}]] \ignorespaces \ignorespaces \ignorespaces \ignorespaces

\tau\Phi(r)=\sum_{i=0}^{\infty}x^{i}q^{i}\tau d_{i}(r)1^{-1}=\sum_{i=0}^{\infty}x^{i}d_{i}\big{(}\tau(r)\big{)}1^{-1}=\Phi\tau(r)

\tau\Phi(r)=\sum_{i=0}^{\infty}x^{i}q^{i}\tau d_{i}(r)1^{-1}=\sum_{i=0}^{\infty}x^{i}d_{i}\big{(}\tau(r)\big{)}1^{-1}=\Phi\tau(r)

d_{n} (b_{j_{1}} \dots b_{j_{m}}) = i = 0 \sum n τ^{n - 1} d_{i} (b_{j_{1}} \dots b_{j_{m - 1}}) d_{n - i} (b_{j_{m}}) \in B

d_{n} (b_{j_{1}} \dots b_{j_{m}}) = i = 0 \sum n τ^{n - 1} d_{i} (b_{j_{1}} \dots b_{j_{m - 1}}) d_{n - i} (b_{j_{m}}) \in B

d_{p} (x_{j_{1}} \dots x_{j_{n}}) = i = 0 \sum p τ^{p - i} d_{i} (x_{j_{1}} \dots x_{j_{n - 1}}) d_{p - i} (x_{j_{n}}) = 0,

d_{p} (x_{j_{1}} \dots x_{j_{n}}) =