The Center of the Quantized Matrix Algebra

Jakobsen, Hans Plesner; Zhang, Hechun

doi:10.1006/jabr.1997.7121

Cited by 20 publications

(17 citation statements)

References 7 publications

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“…Assuming that q ∈ k is a primitive mth root of unity, and recalling Theorem 1.3, the cardinality of the image in (Z/mZ) n 2 is m n 2 −n if m is odd. Thus we conclude that PIdeg O q M n (k) = m n(n−1) 2 , recovering the result of Jakobsen and Zhang [15] in characteristic zero. By similar methods, one can show that…”

Section: The Multiparameter Coordinate Ring Of Quantum N × N Matricessupporting

confidence: 71%

“…When k has characteristic zero and q is a primitive mth root of unity for m odd, Jakobsen and Zhang found in [15] that PIdeg O q (M n (k)) = m n(n−1) 2 by using De Concini and Procesi's tool given in Theorem 1.3. This result is reproved in [14] using results of De Concini and Procesi and also Jøndrup's work from [16].…”

Section: The Multiparameter Coordinate Ring Of Quantum N × N Matricesmentioning

confidence: 97%

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PI degree parity in q-skew polynomial rings

Haynal

2008

Journal of Algebra

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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 ; τ 1 , δ 1 ] · · · [x n ; τ n , δ n ] agrees with the PI degree of R[x 1 ; τ 1 ] · · · [x n ; τ n ] when each (τ i , δ i ) satisfies a q i -skew relation for q i ∈ k × and extends to a higher q i -skew τ 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.

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Section: The Multiparameter Coordinate Ring Of Quantum N × N Matricessupporting

confidence: 71%

Section: The Multiparameter Coordinate Ring Of Quantum N × N Matricesmentioning

confidence: 97%

PI degree parity in q-skew polynomial rings

Haynal

2008

Journal of Algebra

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“…But for an algebra of the above type considered by De Concini and Procesi, it can be quite complicated to find this rank (cf. [3] and [4]). …”

Section: Introductionmentioning

confidence: 95%

Representations of skew polynomial algebras

Jøndrup¹

1999

Proc. Amer. Math. Soc.

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Abstract. C. De Concini and C. Procesi have proved that in many cases the degree of a skew polynomial algebra is the same as the degree of the corresponding quasi polynomial algebra. We prove a slightly more general result. In fact we show that in case the skew polynomial algebra is a P.I. algebra, then its degree is the degree of the quasi polynomial algebra.Our argument is then applied to determine the degree of some algebras given by generators and relations.

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“…We call these quantum minors covariant quantum minors. The same argument as in [5,Theorem 4.3] shows that these covariant quantum minors q-commute with all of the generators x ij . Furthermore, similar to the proof as in [20, is a dual canonical basis element up to a power of q.…”

Section: Construction Of Basesmentioning

confidence: 73%

Dual canonical bases for the quantum general linear supergroup

Zhang¹,

Zhang²

2006

Journal of Algebra

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Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup O q (SL m|n ). We apply the dual canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n = 1, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel-Weil theorem.

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The Center of the Quantized Matrix Algebra

Abstract: We compute the center in the case where q is a root of unity. The main steps are to compute the degree of an associated quasipolynomial algebra and to compute the dimensions of some interesting irreducible modules. ᮊ 1997 Academic

Cited by 20 publications

References 7 publications

PI degree parity in q-skew polynomial rings

PI degree parity in q-skew polynomial rings

Representations of skew polynomial algebras

Dual canonical bases for the quantum general linear supergroup

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