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Copy file name to clipboardexpand all lines: QuantumDividedPowerAlgebra.md
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The [QuantumDividedPowerAlgebra](QuantumDividedPowerAlgebra) is a graded algebra over a ring R[q]. The component in degree n is the free R[q]-module with basis x^n^. The multiplication is defined on basis elements by x^r^.x^s^ = [r+s,r]_q x^r+s^ where [r+s,r]_q is the quantum binomial coefficient.
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The [QuantumDividedPowerAlgebra](QuantumDividedPowerAlgebra) is a graded algebra over a ring R[q]. The component in degree n is the free R[q]-module with basis x<sup>n</sup>. The multiplication is defined on basis elements by x<sup>r</sup>.x<sup>s</sup> = [r+s,r]_q x<sup>r+s</sup> where [r+s,r]_q is the quantum binomial coefficient.
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The [DividedPowerAlgebra](http://trac.sagemath.org/sage_trac/search/opensearch?q=wiki%3AQuantumDividedPowerAlgebra) is a graded algebra over a ring R. The component in degree n is the free R-module with basis x^n^. The multiplication is defined on basis elements by x^r^.x^s^ = [r+s,r]_q x^r+s^ where [r+s,r] is the binomial coefficient.
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The [DividedPowerAlgebra](http://trac.sagemath.org/sage_trac/search/opensearch?q=wiki%3AQuantumDividedPowerAlgebra) is a graded algebra over a ring R. The component in degree n is the free R-module with basis x<sup>n</sup>. The multiplication is defined on basis elements by x<sup>r</sup>.x<sup>s</sup> = [r+s,r]_q x<sup>r+s</sup> where [r+s,r] is the binomial coefficient.
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The divided power algebra is a Hopf algebra and is the dual Hopf algebra to R[x]. The coproduct on the divided power Hopf algebra is x^k^ |--> x^k^ x 1 + x^k-1^ x x + ... 1 x x^k^ (where I have used x as an indeterminate and as a tensor product symbol).
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The divided power algebra is a Hopf algebra and is the dual Hopf algebra to R[x]. The coproduct on the divided power Hopf algebra is x<sup>k</sup> |--> x<sup>k</sup> x 1 + x<sup>k-1</sup> x x + ... 1 x x<sup>k</sup> (where I have used x as an indeterminate and as a tensor product symbol).
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See ticket [#11979](https://trac.sagemath.org/ticket/11979)
Copy file name to clipboardexpand all lines: ReleaseTours sage-9.6.md
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### Miscellaneous improvements
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* Ideal membership over quotient rings can now be decided (by reducing to ideal membership in the parent ring). [#33237](https://trac.sagemath.org/ticket/33237)
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* Iterating over (some) infinite modules (including ℤ_^n^_) now enumerates the entire module, in a "natural" order. [#33287](https://trac.sagemath.org/ticket/33287)
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* Iterating over (some) infinite modules (including ℤ_<sup>n</sup>_) now enumerates the entire module, in a "natural" order. [#33287](https://trac.sagemath.org/ticket/33287)
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*`BinaryQF.solve_integer()` now also works for quadratic forms of square discriminant. [#33026](https://trac.sagemath.org/ticket/33026)
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*`Quaternion fractional ideals` (including orders) now support the usual operations (e.g., `a*I`, `I*a`, `I+J`). [#32264](https://trac.sagemath.org/ticket/32264)
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*`AdditiveAbelianGroupWrapper` now exposes `.discrete_log()` for (multi-dimensional) logarithms in finite abelian groups. [#32384](https://trac.sagemath.org/ticket/32384)
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