The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces

Krähmer, Ulrich

Tabiri, Angela Ankomaah

English

Ulrich Krähmer

Angela Ankomaah Tabiri

Crossref

Algebr Represent Theor (2017) 20:655–658DOI 10.1007/s10468-016-9658-8The Nodal Cubic is a Quantum Homogeneous SpaceUlrich Kra¨hmer1 ·Angela Ankomaah Tabiri1Received: 5 September 2016 / Accepted: 4 November 2016 / Published online: 7 January 2017© The Author(s) 2017. This article is published with open access at Springerlink.comAbstract The cusp was recently shown to admit the structure of a quantum homogeneousspace, that is, its coordinate ring B can be embedded as a right coideal subalgebra into aHopf algebra A such that A is faithfully flat as a B-module. In the present article such aHopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus furthermotivating the question which affine varieties are quantum homogeneous spaces.Keywords Hopf algebra · Quantum homogeneous space · Singular curve ·Noncommutative Galois extensionMathematics Subject Classification (2010) 16T05 · 16T20 · 14H50Just as quantum groups (Hopf algebras) generalise affine algebraic groups, quantum homo-geneous spaces as studied e.g. in [2, 7, 11, 14–19, 21] generalise affine varieties with atransitive action of an algebraic group:Definition A quantum homogeneous space is a right coideal subalgebra B of a Hopfalgebra A which is faithfully flat as a left B-module.There is also an analytic theory of transitive or more generally ergodic actions of compactor locally compact quantum groups, see e.g. [8, 13] and the references therein.Presented by Paul Smith. Ulrich Kra¨hmerulrich.kraehmer@glasgow.ac.ukAngela Ankomaah Tabiria.tabiri.1@research.gla.ac.uk1 School of Mathematics and Statistics, 15 University Gardens, University of Glasgow, Glasgow,G12 8QW, UK656 U. Kra¨hmer, A. TabiriThe best studied examples are deformation quantisations of affine homogeneous spaces,and in particular Podles´’ quantum 2-spheres [20]. However, even if A is noncommutative,B can be a commutative algebra, so a natural question to ask is:Question Which affine varieties are quantum homogeneous spaces?In the operator algebraic setting, the analogue of this question has been raised and stud-ied for example in [12]. Here we consider the purely algebraic setting, working over analgebraically closed field k of characteristic 0.Unlike a homogeneous space, an affine variety which is a quantum homogeneous spacecan be singular as the example of the cusp shows (see [15, Section 2.11] and [10, Construc-tion 1.2]). Note that by the results from [9], noncommutative Hopf algebras coacting oncommutative algebras are quite restricted. Still, we conjecture that every plane curve can begiven the structure of a quantum homogeneous space, and our aim here is to point this outfor the nodal cubic given by the equation y2 = x2 + x3:Theorem Fix (q, p) ∈ κ2 satisfying p2 = q2 + q3. Then the unital associative κ-algebraA with generators x, y, a, a−1, b satisfying the relationsaa−1 = a−1a = 1, y2 = x2 + x3, b2 = a3,ba = ab, ya = ay, bx = xb, yx = xy, by = −yb + 2pb2,a2x = −xa2 − axa − a2 + (1 + 3q) a3,ax2 = −ax − xa − x2a − xax + (2 + 3q)qa3.admits a Hopf algebra structure whose coproduct , counit ε and antipode S satisfy(x) = 1 ⊗ (x − qa) + x ⊗ a, (y) = 1 ⊗ (y − pb) + y ⊗ b,(a) = a ⊗ a, (b) = b ⊗ b, ε(x) = q, ε(y) = p, ε(a) = ε(b) = 1,S(x) = q − (x − q) a−1, S(y) = p − (y − p) b−1, S(a) = a−1, S(b) = b−1.Furthermore, the right coideal subalgebra B ⊂ A generated by x, y is the coordinate ringof the nodal cubic, and A is free and in particular faithfully flat as a B-module.Observe that the commutation relations in A are chosen in such a way that(y − pb)2 = y2 − p2b2, (x − qa)2 + (x − qa)3 = x2 + x3 − (q2 + q3)a3so that(y − pb)2 = (x − qa)2 + (x − qa)3.Thus informally speaking each coordinate on the curve becomes perturbed by some addi-tional group-like “quantum coordinate” and the perturbed coordinates still satisfy thedefining relation of the curve. As x −qa and y −pb are twisted primitive, the Hopf algebraA is generated by group-likes and twisted primitives. This implies:Proposition The Hopf algebra A is pointed.The point (q, p) on the curve is the point the quantum orbit of which it is presented as. Touse these observations as starting point of a general study of quantum homogeneous spacestructures on affine varieties seems a promising future research direction.The Nodal Cubic is a Quantum Homogeneous Space 657The proof of the theorem consists of a straightforward (albeit tedious) verification thatthe formulas for the coproduct, counit and antipode are compatible with the defining rela-tions of A, followed by a similarly straigthforward application of Bergman’s diamondlemma [1] yielding a vector space basis of A that implies the freeness over B:Proposition The set{xiyj (ax)lambn | i, l ∈ N, j ∈ {0, 1},m ∈ Z, n ∈ {0, 1}}is a vector space basis of A, and the GK-dimension of A equals 3.Using this basis, one also easily observes that like the nonstandard Podles´ spheres, thealgebra extension B ⊂ A is an example of a coalgebra Galois extension [3–6] rather thanof a Hopf-Galois extension: the coalgebra is C := A/B+A, where B+ := B ∩ ker ε. Thecanonical projection π : A → C defines a left C-coactionλ : A → C ⊗ A, f → f(−1) ⊗ f(0) := π(f(1)) ⊗ f(2)and we have (as a consequence of the faithful flatness of A over B)B = {f ∈ A | f(−1) ⊗ f(0) = π(1) ⊗ f }.That C is not a Hopf algebra quotient of A follows from B+A = AB+(cf. [18, Lemma 1.4]); for example, we haveAB+ 	 a2(x − q) = −xa2 − axa − (1 + q)a2 + (1 + 3q)a3 /∈ B+A.We finally remark that some properties of the algebra A are better understood when usinga slightly different set of generators: if we abbreviatec := 3x − (1 + 3q)a + 1, d := 3y − 6pb, e := ac + rcawhere r is a primitive 6th root of 1 (so that r + r−1 = 1), then the defining relations of A interms of the generators a±1, b, c, d, e readaa−1 = a−1a = 1, ab = ba, ac + rca = e, ad = da, ae + r−1ea = 0,bc = cb, bd = −db, be = eb, b2 = a3, cd = dc, r−1ce + ec = 3(a − a3),de = ed, 3d2 = c3 − 3c + 2 + (1 + 3q)(−2 + 6q + 9q2)a3.Using these generators, one easily verifies for example:Proposition The units in A are of the form αaibj , α ∈ k, i ∈ Z, j ∈ {0, 1}.Acknowledgments We thank Ken Brown and Tomasz Brzezin´ski for helpful comments on a first draft ofthis paper. A.T. is funded by a Faculty for the Future Fellowship of the Schlumberger Foundation.Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter-national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons license, and indicate if changes were made.References1. Bergman, G.M.: The diamond lemma for ring theory. Adv. in Math. 29(2), 178–218 (1978).doi:10.1016/0001-8708(78)90010-5. MR506890658 U. Kra¨hmer, A. Tabiri2. Brown, K., Gilmartin, P.: Quantum homogeneous spaces of connected hopf algebras. J. Algebra 454,400–432 (2016). doi:10.1016/j.jalgebra.2016.01.030. MR34734333. Brzezin´ski, T.: On modules associated to coalgebra galois extensions. J. Algebra 215(1), 290–317(1999). doi:10.1006/jabr.1998.7738. MR16841584. Brzezin´ski, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of galois type. Comm.Algebra 27(3), 1347–1367 (1999). doi:10.1080/00927879908826498. MR16690955. Brzezin´ski, T., Hajac, P.M.: The chern-galois character. English, with English and French summaries. C.R. Math. Acad. Sci. Paris 338(2), 113–116 (2004). doi:10.1016/j.crma.2003.11.009. MR20382786. Brzezin´ski, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191(2), 467–492 (1998).doi:10.1007/s002200050274. MR16043407. De Commer, K.: On the construction of quantum homogeneous spaces from *-galois objects. Algebr.Represent. Theory 15(4), 795–815 (2012). doi:10.1007/s10468-011-9265-7. MR29444428. De Commer, K., Yamashita, M.: Tannaka-krei˘n duality for compact quantum homogeneous spaces. i.General theory. Theory Appl. Categ. 28(31), 1099–1138 (2013). MR31216229. Etingof, P., Debashish, G., Mandal, A., Walton, C.: Hopf coactions on commutative algebras generatedby a quadratically independent comodule. Comm. Algebra. to appear10. Goodearl, K.R., Zhang, J.J.: Noetherian hopf algebra domains of gelfand-kirillov dimension two. J.Algebra 324(11), 3131–3168 (2010). doi:10.1016/j.jalgebra.2009.11.001. MR273299111. Heckenberger, I., Kolb, S.: Differential calculus on quantum homogeneous spaces. Lett. Math. Phys.63(3), 255–264 (2003). doi:10.1023/A:1024496304742. MR199289012. Huang, H.: Faithful compact quantum group actions on connected compact metrizable spaces. J. Geom.Phys. 70, 232–236 (2013). doi:10.1016/j.geomphys.2013.03.027. MR305429713. Kasprzak, P., Sołtan, P.M.: Embeddable quantum homogeneous spaces. J. Math. Anal. Appl. 411(2),574–591 (2014). doi:10.1016/j.jmaa.2013.07.084. MR312841514. Kolb, S., Pellegrini, J.: Braid group actions on coideal subalgebras of quantized enveloping algebras. J.Algebra 336, 395–416 (2011). doi:10.1016/j.jalgebra.2011.04.001. MR280255215. Kra¨hmer, U.: On the hochschild (co)homology of quantum homogeneous spaces. Israel J. Math. 189,237–266 (2012). doi:10.1007/s11856-011-0168-4. MR293139616. Liu, L., Wu, Q.: Twisted calabi-yau property of right coideal subalgebras of quantized envelopingalgebras. J. Algebra 399, 1073–1085 (2014). doi:10.1016/j.jalgebra.2013.10.019. MR314462717. Liu, L., Wu, Q.: Rigid dualizing complexes over quantum homogeneous spaces. J. Algebra 353, 121–141(2012). doi:10.1016/j.jalgebra.2011.12.007. MR287243918. Mu¨ller, E.F., Schneider, H.-J.: Quantum homogeneous spaces with faithfully flat module structures.Israel J. Math. 111, 157–190 (1999). doi:10.1007/BF02810683. MR171073719. O´ Buachalla, R.: Noncommutative complex structures on quantum homogeneous spaces. J. Geom. Phys.99, 154–173 (2016). doi:10.1016/j.geomphys.2015.10.003. MR342836220. Podles´, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987). doi:10.1007/BF00416848.MR91932221. Schneider, H.-J.: Principal homogeneous spaces for arbitrary hopf algebras Hopf algebras. Israel J. Math.72(1-2), 167–195 (1990). doi:10.1007/BF02764619. MR1098988

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The Nodal Cubic is a Quantum Homogeneous Space

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The nodal cubic is a quantum homogeneous space

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Algebr Represent Theor (2017) 20:655–658
DOI 10.1007/s10468-016-9658-8
The Nodal Cubic is a Quantum Homogeneous Space
Ulrich Kra¨hmer1 ·Angela Ankomaah Tabiri1
Received: 5 September 2016 / Accepted: 4 November 2016 / Published online: 7 January 2017
© The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract The cusp was recently shown to admit the structure of a quantum homogeneous
space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a
Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a
Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further
motivating the question which affine varieties are quantum homogeneous spaces.
Keywords Hopf algebra · Quantum homogeneous space · Singular curve ·
Noncommutative Galois extension
Mathematics Subject Classification (2010) 16T05 · 16T20 · 14H50
Just as quantum groups (Hopf algebras) generalise affine algebraic groups, quantum homo-
geneous spaces as studied e.g. in [2, 7, 11, 14–19, 21] generalise affine varieties with a
transitive action of an algebraic group:
Definition A quantum homogeneous space is a right coideal subalgebra B of a Hopf
algebra A which is faithfully flat as a left B-module.
There is also an analytic theory of transitive or more generally ergodic actions of compact
or locally compact quantum groups, see e.g. [8, 13] and the references therein.
Presented by Paul Smith.
 Ulrich Kra¨hmer
ulrich.kraehmer@glasgow.ac.uk
Angela Ankomaah Tabiri
a.tabiri.1@research.gla.ac.uk
1 School of Mathematics and Statistics, 15 University Gardens, University of Glasgow, Glasgow,
G12 8QW, UK
656 U. Kra¨hmer, A. Tabiri
The best studied examples are deformation quantisations of affine homogeneous spaces,
and in particular Podles´’ quantum 2-spheres [20]. However, even if A is noncommutative,
B can be a commutative algebra, so a natural question to ask is:
Question Which affine varieties are quantum homogeneous spaces?
In the operator algebraic setting, the analogue of this question has been raised and stud-
ied for example in [12]. Here we consider the purely algebraic setting, working over an
algebraically closed field k of characteristic 0.
Unlike a homogeneous space, an affine variety which is a quantum homogeneous space
can be singular as the example of the cusp shows (see [15, Section 2.11] and [10, Construc-
tion 1.2]). Note that by the results from [9], noncommutative Hopf algebras coacting on
commutative algebras are quite restricted. Still, we conjecture that every plane curve can be
given the structure of a quantum homogeneous space, and our aim here is to point this out
for the nodal cubic given by the equation y2 = x2 + x3:
Theorem Fix (q, p) ∈ κ2 satisfying p2 = q2 + q3. Then the unital associative κ-algebra
A with generators x, y, a, a−1, b satisfying the relations
aa−1 = a−1a = 1, y2 = x2 + x3, b2 = a3,
ba = ab, ya = ay, bx = xb, yx = xy, by = −yb + 2pb2,
a2x = −xa2 − axa − a2 + (1 + 3q) a3,
ax2 = −ax − xa − x2a − xax + (2 + 3q)qa3.
admits a Hopf algebra structure whose coproduct , counit ε and antipode S satisfy
(x) = 1 ⊗ (x − qa) + x ⊗ a, (y) = 1 ⊗ (y − pb) + y ⊗ b,
(a) = a ⊗ a, (b) = b ⊗ b, ε(x) = q, ε(y) = p, ε(a) = ε(b) = 1,
S(x) = q − (x − q) a−1, S(y) = p − (y − p) b−1, S(a) = a−1, S(b) = b−1.
Furthermore, the right coideal subalgebra B ⊂ A generated by x, y is the coordinate ring
of the nodal cubic, and A is free and in particular faithfully flat as a B-module.
Observe that the commutation relations in A are chosen in such a way that
(y − pb)2 = y2 − p2b2, (x − qa)2 + (x − qa)3 = x2 + x3 − (q2 + q3)a3
so that
(y − pb)2 = (x − qa)2 + (x − qa)3.
Thus informally speaking each coordinate on the curve becomes perturbed by some addi-
tional group-like “quantum coordinate” and the perturbed coordinates still satisfy the
defining relation of the curve. As x −qa and y −pb are twisted primitive, the Hopf algebra
A is generated by group-likes and twisted primitives. This implies:
Proposition The Hopf algebra A is pointed.
The point (q, p) on the curve is the point the quantum orbit of which it is presented as. To
use these observations as starting point of a general study of quantum homogeneous space
structures on affine varieties seems a promising future research direction.
The Nodal Cubic is a Quantum Homogeneous Space 657
The proof of the theorem consists of a straightforward (albeit tedious) verification that
the formulas for the coproduct, counit and antipode are compatible with the defining rela-
tions of A, followed by a similarly straigthforward application of Bergman’s diamond
lemma [1] yielding a vector space basis of A that implies the freeness over B:
Proposition The set
{xiyj (ax)lambn | i, l ∈ N, j ∈ {0, 1},m ∈ Z, n ∈ {0, 1}}
is a vector space basis of A, and the GK-dimension of A equals 3.
Using this basis, one also easily observes that like the nonstandard Podles´ spheres, the
algebra extension B ⊂ A is an example of a coalgebra Galois extension [3–6] rather than
of a Hopf-Galois extension: the coalgebra is C := A/B+A, where B+ := B ∩ ker ε. The
canonical projection π : A → C defines a left C-coaction
λ : A → C ⊗ A, f → f(−1) ⊗ f(0) := π(f(1)) ⊗ f(2)
and we have (as a consequence of the faithful flatness of A over B)
B = {f ∈ A | f(−1) ⊗ f(0) = π(1) ⊗ f }.
That C is not a Hopf algebra quotient of A follows from B+A = AB+
(cf. [18, Lemma 1.4]); for example, we have
AB+ 	 a2(x − q) = −xa2 − axa − (1 + q)a2 + (1 + 3q)a3 /∈ B+A.
We finally remark that some properties of the algebra A are better understood when using
a slightly different set of generators: if we abbreviate
c := 3x − (1 + 3q)a + 1, d := 3y − 6pb, e := ac + rca
where r is a primitive 6th root of 1 (so that r + r−1 = 1), then the defining relations of A in
terms of the generators a±1, b, c, d, e read
aa−1 = a−1a = 1, ab = ba, ac + rca = e, ad = da, ae + r−1ea = 0,
bc = cb, bd = −db, be = eb, b2 = a3, cd = dc, r−1ce + ec = 3(a − a3),
de = ed, 3d2 = c3 − 3c + 2 + (1 + 3q)(−2 + 6q + 9q2)a3.
Using these generators, one easily verifies for example:
Proposition The units in A are of the form αaibj , α ∈ k, i ∈ Z, j ∈ {0, 1}.
Acknowledgments We thank Ken Brown and Tomasz Brzezin´ski for helpful comments on a first draft of
this paper. A.T. is funded by a Faculty for the Future Fellowship of the Schlumberger Foundation.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter-
national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if changes were made.
References
1. Bergman, G.M.: The diamond lemma for ring theory. Adv. in Math. 29(2), 178–218 (1978).
doi:10.1016/0001-8708(78)90010-5. MR506890
658 U. Kra¨hmer, A. Tabiri
2. Brown, K., Gilmartin, P.: Quantum homogeneous spaces of connected hopf algebras. J. Algebra 454,
400–432 (2016). doi:10.1016/j.jalgebra.2016.01.030. MR3473433
3. Brzezin´ski, T.: On modules associated to coalgebra galois extensions. J. Algebra 215(1), 290–317
(1999). doi:10.1006/jabr.1998.7738. MR1684158
4. Brzezin´ski, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of galois type. Comm.
Algebra 27(3), 1347–1367 (1999). doi:10.1080/00927879908826498. MR1669095
5. Brzezin´ski, T., Hajac, P.M.: The chern-galois character. English, with English and French summaries. C.
R. Math. Acad. Sci. Paris 338(2), 113–116 (2004). doi:10.1016/j.crma.2003.11.009. MR2038278
6. Brzezin´ski, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191(2), 467–492 (1998).
doi:10.1007/s002200050274. MR1604340
7. De Commer, K.: On the construction of quantum homogeneous spaces from *-galois objects. Algebr.
Represent. Theory 15(4), 795–815 (2012). doi:10.1007/s10468-011-9265-7. MR2944442
8. De Commer, K., Yamashita, M.: Tannaka-krei˘n duality for compact quantum homogeneous spaces. i.
General theory. Theory Appl. Categ. 28(31), 1099–1138 (2013). MR3121622
9. Etingof, P., Debashish, G., Mandal, A., Walton, C.: Hopf coactions on commutative algebras generated
by a quadratically independent comodule. Comm. Algebra. to appear
10. Goodearl, K.R., Zhang, J.J.: Noetherian hopf algebra domains of gelfand-kirillov dimension two. J.
Algebra 324(11), 3131–3168 (2010). doi:10.1016/j.jalgebra.2009.11.001. MR2732991
11. Heckenberger, I., Kolb, S.: Differential calculus on quantum homogeneous spaces. Lett. Math. Phys.
63(3), 255–264 (2003). doi:10.1023/A:1024496304742. MR1992890
12. Huang, H.: Faithful compact quantum group actions on connected compact metrizable spaces. J. Geom.
Phys. 70, 232–236 (2013). doi:10.1016/j.geomphys.2013.03.027. MR3054297
13. Kasprzak, P., Sołtan, P.M.: Embeddable quantum homogeneous spaces. J. Math. Anal. Appl. 411(2),
574–591 (2014). doi:10.1016/j.jmaa.2013.07.084. MR3128415
14. Kolb, S., Pellegrini, J.: Braid group actions on coideal subalgebras of quantized enveloping algebras. J.
Algebra 336, 395–416 (2011). doi:10.1016/j.jalgebra.2011.04.001. MR2802552
15. Kra¨hmer, U.: On the hochschild (co)homology of quantum homogeneous spaces. Israel J. Math. 189,
237–266 (2012). doi:10.1007/s11856-011-0168-4. MR2931396
16. Liu, L., Wu, Q.: Twisted calabi-yau property of right coideal subalgebras of quantized enveloping
algebras. J. Algebra 399, 1073–1085 (2014). doi:10.1016/j.jalgebra.2013.10.019. MR3144627
17. Liu, L., Wu, Q.: Rigid dualizing complexes over quantum homogeneous spaces. J. Algebra 353, 121–141
(2012). doi:10.1016/j.jalgebra.2011.12.007. MR2872439
18. Mu¨ller, E.F., Schneider, H.-J.: Quantum homogeneous spaces with faithfully flat module structures.
Israel J. Math. 111, 157–190 (1999). doi:10.1007/BF02810683. MR1710737
19. O´ Buachalla, R.: Noncommutative complex structures on quantum homogeneous spaces. J. Geom. Phys.
99, 154–173 (2016). doi:10.1016/j.geomphys.2015.10.003. MR3428362
20. Podles´, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987). doi:10.1007/BF00416848.
MR919322
21. Schneider, H.-J.: Principal homogeneous spaces for arbitrary hopf algebras Hopf algebras. Israel J. Math.
72(1-2), 167–195 (1990). doi:10.1007/BF02764619. MR1098988


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The nodal cubic is a quantum homogeneous space

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