Professor Stéphane Launois

Professor of Pure Mathematics

Stéphane has served as the School's Director of Research, chairing the School's Research and Enterprise Committee, and was Head of the Mathematics group before handing over to Clare Dunning in autumn 2018.

Research interests

Noncommutative algebra and noncommutative geometry; in particular, quantum algebras and their links with their (semi-)classical counterparts: enveloping algebras and Poisson algebras.Representation TheoryCombinatoricsHomological algebra.

Supervision

Isaac OppongAlexandra Rogers

Publications

Article

• Launois, S. and Topley, L. (2019). Transfer results for Frobenius extensions. Journal of Algebra [Online] 524:35-58. Available at: https://doi.org/10.1016/j.jalgebra.2019.01.006.
We study Frobenius extensions which are free-filtered by a totally ordered, finitely generated abelian group, and their free-graded counterparts. First we show that the Frobenius property passes up from a free-graded extension to a free-filtered extension, then also from a free-filtered extension to the extension of their Rees algebras. Our main theorem states that, under some natural hypotheses, a free-filtered extension of algebras is Frobenius if and only if the associated graded extension is Frobenius. In the final section we apply this theorem to provide new examples and non-examples of Frobenius extensions.
• Goodearl, K., Launois, S. and Lenagan, T. (2019). Tauvel’s height formula for quantum nilpotent algebras. Communications in Algebra [Online]. Available at: https://doi.org/10.1080/00927872.2019.1581210.
Tauvel’s height formula, which provides a link between the height of a prime ideal and
the Gelfand-Kirillov dimension of the corresponding factor algebra, is verified for quantum
nilpotent algebras.
• Launois, S. and Sanchez, O. (2019). On the Dixmier-Moeglin equivalence for Poisson-Hopf algebras. Advances in Mathematics [Online] 346:48-69. Available at: https://doi.org/10.1016/j.aim.2019.01.036.
We prove that the Poisson version of the Dixmier-Moeglin equivalence
holds for cocommutative a?ne Poisson-Hopf algebras. This is a ?rst step
towards understanding the symplectic foliation and the representation theory
of (cocommutative) a?ne Poisson-Hopf algebras. Our proof makes substantial
use of the model theory of ?elds equipped with ?nitely many possibly noncommuting
derivations. As an application, we show that the symmetric algebra of
a ?nite dimensional Lie algebra, equipped with its natural Poisson structure,
satis?es the Poisson Dixmier-Moeglin equivalence.
• Kitchin, A. and Launois, S. (2019). On the automorphisms of quantum Weyl algebras. Journal of Pure and Applied Algebra [Online] 223:1514-1530. Available at: https://doi.org/10.1016/j.jpaa.2018.06.016.
Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U+q(so5).
• Bell, J., Launois, S. and Nolan, B. (2017). A strong Dixmier-Moeglin equivalence for quantum Schubert cells. Journal of Algebra [Online] 487:269-293. Available at: http://dx.doi.org/10.1016/j.jalgebra.2017.06.005.
Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence.

We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence . Using the example of the universal enveloping algebra of sl2(C), we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence.

For a simple complex Lie algebra g, a non-root of unity q?0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra Uq[w] of the quantised enveloping K-algebra Uq(g). These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.
• Bell, J., Launois, S., Leon Sanchez, O. and Moosa, R. (2017). Poisson algebras via model theory and differential-algebraic geometry. Journal of The European Mathematical Society [Online] 19:2019-2049. Available at: http://dx.doi.org/10.4171/JEMS/712.
Brown and Gordon asked whether the Poisson Dixmier–Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier–Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier–Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.
• Launois, S. and Lecoutre, C. (2016). Poisson Deleting Derivations Algorithm and Poisson Spectrum. Communications in Algebra [Online] 45:1294-1313. Available at: http://dx.doi.org/10.1080/00927872.2016.1175619.
Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.
[CrossRef], [Web of Science ®]
] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.
[CrossRef], [Web of Science ®]
]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.
[CrossRef], [Web of Science ®]
]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class
• Launois, S. and Lecoutre, C. (2016). A quadratic Poisson Gel’fand-Kirillov problem in prime characteristic. Transactions of the American Mathematical Society [Online] 368:755-785. Available at: http://dx.doi.org/10.1090/tran/6352.
• Grabowski, J. and Launois, S. (2014). Graded quantum cluster algebras and an application to quantum Grassmannians. Proceedings of the London Mathematical Society [Online] 109:697-732. Available at: http://plms.oxfordjournals.org/content/109/3/697.
• Kitchin, A. and Launois, S. (2014). Endomorphisms of Quantum Generalized Weyl Algebras. Letters in Mathematical Physics [Online] 104:837-848. Available at: http://dx.doi.org/10.1007/s11005-014-0691-4.
We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra Uq(sl2) and certain minimal primitive quotients of the positive part of Uq(so5)
• Bell, J., Casteels, K. and Launois, S. (2014). Primitive ideals in quantum Schubert cells: Dimension of the strata. Forum Mathematicum [Online] 26:703-721. Available at: http://dx.doi.org/10.1515/forum-2011-0155.
The aim of this paper is to study the representation theory of quantum Schubert cells. Let be a simple complex Lie algebra. To each element w of the Weyl group W of , De Concini, Kac and Procesi have attached a subalgebra Uq[w] of the quantised enveloping algebra Uq(). Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of Uq[w]. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of Uq[w] admits a stratification indexed by those elements vW with vw in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair.
• Launois, S. and Lenagan, T. (2014). Efficient Recognition of Totally Nonnegative Matrix Cells. Foundations of Computational Mathematics [Online] 14:371-387. Available at: http://dx.doi.org/10.1007/s10208-013-9169-5.
The space of m×p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegative cells.
• Launois, S. and Lopes, S. (2013). Classification of factorial generalized down–up algebras. Journal of Algebra [Online] 396:184-206. Available at: http://dx.doi.org/10.1016/j.jalgebra.2013.08.012.
We determine when a generalized down–up algebra is a Noetherian unique factorisation domain or a Noetherian unique factorisation ring.
• Launois, S. and Lenagan, T. (2013). Automorphisms of quantum matrices. Glasgow Mathematical Journal [Online] 55A:89-100. Available at: http://dx.doi.org/10.1017/S0017089513000529.
• Bell, J., Casteels, K. and Launois, S. (2012). Enumeration of H-strata in quantum matrices with respect to dimension. Journal of Combinatorial Theory, Series A [Online] 119:83-98. Available at: http://dx.doi.org/10.1016/j.jcta.2011.07.007.
We present a combinatorial method to determine the dimension of H-strata in the algebra of m x n quantum matrices O(q)(M(m,n)(K)) as follows. To a given H-stratum we associate a certain permutation via the notion of pipe dreams. We show that the dimension of the H-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the d-dimensional H-strata in Q(q)(M(m,n)(K)). Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors (Bell and Launois (2010) [3], Bell, Launois and Lutley (2010) [4]) regarding the asymptotic proportion of d-dimensional H-strata in Q(q)(M(m,n) (K)).
• Goodearl, K., Launois, S. and Lenagan, T. (2011). Totally nonnegative cells and Matrix Poisson varieties. Advances in Mathematics [Online] 226:779-826. Available at: http://dx.doi.org/10.1016/j.aim.2010.07.010.
We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.
• Launois, S. and Lenagan, T. (2011). Twisting the quantum Grassmannian. Proceedings of the American Mathematical Society [Online] 139:99-110. Available at: http://dx.doi.org/10.1090/S0002-9939-2010-10478-1.
In contrast to the classical and semiclassical settings, the Coxeter element (12...n) which cycles the columns of an m x n matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained by means of a cocycle twist. A consequence is that the torus invariant prime ideals of the quantum grassmannian are permuted by the action of the Coxeter element (12...n). We view this as a quantum analogue of the recent result of Knutson, Lam and Speyer, where the Lusztig strata of the classical grassmannian are permuted by (12...n).
• Grabowski, J. and Launois, S. (2011). Quantum Cluster Algebra Structures on Quantum Grassmannians and their Quantum Schubert Cells: The Finite-type Cases. International Mathematics Research Notices [Online]:2230-2262. Available at: http://dx.doi.org/10.1093/imrn/rnq153.
• Goodearl, K. and Launois, S. (2011). The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras. Bulletin de la Société Mathématique de France [Online] 139:1-39. Available at: http://smf4.emath.fr/en/Publications/Bulletin/139/html/smf_bull_139_1-39.php.
The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces, quantum symmetric and antisymmetric matrices. For a similarly large class of Poisson polynomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson prime factor ring) is a rational function field F(x(1), ..., x(n)) over the base field (respectively, over an extension field of the base field) with {x(i),x(j)} = lambda(ij)x(i)x(j)for suitable scalars lambda(ij), thus establishing a quadratic Poisson version of the Gel'fand-Kirillov problem. Finally, partial solutions to the isomorphism problem for Poisson fields of the type just mentioned are obtained.
• Goodearl, K., Launois, S. and Lenagan, T. (2011). Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Mathematische Zeitschrift [Online] 269:29-45. Available at: http://dx.doi.org/10.1007/s00209-010-0714-5.
The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over Q.
• Bell, J., Launois, S. and Lutley, J. (2010). An automaton-theoretic approach to the representation theory of quantum algebras. Advances in Mathematics [Online] 223:476-510. Available at: http://dx.doi.org/10.1016/j.aim.2009.08.013.
We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatories. We show that for a fixed number in, the torus-invariant primitive ideals in m x n quantum matrices can be seen as a regular language in a natural way. Using this description and a semigroup approach to the set of Cauchon diagrams, a combinatorial object that parameterizes the primes that are torus-invariant, we show that for m fixed, the number P(m, n) of torus-invariant primitive ideals in m x n quantum matrices satisfies a linear recurrence in n over the rational numbers. In the 3 x n case we give a concrete description of the torus-invariant primitive ideals and use this description to give an explicit formula for the number P(3, n).
• Bell, J. and Launois, S. (2010). On the dimension of H-strata in quantum matrices. Algebra and Number Theory [Online] 4:175-200. Available at: http://dx.doi.org/10.2140/ant.2010.4.175.
We study the topology of the prime spectrum of an algebra supporting a rational torus action. More precisely, we study inclusions between prime ideals that are torus-invariant using the H-stratification theory of Goodearl and Letzter on the one hand, and the theory of deleting derivations of Cauchon on the other. We also give a formula for the dimensions of the H-strata described by Goodearl and Letzter. We apply the results obtained to the algebra of m × n generic quantum matrices to show that the dimensions of the H-strata are bounded above by the minimum of m and n, and that all values between 0 and this bound are achieved.
• Bell, J., Launois, S. and Nguyen, N. (2009). Dimension and enumeration of primitive ideals in quantum algebras. Journal of Algebraic Combinatorics [Online] 29:269-294. Available at: http://dx.doi.org/10.1007/s10801-008-0132-5.
In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2xn quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the "variety of 2xn quantum matrices".
• Launois, S. and Richard, L. (2009). Poisson(co)homology of truncated polynomial algebras in two variables. Comptes Rendus Mathematique [Online] 347:133-138. Available at: http://dx.doi.org/10.1016/j.crma.2008.12.005.
We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection. To cite this article: S. Launois, L. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
• Launois, S., Lenagan, T. and Rigal, L. (2008). Prime ideals in the quantum grassmanian. Selecta Mathematica - New Series [Online] 13:697-725. Available at: http://dx.doi.org/10.1007/s00029-008-0054-z.
We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a natural torus action of H = (k*)(n) on the quantum grassmannian O-q(G(m,n)(k)) and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the nonnegative cells in recent studies concerning the totally nonnegative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.
• Launois, S. and Lopes, S. (2007). Automorphisms and derivations of U_q(sl_4^+). Journal of Pure and Applied Algebra [Online] 211:249-264. Available at: http://dx.doi.org/10.1016/j.jpaa.2007.01.003.
We compute the automorphism group of the q-enveloping algebra U-q(sl(+)(4)) of the nilpotent Lie algebra of strictly upper 4 triangular matrices of size 4. The result obtained gives a positive answer to a conjecture of Andruskiewitsch and Dumas. We also compute the derivations of this algebra and then show that the Hochschild cohomology group of degree I of this algebra is a free (left) module of rank 3 (which is the rank of the Lie algebra Sl(4)) over the center of U-q(sl(4)(+))
• Launois, S. and Lenagan, T. (2007). Primitive ideals and automorphisms of quantum matrices. Algebras and Representation Theory [Online] 10:339-365. Available at: http://dx.doi.org/10.1007/s10468-007-9059-0.
Let K be a field and q be a nonzero element of K that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O-q(M-m,M-n) of quantum matrices. Next, we describe all height one primes of these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O-q(M-m,M-n) whenever O-q(M-m,M-n) has only finitely many height one primes. Finally, we compute the automorphism group of O-q(M-m,M-n) in the case where m not equal n. In order to do this, we first study the action of this group on the prime spectrum of O-q(M-m,M-n). Then, by using the preferred basis of O-q(M-m,M-n) and PBW bases, we prove that the automorphism group of O-q(M-m,M-n) is isomorphic to the torus (K*)(m+n=1) when m not equal n and (m, n) not equal (1, 3) (3, 1).
• Launois, S. and Lenagan, T. (2007). Quantised coordinate rings of semisimple groups are unique factorisation domains. Bulletin of the London Mathematical Society [Online] 39:439-446. Available at: http://dx.doi.org/10.1112/blms/bdm025.
We show that the quantum coordinate ring of a semisimple group is a unique factorisation domain in the sense of Chatters and Jordan in the case where the deformation parameter q is a transcendental element.
• Launois, S. (2007). Combinatorics of H-primes in quantum matrices. Journal of Algebra [Online] 309:139-167. Available at: http://dx.doi.org/10.1016/j.jalgebra.2006.10.023.
For q epsilon C transcendental over Q, we give an algorithmic construction of an order-isomorphism between the set of H-primes of O-q (M-n (C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S-2n consisting of those permutations that move any integer by no more than it positions. Further, we describe the permutations that correspond via this bijection to rank t H-primes. More precisely, we establish the following result. Imagine that there is a barrier between positions n and it + 1. Then a 2n-permuation sigma epsilon S corresponds to a rank t H-invariant prime ideal Of O-q (M-n (Q) if and only if the number of integers that are moved by sigma from the right to the left of this barrier is exactly n - t. The existence of such an order-isomorphism was conjectured by Goodearl and Lenagan.
• Launois, S. and Richard, L. (2007). Twisted Poincare duality for some quadratic Poisson algebras. Letters in Mathematical Physics [Online] 79:161-174. Available at: http://dx.doi.org/10.1007/s11005-006-0133-z.
We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R=C[X1Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier.
• Launois, S. (2007). Primitive ideals and automorphism group of Uq+(B2). Journal of Algebra and its Applications 6:21-47.
Let g be a complex simple Lie algebra of type B-2 and q be a nonzero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of Gelfand-Kirillov dimension 2 of the positive part U+(g) of the enveloping algebra of g are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U+ q (g) of the quantized enveloping algebra of g and then we study the simple factor algebras of Gelfand-Kirillov dimension 2 of U+ q (g). In particular, we show that the centers of such simple factor algebras are reduced to the ground field C and we compute their group of invertible elements. These computations allow us to prove that the automorphism group of U-q(+) (g) is isomorphic to the torus (C*)(2), as conjectured by Andruskiewitsch and Dumas.
• Launois, S. and Lenagan, T. (2007). The first Hochschild cohomology group of quantum matrices and the quantum special linear group. Journal of Noncommutative Geometry 1:281-309.
We calculate the first Hochschild cohomology group of quantum matrices, the quantum general linear group and the quantum special linear group in the generic case when the deformation parameter is not a root of unity. As a corollary, we obtain information about twisted Hochschild homology of these algebras.
• Launois, S., Lenagan, T. and Rigal, L. (2006). Quantum unique factorisation domains. Journal of the London Mathematical Society [Online] 74:321-340. Available at: http://dx.doi.org/10.1112/S0024610706022927.
We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl-Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups O-q (GL(n)) and O-q (SLn).
• Launois, S. (2005). Rank t H-primes in quantum matrices. Communications in Algebra [Online] 33:837-854. Available at: http://dx.doi.org/10.1081/AGB-200051150.
Let K be a (commutative) field and consider a nonzero element q in K that is not a root of unity. Goodearl and Lenagan (2002) have shown that the number of H-primes in R = O-q (M-n (K)) that contain all (t + 1) x (t + 1) quantum minors but not all t x t quantum minors is a perfect square. The aim of this paper is to make precise their result. we prove that this number is equal to (t!)S-2(n + 1, t + 1)(2), where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (1997) and Arakawa and Kaneko (1999).
• Launois, S. (2004). Les ideaux premiers invariants de Oq(Mm,p(C)). Journal of Algebra [Online] 272:191-246. Available at: http://dx.doi.org/10.1016/j.jalgebra.2003.05.005.
For q generic we give a positive answer to a conjecture of Goodearl and Lenagan: the -invariant prime ideals of are generated by quantum minors.
• Launois, S. (2004). Generators for H-invariant prime ideals in O-q(M-m,M-p(C)). Proceedings of The Edinburgh Mathematical Society [Online] 47:163-190. Available at: http://dx.doi.org/10.1017/S0013091502000718.
It is known that, for generic q, the H-invariant prime ideals in O-q(M-m,M-p(C)) are generated by quantum minors (see S. Launois, Les ideaux premiers invariants de O-q(M-m,M-p(C)), J. Alg., in press). In this paper, m and p being given, we construct an algorithm which computes a generating set of quantum minors for each H-invariant prime ideal in O-q(M-m,M-p(C)). We also describe, in the general case, an explicit generating set of quantum minors for some particular H-invariant prime ideals in O-q(M-m,M-p(C)). In particular, if (Y-i,Y-alpha)((i,alpha)is an element of[1,m]x[1,p]) denotes the matrix of the canonical generators of O-q(M-m,M-p(C)), we prove that, if u greater than or equal to 3, the ideal in O-q(M-m,M-p(C)) generated by Y-1,Y-p and the u x u quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of O-q(M-m,M-p(C)) are maximal orders (see T. H. Lenagan and L.

Book section

• Bell, J., Casteels, K. and Launois, S. (2012). Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type B_n. In: Ara, P., Brown, K., Lenagan, T., Stafford, J. and Zhang, J. eds. New Trends in Noncommutative Algebra. American Mathematical Society, pp. 27-40. Available at: http://dx.doi.org/10.1090/conm/562.

Conference or workshop item

• Launois, S. (2006). On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras. In: From Lie Algebras to Quantum Groups. pp. 125-143.
We investigate the automorphism group of the quantised enveloping algebra U of the positive nilpotent part of certain simple complex Lie algebras g in the case where the deformation parameter q \in \mathbb{C}^* is not a root of unity. Studying its action on the set of minimal primitive ideals of U we compute this group in the cases where g=sl_3 and g=so_5 confirming a Conjecture of Andruskiewitsch and Dumas regarding the automorphism group of U. In the case where g=sl_3, we retrieve the description of the automorphism group of the quantum Heisenberg algebra that was obtained independently by Alev and Dumas, and Caldero. In the case where g=so_5, the automorphism group of U was computed in [16] by using previous results of Andruskiewitsch and Dumas. In this paper, we give a new (simpler) proof of the Conjecture of Andruskiewitsch and Dumas in the case where g=so_5 based both on the original proof and on graded arguments developed in [17] and [18].

Review

• Launois, S. (2013). Review of the book ’Lie superalgebras and enveloping algebras’ (Graduate Studies in Mathematics 131) By Ian M. Musson. Bulletin of the London Mathematical Society [Online] 45:666-667. Available at: http://dx.doi.org/10.1112/blms/bdt004.
• Launois, S. (2009). Book Review:Quantum groups: A path to current algebra. Bulletin of the London Mathematical Society [Online] 41:571-572. Available at: http://dx.doi.org/10.1112/blms/bdp048.

Thesis

• Nolan, B. (2017). A Strong Dixmier-Moeglin Equivalence for Quantum Schubert Cells and an Open Problem for Quantum Plücker Coordinates.
In this thesis, the algebras of primary interest are the quantum Schubert cells and the quantum Grassmannians, both of which are known to satisfy a condition on primitive ideals known as the Dixmier-Moeglin equivalence.
A stronger version of the Dixmier-Moeglin equivalence is introduced - a version which deals with all prime ideals of an algebra rather than just the primitive ideals. Quantum Schubert cells are shown to satisfy the strong Dixmier-Moeglin equivalence.
Until now, given a torus-invariant prime ideal of the quantum Grassmannian, one
could not decide which quantum Plücker coordinates it contains. Presented here is a graph-theoretic method for answering this question. This may be useful for providing a full description of the inclusions between the torus-invariant prime ideals of the quantum Grassmannian and may lead to a proof that quantum Grassmannians satisfy the strong Dixmier-Moeglin equivalence.
• Lecoutre, C. (2014). Polynomial Poisson Algebras: Gel’fand-Kirillov Problem and Poisson Spectra.
We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras.

First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator.

Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.
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