386 research outputs found
On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras
No full textWe 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]
Poisson(co)homology of truncated polynomial algebras in two variables
No full textWe 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)
Combinatorics of H-primes in quantum matrices
No full textFor 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
Quantised coordinate rings of semisimple groups are unique factorisation domains
No full textWe 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
Primitive ideals and automorphism group of Uq+(B2)
No full textLet 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
Poisson Deleting Derivations Algorithm and Poisson Spectrum
No full textCauchon [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 clas
Generators for H-invariant prime ideals in O-q(M-m,M-p(C))
No full textIt 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
Enumeration of H-strata in quantum matrices with respect to dimension
No full textWe 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))
Primitive ideals and automorphisms of quantum matrices.
No full textLet 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)
Stephane Mallarme: A synthesis of romanticism and parnassianism, 1970
No full textThe purpose of this paper is to analyse works of Stephane Mallarme, father of Symbolism, pointing out romantic and parnassian elements. Symbolism, like Romanticism, attempted to express the interior thoughts of man. The symbolist movement then, was not only a revolt against Parnassianism but also a return to Romanticism. On the other hand, one would not be incorrect in saying that Romanticism reached its culmination in the works of the symbolists poets. For this reason, an attempt will be made to show that the works of Mallarme, father of Symbolism, can be considered as a synthesis of Romanticism and Parnassianism. This thesis contains three chapters. The first chapter is devoted to a discussion of Romanticism and of Parnassianism. Special attention is given to the origin, development, characteristics and influences of each school. The relationship of one School with the other is also pointed out. The second chapter consists of a biographical sketch of Stephane Mallarme. Special emphasis is placed on factors and events in his life which may have influenced or determined the elements of Romanticism and Parnassianism in his poetry. The third chapter is devoted to an analysis of some of the poems of Stephane Mallarme", "Les Fenetres," V Apparition," "L'Azur," "Toast Funebre," "Le Vierge," "L'Apres-Midi d'un Faune." In these analyses special attention is given to the romantic and parnassian tendencies of the poems. Since these romantic-parnaassian elements occur frequently throughout his works, it has been concluded that Mallarme's poetry can be considered as a synthesis of the two poetic schools
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