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    Irredundant Triangular Decomposition

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    Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set WW defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success

    Triangular decomposition of right coideal subalgebras

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    Let g\mathfrak g be a Kac-Moody algebra. We show that every homogeneous right coideal subalgebra UU of the multiparameter version of the quantized universal enveloping algebra Uq(g),U_q(\mathfrak{g}), qmβ‰ 1q^m\neq 1 containing all group-like elements has a triangular decomposition U=Uβˆ’βŠ—k[F]k[H]βŠ—k[G]U+U=U^-\otimes_{{\bf k}[F]} {\bf k}[H] \otimes_{{\bf k}[G]} U^+, where Uβˆ’U^- and U+ U^+ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if U1 U_1 and U2 U_2 are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular composition U2βŠ—k[F]k[H]βŠ—k[G]U1 U_2\otimes_{{\bf k}[F]} {\bf k}[H]\otimes_{{\bf k}[G]} U_1 is a right coideal but not necessary a subalgebra. Using a recent combinatorial classification of right coideal subalgebras of the quantum Borel algebra Uq+(so2n+1),U_q^+(\mathfrak{so}_{2n+1}), we find a necessary condition for the triangular composition to be a right coideal subalgebra of Uq(so2n+1).U_q(\mathfrak{so}_{2n+1}). If qq has a finite multiplicative order t>4,t>4, similar results remain valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum groups uq(g),u_q({\frak g}), $u_q(\frak{so}_{2n+1}).
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