110,207 research outputs found
Irredundant Triangular Decomposition
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 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
Let be a Kac-Moody algebra. We show that every homogeneous
right coideal subalgebra of the multiparameter version of the quantized
universal enveloping algebra containing all
group-like elements has a triangular decomposition , where and are right coideal
subalgebras of negative and positive quantum Borel subalgebras. However if and are arbitrary right coideal subalgebras of respectively
positive and negative quantum Borel subalgebras, then the triangular
composition is a
right coideal but not necessary a subalgebra. Using a recent combinatorial
classification of right coideal subalgebras of the quantum Borel algebra
we find a necessary condition for the triangular
composition to be a right coideal subalgebra of
If has a finite multiplicative order similar results remain valid
for homogeneous right coideal subalgebras of the multiparameter version of the
small Lusztig quantum groups $u_q(\frak{so}_{2n+1}).
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