We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

Introduction to Quantum Groups

1 S-polynomials eliminating the leading term Buchberger's criterion and algorithm 2 Wavelet Design construct wavelet filters 3 Proof of the Buchberger Criterion two lemmas proof of the Buchberger criterion termination and elimination

Gröbner Bases와 응용

An introduction to commutative and noncommutative Gro¨bner bases

Comprehensive Gro¨bner bases

Non-commutative Gröbner bases in algebras of solvable type

An extension of Buchberger's algorithm and calculationsin enveloping fields of lie algebras

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed as a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.

We will study this conjecture for modules M e R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanley's Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

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On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed in a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.

We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will derive a criterion which allows us to extract an upper bound on depth M from particular involutive bases. As a corollary we obtain that any monomial ideal M which possesses an involutive basis of this type satisfies Stanley's Conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of M. Moreover, we will show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithmic part for the computation of Stanley decompositions in these situations.

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Joachim Apel

Papers

An extension of Buchberger's algorithm and calculationsin enveloping fields of lie algebras

The Theory of Involutive Divisions and an Application to Hilbert Function Computations

On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals

On a Conjecture of R. P. Stanley; Part I—Monomial Ideals

Computational model of mesoscopic structure of concrete for simulation of fracture processes