# String bases for quantum groups of type ᵣ

@inproceedings{Berenstein1993StringBF, title={String bases for quantum groups of type ᵣ}, author={Arkady Berenstein and Andrei Zelevinsky}, year={1993} }

This is the quantum deformation (or q−deformation) of the algebra of polynomial functions on the group Nr+1 of upper unitriangular (r + 1) × (r + 1) matrices. In this paper we introduce and study a class of bases in Ar which we call string bases. The main example of a string basis is given as follows. Let U+ = U+,r be the quantized universal enveloping algebra of the Lie algebra nr+1 of Nr+1 (see e.g., [10]). Then Ar is seen to be the graded dual of U+, and the basis in Ar dual to the Lusztig’s…

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## References

SHOWING 1-10 OF 11 REFERENCES

On crystal bases of the $Q$-analogue of universal enveloping algebras

- Mathematics
- 1991

0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the…

Canonical bases arising from quantized enveloping algebras

- Mathematics
- 1990

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see…

Verma bases for representations of classical simple Lie algebras

- Mathematics
- 1986

Complete bases are constructed for all finite‐dimensional irreducible representations of the simple Lie algebras over C of the types An (n≥1), Bn and Cn (2≤n≤6), Dn (4≤n≤6), and G2. Each basis vector…

Quivers, perverse sheaves, and quantized enveloping algebras

- Mathematics
- 1991

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of…

Quantum Groups

- Mathematics
- 1994

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups…

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1958

The author has always understood the multiplicity question to mean the following: Let I be the set of all integral linear forms on fi and m be the function in / which assigns to each integral linear form vE I the mul-tiplicity m\(v) of its occurrence as a weight of w.

Good bases for G-modules

- Mathematics
- 1990

We prove a conjecture of I. M. Gelfand, A. V. Zelevinsky and K. Baclawski about the existence of good bases for G-modules. We deduce the result from a previously proved theorem [21] about weak…

Special bases forSN and GL(n)

- Mathematics
- 1981

The special basis in spaces of finite dimensional representation ofSN and GL(n) is constructed and its properties are studied.