## How is the weight of a representation related to its weight?

A weight of the representation V is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called weight vectors. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of, with the corresponding eigenvalues given by λ.

### Is the weight of a representation an eigenvalue?

In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space .

#### What are the properties of a relational decomposition?

The properties of a relational decomposition are listed below : Using functional dependencies the algorithms decompose the universal relation schema R in a set of relation schemas D = { R1, R2, …..

**How to check if a decomposition is not dependency preserving?**

If a decomposition is not dependency preserving some dependency is lost in decomposition. To check this condition, take the JOIN of 2 or more relations in the decomposition. R = (A, B, C) F = {A ->B, B->C} Key = {A} R is not in BCNF. Decomposition R1 = (A, B), R2 = (B, C)

**Which is the subspace of the representation V?**

Then the weight space of V with weight λ is the subspace . A weight of the representation V is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called weight vectors. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of

## Which is the highest weight representation of G?

A representation (not necessarily finite dimensional) V of g {\\displaystyle {\\mathfrak {g}}} is called highest-weight module if it is generated by a weight vector v ∈ V that is annihilated by the action of all positive root spaces in g {\\displaystyle {\\mathfrak {g}}} .

### Why are weights important in abelian Lie algebras?

Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations. If G is a Lie group or an algebraic group, then a multiplicative character θ: G → F× induces a weight χ = dθ: g → F on its Lie algebra by differentiation.

#### Is the group generated by transvections a subgroup of SL?

These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [GL, GL] ≤ SL), but in general do not coincide with it. The group generated by transvections is denoted E (n, A) (for elementary matrices) or TV (n, A).

**How are the components of the special linear group identified?**

Over the real numbers, SL±(n, R) has two connected components, corresponding to SL (n, R) and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant −1 ). In odd dimension these are naturally identified by