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If working over a ring where SL is generated by transvections (such as a field or Euclidean domain), one can give a presentation of SL using transvections with some relations. Transvections satisfy the Steinberg relations , but these are not sufficient: the resulting group is the Steinberg group , which is not the special linear group, but ...
In chemistry, biochemistry, and pharmacology, a dissociation constant (K D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions.
The algebra plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ball.
Dissociation in chemistry is a general process in which molecules (or ionic compounds such as salts, or complexes) separate or split into other things such as atoms, ...
SL(2, R) is the group of all linear transformations of R 2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL ± (2, R) preserves unoriented area: it may reverse orientation.
The structure constants often make an appearance in the approximation to the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group. For small elements X , Y {\displaystyle X,Y} of the Lie algebra, the structure of the Lie group near the identity element is given by
It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL(2, R). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ 2. Thus in the case of the Lie algebra sl 2, the infinitesimal character of an irreducible representation is specified by one complex number.
It is a consequence of the Weyl character formula, and for the Lie algebra sl 2 it is essentially the Clebsch–Gordan formula. Steinberg's formula states that the multiplicity of the irreducible representation of highest weight ν in the tensor product of the irreducible representations with highest weights λ and μ is given by