Representation of a group or algebra in terms of an algebra with quaternionic structure
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map
which satisfies
Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and
Quaternionic representations of associative and Lie algebras can be defined in a similar way.
and 20 Related for: Quaternionic representation information
field of representation theory, a quaternionicrepresentation is a representation on a complex vector space V with an invariant quaternionic structure...
A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication...
mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure...
argument), one can show that any complex symplectic representation is a quaternionicrepresentation. Quaternionic representations of finite or compact groups...
pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionicrepresentation: it admits an invariant quaternionic structure...
dealing with quaternionic groups it is convenient to represent quaternions using complex 2×2-matrices, With this representation, quaternionic multiplication...
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of...
representations of G . {\displaystyle G.} Definition. A quaternionicrepresentation is a (complex) representation V , {\displaystyle V,} which possesses a G {\displaystyle...
term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words...
v+W\mapsto gv+W} . quaternionic A quaternionicrepresentation of a group G is a complex representation equipped with a G-invariant quaternionic structure. quiver...
The Lie algebra of Sp(n) is given by the quaternionic skew-Hermitian matrices, the set of n-by-n quaternionic matrices that satisfy A + A † = 0 {\displaystyle...
angle φ. In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and quaternionic. The existence of spinors in 3 dimensions follows from...
that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure. There is an invariant complex antilinear...
U(N) to U(N – 1) states that Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations...
characters of holomorphic discrete series representations. Quaternionic discrete series representation Bargmann, V (1947), "Irreducible unitary representations...
indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows. Let...
binary icosahedral group. A hyperbolic versor is a generalization of quaternionic versors to indefinite orthogonal groups, such as Lorentz group. It is...